18
---------------------- ---------------- NAME (PRINT): ______--==;----;=________ Last/Surname First IGiven Name STUDENT#: SIGNATURE: UNIVERSITY OF TORONTO MISSISSAUGA JUNE 2012 FINAL EXAMINATION STA220H5F The Practice of Statistics I Ramya Thinniyam Duration -3 hours Aids: Non-Programmable Calculator; 1 page of double-sided Letter (8-1/2 x 11) sheet (hand written); Statistical Tables (provided) The University of Toronto Mississauga and you, as a student, share a commitment to academic integrity. You arE1 reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam, including but not limited to any electronic devices with storage, such as cell phones, pagers, personal digital assistants (PDAs), iPods, and MP3 players. Unauthorized calculators and notes are also not permitted. Do not have any of these items in your possession in the area of your desk. Please tum the electronics off and put all unauthorized aids with your belongings at the front of the room before the examination begins. If any of these items are kept with you during the writing of your exam, you may be charged with an academic offence. A typical penalty may cause you to fail the course. Please note, you CANNOT petition to re-write an examination once the exam has . begun. i INSTRUCTIONS: -There are 9 questions - answer all questions. -For all true/false and fill in the blank questions - circle or put your final answers in blanks as instructed. Only final answers will be marked. -For all other questions, show your work to earn full marks and then circle the final answer. Correct answers with no justifications will not receive any marks. -For questions with Minitab output given, you may copy the appropriate numbers without calculating yourself. Do your own calculations when specified to do from scratch. -Final answers should be in reduced form. Round your answers to 4 decimal places where appropriate. BEST WISHES! @ Question 1 2 3 4 5 6 7 8 9 TOTAL I Value 10 10 10 10 20 10 10 10 10 100 Mark Page 1 of 12

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---------------------- ----------------

NAME (PRINT) ----~~______--==----=________ LastSurname First IGiven Name

STUDENT SIGNATURE

UNIVERSITY OF TORONTO MISSISSAUGA JUNE 2012 FINAL EXAMINATION

STA220H5F The Practice of Statistics I

Ramya Thinniyam Duration - 3 hours

Aids Non-Programmable Calculator 1 page of double-sided Letter (8-12 x 11) sheet (hand written) Statistical Tables (provided)

The University of Toronto Mississauga and you as a student share a commitment to academic integrity You arE1 reminded that you may be charged with an academic offence for possessing any unauthorized aids during the writing of an exam including but not limited to any electronic devices with storage such as cell phones pagers personal digital assistants (PDAs) iPods and MP3 players Unauthorized calculators and notes are also not permitted Do not have any of these items in your possession in the area of your desk Please tum the electronics off and put all unauthorized aids with your belongings at the front of the room before the examination begins Ifany of these items are kept with you during the writing ofyour exam you may be charged with an academic offence A typical penalty may cause you to fail the course

Please note you CANNOT petition to re-write an examination once the exam has begun i

INSTRUCTIONS -There are 9 questions - answer all questions -For all truefalse and fill in the blank questions - circle or put your final answers in blanks as instructed Only final answers will be marked -For all other questions show your work to earn full marks and then circle the final answer Correct answers with no justifications will not receive any marks -For questions with Minitab output given you may copy the appropriate numbers without calculating yourself Do your own calculations when specified to do from scratch -Final answers should be in reduced form Round your answers to 4 decimal places where appropriate

BEST WISHES

Question 1 2 3 4 5 6 7 8 9 TOTAL I

Value 10 10 10 10 20 10 10 10 10 100

Mark

Page 1 of 12

Full Name Student Number

[10 marks - leach] 1 TRUEIFALSE Ifthe statement is true under all conditions circle T otherwise circle F

(a) The p-value is the probability that the null hypothesis is true T F

(b) For any events A B in a sample space PCA) + PCB) = 1 T F

(c) For any continuous random variable X PCX = 3) = O T F

(d) The range is a more resistant measure of the spread than IQR T F

(e) The margin of error is equal to the width of a confidence interval T F

(1) A hypothesis test with a 1 significance level has a probability of at most 001 of incorrectly rejecting Ho T F

(g) Mutually exclusive events must be independent T F

(h) The best way to deal with outliers is to remove them from the data set T F

(i) The smallerthe p-value the more evidence in favour of Ha T F

(j) The Central Limit Theorem says that the sample mean will be unbiased for the population mean only if the sample size is large T F

[10 marks] 2 A printer manufacturer tests 15 randomly selected printer heads and records the number of

characters printed until failure Below is a stem-and-Ieaf plot of the number of characters (in millions) until failure

Stern-and-Leaf Display nurn_char

Stem-and-leaf of num char N - 15 Leaf Unit - 0010 shy

1 8 5 2 9 2 3 10 7 5 11 38

(4) 12 0259 6 13 236 3 14 38 1 15 5

Page 2 of 12

Full Name Student Number

Fill in the blanks with the final answers

[1 m] (a) Find the range

[1 m] (b) Find the inter-quartile range IQR

~~ (c) If Lt~l Xi =1858 and Lt~l xl =235368 find the sample mean and sample standard deviation

s= ________X=------ shy

[1 m] (d) Find a 99 confidence interval for the mean

[1 m] (e) With 99 confidence the manufacturer can advertise that the mean lifetime of their printers is

at least characters

[1 m] (I) Suppose the popUlation standard deviation is known to be 21 million characters

Find a 99 confidence interval for the mean

[3 m] (g) What assumptions were required to use the CIs in (d) and (f) Assess whether or not these

conditions are met

Page 3 of 12

Full Name Student Number

[10 marks] 3 The table below shows results of a 1998 study on internet communication in particular flaming in chat rooms The investigators were interested in whether being personally criticized by 0 thers wou ld rnake one more lk1 ely to cntlclze 1 0 thers

Personally Criticized

Not Been Personally Criticized Total

Criticized Others 19 8 127 Has Not Criticized Others 23 143 166 Total 42 151 193

Now suppose a chat room user is randomly selected

[2m] (a) Label with letters and name the two events of interest

For each ofthe following write the events in question with proper notation and then estimate the probabilities Put theflnal answer on the blank where indicated

[1m] (b) Probability that the individual has criticized others

~~ (c) Probability of being criticized or criticizing others

[1m] (d) Probability of being criticized and criticizing others

[2 m] (e) The user has been personally criticized by someone else What is the probability that he will

criticize another individual

[3 m] (1) Are criticizing others and being personally criticized independent events Justify

Page 4 of 12

Full Name Student Number

[10 marks - 1 each blanksentence] 4 Religion Studies researchers were interested in making inferences about the true proportion of

Canadians who believe in some fonnmiddotofGod They randomly surveyed 100 Canadians and asked them whether or not they believed in God (Yes or No) Here is the Minitab output

Test and CI for One Proportion

Test of p = 05 vs p not = 05

Event = Yes

Sample X N Sample p 95 CI Z-Value P-Value 1 61 100 0610000 (0514403 0705597) 220 0028

Use the output to fill in the blanks

(a) people in the survey said they believed in God

(b) As given in the output the null hypothesis is and the alternative hypothesis is [use proper notation and symbols]

(c) The value of the test statistic is ________

(d) If Ho is true the distribution of the test statistic is _______

(e) The 95 confidence interval for the true proportion of Canadians Who do NOT believe in some fonn of God is _____________

(f) Using a confidence interval decide on whether or not you agree with this claim More than 34 ofCanadians believe in some form ofGod Fill in agree or disagree _______

(g) If the researchers are interested in answering the question Do the majority ofCanadians believe in some form ofGoel the appropriate p-value is At the 5 significance level the conclusion would be [reject or FTR Ho ]

(b) Give a one sentence conclusion in plain English to answer the question of interest from (g)

Page 5 of 12

Full Name Student Number

[20 marks] 5 An intersection in Mississauga has an average of 6 accidents per month

Before answering each part label the random variable ofinterest and identify its distribution with correct parameters Show your work and circle your final answers

[3 m] (a) What is the probability that there are exactly 2 accidents in September at this intersection

[3mr (b) What is the probability that in the next year 3 months will have exactly 2 accidents in them

each

[4 m] (c) Approximate the probability that in the next two years at least 10 months will have 2 accidents

in them each

Page 6 of 12

Full Name Student Number

[2 m] (d) On average how long would you have to wait for an accident to occur at this intersection

(Include the units in your answer)

[3 m] (e) An accident just occurred today What is the probability that the next accident is more than 12

a month from today

[5 m] (f) You take a random sample of 100 months and record the number of accidents that occur What

is the probability that the average number exceeds 6S accidents

Page 7 of 12

Full Name Student Number

lOmarksj 6 A new toothpaste has been put on the market that claims to reduce the number of teeth that have

problems (cavities sensitive teeth etc) In order to test the effect ofthe new toothpaste researchers randomly select participants and randomly assign each person to use either the new toothpaste or a standard toothpaste (a popular brand that has been used for several years) After 6 months of use each participant reports the number of teeth they have with problems It can be assumed that both populations are Normal Partial Minitab output is given

DESCRIPTIVE STATISTICS standard DESCRIPTIVE STATISTICS new

Variable N Mean StDev SE MeanVariable N Mean StDev SE Mean new 20 330 213 048standard 15 370 198 051

Test for Equal Variances

95 Bonferroni confidence intervals for standard deviations

Sample N Lower StDev Upper standard 15 138964 1 98 335600 new 20 156050 213 330000

F-Test (normal distribution) Test statistic = 086 p-value = 079

[2 m)

(a) Using a=O05 test for the equality of the variances of the standard and new toothpaste populations Include all steps requiredfor a Hypothesis Test (you may copy any relevant parts from the output without calculating them yourself)

Page 8 of 12

Full Name Student Number

[5 m] (b) Which type of hypothesis test would you use to test if the new toothpaste is effective as

claimed by the manufacturer using a005 Why Test the relevant hypothesis from scratch Include all steps requiredfor a Hypothesis Test

[3 m] (c) The manufacturer of the new toothpaste now decides to collect data on consumers ofthe new

toothpaste separately in order to improve their study How large of sample is required to estimate the mean number of teeth with problems within 05 and retain 95 confidence

Page 9 of 12

Full Name Student Number

flOmarksJ 7 Is insomnia related to education Researchers conducted a study to determine if insomnia is related to the education of Tennessee residents 575 adults in Tennessee were selected to participate in the study using a random-digit telephone procedure Thenumber of years of education and insomnia status (normal sleeper or chronic insomniac) were recorded for each participant The researchers discovered that the fewer the years of education the more likely the person is to have chronic insomnia

~Insomnia is a sleep disorder (problems falling asleepstaying asleep throughout the night) In a random-digit telephone procedure telephone numbers are generated at random and the people with those phone numbers are contacted to participate in the study

Fill in the blanks or give a one-sentence brief answer where appropriate to following

[1 m] (a) Identify the population of interest

[1 m] (b) Identify the sample

[4 m] (c) List the variables measured and whether they are categorical or quantitative

[1 m] (d) What inference(s) did the researchers make

[1 m] (e) In general conclusions about a sample can be generalized to the population if the sample is

___________ of the population

[2 m] (f) In this case do the selected participants form a random sample Explain why or why not

Page 10 of 12

Full Name Student Number

[10marksj 8 You and your friend playa dice-rolling game You each roll a fair die once and observe the

number that appears on the up face for each person Ifboth ofyou get the same number nobody wins or loses any money If you get different numbers whoever gets the lower number has to pay the other person $1 Let X represent your winnings (in $) from this game

[4 m] (a) Find the probability distribution for X and display it in table form

[3 m] (b) Calculate the mean ofX and interpret your answer in terms of the game Show your

calculations and then circle your final answer

[3 m] (c) Compute the variance ofX Show your work and circle thefinal answer

Page 11 of 12

Full Name Student Number

10 marks 9 A public school is considering adding to the curriculum a course that it believes will improve the reading comprehension skills of its students The school has decided to conduct an experiment to evaluate the course before it is offered to the general student body Six students are randomly selected and their reading scores are recorded before and after the course These scores are composite measurements ofboth speed and comprehension (where higher scores indicate higher

h )Th I dblcompre enSlOn e resu ts are summanze e ow Student Number Score After Course Score Before Course

1 221 211 2 231 216 3 203 191 4 216 224 5 207 201 6 203 178

The school will implement the new course if there is sufficient evidence at the 5 significance level that the course improves reading comprehension

[2 m) (a) Which type of hypothesis test is appropriate for this situation and why

[6m (b) Conduct the hypothesis test of interest Include all steps requiredfor a Hypothesis Test

[2 m (c) List the assumption(s) required to use this type ofhypothesis test

Page 12 of 12 --END OF EXAM--shy

794 Appendix A Tables

TABLE IV Normal Curve Areas

o 1

00 01 bull02z 03 OS04 06 07 OS 09

0 0000 0040 0080 0120 01990160 0239 0279 0319 03591 0398 0438 0478 0517 05960557 0636 0675 0714 0753

07932 0832 0871 0910 0948 0987 1026 1064 1103 1141 3 1179 1217 1255 1293 13681331 1406 1443 1480 1517

15544 1591 1628 1664 1700 1736 1772 1808 1844 1879

1915oS 1950 1985 2019 2054 2088 2123 2157 2190 2224

22576 2291 2324 2357 24222389 2454 2486 2517 2549

25807 2611 2642 2673 2704 2734 2764 2794 2823 2852

28818 2910 2939 2967 2995 3023 3051 3078 3106 3133 3212 9 y3159 3186 3238 3264 3289 3315 3340 3365 3389

10 3413 3438 3461 3485 35313508 3554 3577 3599 3621 372911 3643 3665 3686 3708 3749 3770 3790 - 3810 3830

3849 386912 3888 3907 3925 3944 3962 3980 3997 4015

4032 4049 13 bulliQ6timiddot 4082 41154099 4131 4147 4162 4177 42071~m224192L4 4236 4251 4265 43194179 4292 4306

4332 4345 4357 4370 4382 4394middot 4406 4418 4429lS 4441 ltIJ

4452L6 4463 4474 4484 4495 4505 4515 4525 4535 4345

4554L7 4564 4573 4582 4591 4599 4608 4616 4625 4633

464118 4656 46644P49 4671 4678 4686 4693 47064699

4713 4761 L9 4719 4732 ~ 47674738 4744 4750 4726 4756 478820 4772 4793 4798 4893 4808 4812 4817~11amp 4783 shy

4821U 4826 4830 4834 48424838 48574846 4850 4854 486122 4864 4868 48874871 4875 4878 48904881 4884

23 4893 4896 4898 49164901 4913 4906 4909 49114904 24 4918 4922492Q 49364925shy 49294927 49344931 4932

4943493825 49524940 4941 4945 4946 49514948 4949 495326 4955 49644956 4957 4959 4960 4961 49634962 4965 497427 49674966 4968 4969 4970 49734971 4972 4974 498128 4975 4976 4977 4977 4978 49804979 4979

29 49864981 4982 4982 4983 4984 4984 49864985~49B5 30 4987 4987middot4987 4988 4988 4989 4990 49904989 4989 i

Source Abridged from Table I ofA Hald Statistical Table$ and Fonnulas (New YorkWiley) 1952 Reproduced by permission of A Hald

796 -~~ - -

o ~~~-TABLE VI Critical Values of t

f(t)

bull

Degrees of -Freedom tJWtlOO t050 tOO51010 tooi tooos

1 63143078 12706 31821 63657 31831 63662 2 1886 2920 4303 6965 9925 22326 31598 3 4541 1638 2353 3-182 5841 10213 12924 4 1533 2132 2776 3747 4604 7173 8610 5 1476 2015 2571 3365 4032 5893 6869 6 19431440 2447 3143 3707 5208 5959 7 1415 23651895 2998 3499 4785 5408

1397 1860 2306i 2896 3355 4501 5041 9 1383 1833 2262 2821 3250 4297 4781

10 18121372 2228 2764 3169 _ 4144 4587 U 1363 1796 2201 2718 3106 4025 4431 12 1356 1782 2179 shy 2681 3055 3930 4318

13~~ ~_ ~-i77113 i1reg 2650 3012 3852 4221 14 1345 21451761 2024 ~ 2977 41403787 15 1341 1753 2131 2602 29fl7 4073 16

373~ 1337 1746 2120 - 29212583 3686 4015

17 1333 1740 2110 2567 -2898 3646 3965 18 1330 1734 2101 t28782552 3610 3922 19 1328 t729 2093 28612539 38833579 - i

li~r~5I - 20 1725 2086 2528 2845 3850 21

3552 t 1323 2080 2518 1721 2831 3527 3819

22 20741321 1717 2508 2819 3505 3~792 23 1319 1714 2069 2500 3767 24

2807 3485 1318 1711 2064 2492 3467 3745

25 2middot797

1316 1708 2060 2485 3725 26

2787 3450 1315 1706 2056 2479 2779 3435 3707

27 1314 1703 2052 2473 2771 36903421 28 1313 170f 1 2048 2763 3674 29

~467 3408 1311 1699 2045 - 2462 2756 36593396

30 1310 1697 2D42 2457 - 2750 3646 40

3385 1303 1684 2021 2423 2704 3551

60 3307

1296 -16n 2000 2390 2660 3460 120

3232 1289 1658 1980 2358 2617 33733160

00 1645 1282 1960 2326 2576 3090 3291

Source 1his table is reproduced with the lcind permtsSioil of~e frustees of Biometrika from E S Pearson and R O Hartley (eds) The Biometrika Tables for Statisticians Vol 1 3d ed Biometrika 1966

r7~~

c

~ ApPENDIX A Tables 801

i~ 1lt TABLE IX Percentage Points of the F-distribution a = 05

~ liZ 1 2

1 1614 1995 2 1851 1900 3 1013 955 4 771 694

II

NUMERATOR DEGREES OF FREEDOM I11

3 4 9 I5 6 7 8

5

2157 2246 2302 2340 1916 1925 1930 1933

928 912 901 894 659 639 626 616

~ 661 579 541 519~ 505 495 6 599 514 476 453 439 428~7 559 474 435 412 397 387

5328 446 407 384 369 358 9 512 426 386 363 348 337~

10 -496 4100 371 348 333 322Q 11 484 7f~i 359398--~ 336 320 309

34938~middot12middot 475 326 300 ~~311~ 13 381467 341 318 303 292 0 fi

14 460 374 334 311 296 285 ~ 15 454 368 329 306 290foil 279 rr

449 363~ 16 324 301 285 274

~ 17 445 )20 296 281359 270 foil 18 441 3-16 293li))~5 277 266Q

438 352middot 313 290= 19 274 263 0 20 435 349 310 287 271 260 ~ 21 432 347 257307 284 268

i 22 430 344 305 282 266 255 303 0 23 428 342 280 264 253

Z 24 426 340 301 278 262 251foil Q 424middot25 339 299 276 260 249

26 423 337 298 274 259 247 27 421 335 296 273 257 246 28 420 334 295 271 245 256 29 418 333 293I 270 255 243 30 417 332 292 269 253 242 40 408 323 284 261 245 234 60 40q 315 276 253 237 225

120 307 392 268 245 229 217 00 384 300 260 237 221 210

2368 2389 1935 1937 889 885 609 604 488 482 421 415 379 373 350 344 329 323 314 307 301 295 291 285

283 277 271 1276 270 265 271 264 259 i266 259 254 1rII261 255 249

251 258 ~i-middot246 f 254 248 242 I

I 251 245 239 249 242 237 246 240 234 244 237 232 242 236 230 240 234 228 239 232 2]7 237 231 225 236 229 224 235 228 222 233 227 221

225 218 212 I

1217 210 204 209 202 196 201 194 188

2405 1938 881 600 477 410 368 339 318 302 290 280 ji

Source From M Merrington and eM Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 19433373-88

i (continued) I

I

I iii 11 iii

II I

I l

iINi II

802 Appendix A Tables

TABLE IX Continued

~1 NUMERATOR DEGREES OF FREEDOM Vl ~r--l-0----12----15-----r--2-0----24---r--30----40----60~--~- ~ OQ

1 2 3 4 S 6 7 8 9

~ 10 11

~ ~ ~ 14 o 15 rIJ (

Iiil 16 ~ 17 ~ 18 ~ 19 = 20

~ 21

i ii shy~ 24 ~ 2S

26 27 28 29 30

middot40 60

120 OQ

2419 1940 879 596 474 406 364 335 ~314

298 285 275 267 260 254 249 245 241 238 235 232 230 227 225 224 222 220 219 218 216

208 199 191 1~

2439 2459 2480 2491 1941 1943 1945 1945 874 870 866 864 591 586 580 577 468 462 456 453 400 394 387 384 357 351 344 341 328 322 315 312 307 301 294 290 291 285 277 274 279 272 265 261 269 262 254 251 260 253 246 242 253 246 239 235 248 240 233 229 242 235 228 224 238 231 223 219 234 227 219 215 231 223 216 211 iJi ~r~ 220 212 208 225 218 210 205 223 215 207 203 220 213 205 201 218 211 203 198 216 209 201 196

215 2Q7 199 195j 2i3 206 197 193 212 204 196 191 210 203 194 190 209 201 193 189

200 192184 179 192 184 175 170 183 i 175 166 161 U5 1m 1~ 1~

2501 1946 862 575 450 381 33S 308 286 270 257 247

238 231 225 219 215 211 207 204 201 198 196 194 192 190 188 187 185 184

174 165 155 146

2511 Imiddotmiddotmiddot

1947 859 572 446 377middot 334 304 283 266 253 243 234 227 220 215 210 206 203

~199 196 f 194 1~1 189 187 185 184 182 181 179

169 159 150 139

2522 1948 857 5~69

443 374 330 301 279 262 249 238 230 222

216 211 206 202 198 195 192 189 186 184 182 180 179 177 175 174

164 153 143 132

2533 2543 1949 1950 855 853 566 563 440 436 370 367 327 323 297 293 275 271 258 254 245 240 234 230 225 221 218 213 211 207 206 201 201 196 197 192 193 188 190 184 1~7 181 184 178

181 176 179 173 177 ~ 171

169 17~ 173 167 171 165

164170 162168

151158 147 139

125135 100122 i

ApPENDIX A Tables~c63 y

TABLE X Percentage Points of the F-distribution a = 025

f(F)

o FfJ25

- -NUMERATOR DEGREES OF FREEDOM ~ Jz 1 2 3 4 5 6 7 8 9

1 6478 7995 8642 8996 9218 9371 9482 9567 9633 2 3851 3900 3917 3925 3930 3933 3936 3937 3939 3 1744 1604 1544 1510 1488 1473 1462 1454 1447 4 1222 1065 998 960 936 920 907 898 890 5 1001 843 776 739 715 698 685 676 668 ( 881 726 660 623 599 582 570 560 552

589 7 807 654 552 529 512 499 490 482 8 757 606 542 505 482 465 453 443 436 9 721 571 508 472 448 432 420 410 403

gt1 10 - 694 483 447 424 407 395 385 378-4~~i 0 003f2~~ 11 672 463 428 404 88 376 366 359fIl

12- 655 510 447 412 389 373 351 344~61f l~ 13 641 497 435 400 377 360 348 339 331

0 ~ 14 630 486 424 389 366 329338-~$P 321 IIl 15 620 477 A15 380 358 34-1 329 320 312fIl

16 61_2 469 408 373 350 334 322 305 31~r-~ ~ 17 462- 4016Cl1) 366 344 328 316 298I j r~ 30(hmiddotfIl 45618 598 395 361 338 322 310 301 293~

592 451 390 356 333= 19 317 305 296 288

~ 20 587 446 386 351 329 313 301 291 284 21 583 442 382 348 325 309 297 287 280 22 - 579 438 378 344 322 305 293 284 276~

0 23 575 435 375 341 318 302 290 281 273 ~ 24 572 432 372 338 315 299 287 278 270 ~ 25 569 429 369 335 313 297 285 275 268

26 566 427 367 333 310 294 282 273 265 27 563 424 365 331 - 292 308 280 271 263 28 561 422 363 329 306 290 278 269 261 29 559 420 361 327 304 288 276 267 259 30 557 418 359 325 303 287 275 265 257

40 542 405 346 290 313 274 262 253 245 60 529 393 334 301 279 263 251 241 233

120 515 380 323 289 267 252 239 230 222 00 502 369 312 279 257 241 229 219 211

SourceFrom M Merrington and C M Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 1943337388

(continued)

4 804 Appendix A Tables

TABLE X Continued

~Vl NUMERATOR DEGREES OF FREEDOM V2~r---l-0--r--U--~-~~--~~20-----~~---~------~~middotmiddot----6-0-----U~O-----~---

9686 9767 9849 9931 9972 1001 1006 _1010 1~014-1 1018 2 3940 3941 3943 3945 3946 3946 3947 3948 394~ 3950 3 1442 1434 1425 1417 1412 1408 1404 1399 1395 1390

884 875 866 856 851 846 841 836 8314 826 662 652 643 633 628 623 618 612 6075 602 546 537 527 517 512 507 501 496 4906 485

7 476 467 457 447 442 436 431 425 420 414 8 430 420 410 400 395 389 384 378 373 367 9 396 387 377 367 361 356 351 345 339 333

372 362 352 342 337 331 326 320 314 308S 10 353 343 333 323 317 312 306 300 294Q 11 288 337 328 318 307 302 296 291 285 279 272 325 315 305 295 289 284 278 272 266 260E ~

)i 14 315 305 295 284 279 273 267 261 255 249 o 15 306 296 286 276 270 264 259 252 246 240rIl ~ 16 232299 289 279 268 263 257 251 245 238 ~ ~17 292 282 272 262 256 250 244 238 232 225

287 277 267 256 250 244 238 232 226~ 18 219 Q 19 282 272 262 251 245 239 233 227 220 21~=

277 268 257 246 241 235 229 222 216 209~ 20middot 273 bull =I~~~ 253 242 237 231 ~ 225 218 211lt 21 204 270 middot260 250 239 233 227 21 214 208 200~ 22 267 257 247 236 230 224 $ 218 211 204 197o 23

z 24 264 254 244 D3 227 221 ~ Ii 215 208 201 194 261 251 241 230 224 218 212 205 198 191~ 25

26 259 249 239 228 222 216 209 203 )95 I 188 27 257 247 236 225 219 213 207 200 i~~3 185 28 2d5tf i 245middot 234 223217 211 205 198 191 183 29 253 243 232 221 215 209 203 196 189 181 30 251 241 231 220 214 207 201 194 187 179

40 164239 229 218 207 201 194 188 180 172 60 227 217 206 194 188 182 174 167 158 148

120 216 205 194 182 176 169 161 153 143 131 I

QO 100205 194 I 183 171 I 164 157 148 139 127

Page 2: STA220_Final_2012S.pdf

Full Name Student Number

[10 marks - leach] 1 TRUEIFALSE Ifthe statement is true under all conditions circle T otherwise circle F

(a) The p-value is the probability that the null hypothesis is true T F

(b) For any events A B in a sample space PCA) + PCB) = 1 T F

(c) For any continuous random variable X PCX = 3) = O T F

(d) The range is a more resistant measure of the spread than IQR T F

(e) The margin of error is equal to the width of a confidence interval T F

(1) A hypothesis test with a 1 significance level has a probability of at most 001 of incorrectly rejecting Ho T F

(g) Mutually exclusive events must be independent T F

(h) The best way to deal with outliers is to remove them from the data set T F

(i) The smallerthe p-value the more evidence in favour of Ha T F

(j) The Central Limit Theorem says that the sample mean will be unbiased for the population mean only if the sample size is large T F

[10 marks] 2 A printer manufacturer tests 15 randomly selected printer heads and records the number of

characters printed until failure Below is a stem-and-Ieaf plot of the number of characters (in millions) until failure

Stern-and-Leaf Display nurn_char

Stem-and-leaf of num char N - 15 Leaf Unit - 0010 shy

1 8 5 2 9 2 3 10 7 5 11 38

(4) 12 0259 6 13 236 3 14 38 1 15 5

Page 2 of 12

Full Name Student Number

Fill in the blanks with the final answers

[1 m] (a) Find the range

[1 m] (b) Find the inter-quartile range IQR

~~ (c) If Lt~l Xi =1858 and Lt~l xl =235368 find the sample mean and sample standard deviation

s= ________X=------ shy

[1 m] (d) Find a 99 confidence interval for the mean

[1 m] (e) With 99 confidence the manufacturer can advertise that the mean lifetime of their printers is

at least characters

[1 m] (I) Suppose the popUlation standard deviation is known to be 21 million characters

Find a 99 confidence interval for the mean

[3 m] (g) What assumptions were required to use the CIs in (d) and (f) Assess whether or not these

conditions are met

Page 3 of 12

Full Name Student Number

[10 marks] 3 The table below shows results of a 1998 study on internet communication in particular flaming in chat rooms The investigators were interested in whether being personally criticized by 0 thers wou ld rnake one more lk1 ely to cntlclze 1 0 thers

Personally Criticized

Not Been Personally Criticized Total

Criticized Others 19 8 127 Has Not Criticized Others 23 143 166 Total 42 151 193

Now suppose a chat room user is randomly selected

[2m] (a) Label with letters and name the two events of interest

For each ofthe following write the events in question with proper notation and then estimate the probabilities Put theflnal answer on the blank where indicated

[1m] (b) Probability that the individual has criticized others

~~ (c) Probability of being criticized or criticizing others

[1m] (d) Probability of being criticized and criticizing others

[2 m] (e) The user has been personally criticized by someone else What is the probability that he will

criticize another individual

[3 m] (1) Are criticizing others and being personally criticized independent events Justify

Page 4 of 12

Full Name Student Number

[10 marks - 1 each blanksentence] 4 Religion Studies researchers were interested in making inferences about the true proportion of

Canadians who believe in some fonnmiddotofGod They randomly surveyed 100 Canadians and asked them whether or not they believed in God (Yes or No) Here is the Minitab output

Test and CI for One Proportion

Test of p = 05 vs p not = 05

Event = Yes

Sample X N Sample p 95 CI Z-Value P-Value 1 61 100 0610000 (0514403 0705597) 220 0028

Use the output to fill in the blanks

(a) people in the survey said they believed in God

(b) As given in the output the null hypothesis is and the alternative hypothesis is [use proper notation and symbols]

(c) The value of the test statistic is ________

(d) If Ho is true the distribution of the test statistic is _______

(e) The 95 confidence interval for the true proportion of Canadians Who do NOT believe in some fonn of God is _____________

(f) Using a confidence interval decide on whether or not you agree with this claim More than 34 ofCanadians believe in some form ofGod Fill in agree or disagree _______

(g) If the researchers are interested in answering the question Do the majority ofCanadians believe in some form ofGoel the appropriate p-value is At the 5 significance level the conclusion would be [reject or FTR Ho ]

(b) Give a one sentence conclusion in plain English to answer the question of interest from (g)

Page 5 of 12

Full Name Student Number

[20 marks] 5 An intersection in Mississauga has an average of 6 accidents per month

Before answering each part label the random variable ofinterest and identify its distribution with correct parameters Show your work and circle your final answers

[3 m] (a) What is the probability that there are exactly 2 accidents in September at this intersection

[3mr (b) What is the probability that in the next year 3 months will have exactly 2 accidents in them

each

[4 m] (c) Approximate the probability that in the next two years at least 10 months will have 2 accidents

in them each

Page 6 of 12

Full Name Student Number

[2 m] (d) On average how long would you have to wait for an accident to occur at this intersection

(Include the units in your answer)

[3 m] (e) An accident just occurred today What is the probability that the next accident is more than 12

a month from today

[5 m] (f) You take a random sample of 100 months and record the number of accidents that occur What

is the probability that the average number exceeds 6S accidents

Page 7 of 12

Full Name Student Number

lOmarksj 6 A new toothpaste has been put on the market that claims to reduce the number of teeth that have

problems (cavities sensitive teeth etc) In order to test the effect ofthe new toothpaste researchers randomly select participants and randomly assign each person to use either the new toothpaste or a standard toothpaste (a popular brand that has been used for several years) After 6 months of use each participant reports the number of teeth they have with problems It can be assumed that both populations are Normal Partial Minitab output is given

DESCRIPTIVE STATISTICS standard DESCRIPTIVE STATISTICS new

Variable N Mean StDev SE MeanVariable N Mean StDev SE Mean new 20 330 213 048standard 15 370 198 051

Test for Equal Variances

95 Bonferroni confidence intervals for standard deviations

Sample N Lower StDev Upper standard 15 138964 1 98 335600 new 20 156050 213 330000

F-Test (normal distribution) Test statistic = 086 p-value = 079

[2 m)

(a) Using a=O05 test for the equality of the variances of the standard and new toothpaste populations Include all steps requiredfor a Hypothesis Test (you may copy any relevant parts from the output without calculating them yourself)

Page 8 of 12

Full Name Student Number

[5 m] (b) Which type of hypothesis test would you use to test if the new toothpaste is effective as

claimed by the manufacturer using a005 Why Test the relevant hypothesis from scratch Include all steps requiredfor a Hypothesis Test

[3 m] (c) The manufacturer of the new toothpaste now decides to collect data on consumers ofthe new

toothpaste separately in order to improve their study How large of sample is required to estimate the mean number of teeth with problems within 05 and retain 95 confidence

Page 9 of 12

Full Name Student Number

flOmarksJ 7 Is insomnia related to education Researchers conducted a study to determine if insomnia is related to the education of Tennessee residents 575 adults in Tennessee were selected to participate in the study using a random-digit telephone procedure Thenumber of years of education and insomnia status (normal sleeper or chronic insomniac) were recorded for each participant The researchers discovered that the fewer the years of education the more likely the person is to have chronic insomnia

~Insomnia is a sleep disorder (problems falling asleepstaying asleep throughout the night) In a random-digit telephone procedure telephone numbers are generated at random and the people with those phone numbers are contacted to participate in the study

Fill in the blanks or give a one-sentence brief answer where appropriate to following

[1 m] (a) Identify the population of interest

[1 m] (b) Identify the sample

[4 m] (c) List the variables measured and whether they are categorical or quantitative

[1 m] (d) What inference(s) did the researchers make

[1 m] (e) In general conclusions about a sample can be generalized to the population if the sample is

___________ of the population

[2 m] (f) In this case do the selected participants form a random sample Explain why or why not

Page 10 of 12

Full Name Student Number

[10marksj 8 You and your friend playa dice-rolling game You each roll a fair die once and observe the

number that appears on the up face for each person Ifboth ofyou get the same number nobody wins or loses any money If you get different numbers whoever gets the lower number has to pay the other person $1 Let X represent your winnings (in $) from this game

[4 m] (a) Find the probability distribution for X and display it in table form

[3 m] (b) Calculate the mean ofX and interpret your answer in terms of the game Show your

calculations and then circle your final answer

[3 m] (c) Compute the variance ofX Show your work and circle thefinal answer

Page 11 of 12

Full Name Student Number

10 marks 9 A public school is considering adding to the curriculum a course that it believes will improve the reading comprehension skills of its students The school has decided to conduct an experiment to evaluate the course before it is offered to the general student body Six students are randomly selected and their reading scores are recorded before and after the course These scores are composite measurements ofboth speed and comprehension (where higher scores indicate higher

h )Th I dblcompre enSlOn e resu ts are summanze e ow Student Number Score After Course Score Before Course

1 221 211 2 231 216 3 203 191 4 216 224 5 207 201 6 203 178

The school will implement the new course if there is sufficient evidence at the 5 significance level that the course improves reading comprehension

[2 m) (a) Which type of hypothesis test is appropriate for this situation and why

[6m (b) Conduct the hypothesis test of interest Include all steps requiredfor a Hypothesis Test

[2 m (c) List the assumption(s) required to use this type ofhypothesis test

Page 12 of 12 --END OF EXAM--shy

794 Appendix A Tables

TABLE IV Normal Curve Areas

o 1

00 01 bull02z 03 OS04 06 07 OS 09

0 0000 0040 0080 0120 01990160 0239 0279 0319 03591 0398 0438 0478 0517 05960557 0636 0675 0714 0753

07932 0832 0871 0910 0948 0987 1026 1064 1103 1141 3 1179 1217 1255 1293 13681331 1406 1443 1480 1517

15544 1591 1628 1664 1700 1736 1772 1808 1844 1879

1915oS 1950 1985 2019 2054 2088 2123 2157 2190 2224

22576 2291 2324 2357 24222389 2454 2486 2517 2549

25807 2611 2642 2673 2704 2734 2764 2794 2823 2852

28818 2910 2939 2967 2995 3023 3051 3078 3106 3133 3212 9 y3159 3186 3238 3264 3289 3315 3340 3365 3389

10 3413 3438 3461 3485 35313508 3554 3577 3599 3621 372911 3643 3665 3686 3708 3749 3770 3790 - 3810 3830

3849 386912 3888 3907 3925 3944 3962 3980 3997 4015

4032 4049 13 bulliQ6timiddot 4082 41154099 4131 4147 4162 4177 42071~m224192L4 4236 4251 4265 43194179 4292 4306

4332 4345 4357 4370 4382 4394middot 4406 4418 4429lS 4441 ltIJ

4452L6 4463 4474 4484 4495 4505 4515 4525 4535 4345

4554L7 4564 4573 4582 4591 4599 4608 4616 4625 4633

464118 4656 46644P49 4671 4678 4686 4693 47064699

4713 4761 L9 4719 4732 ~ 47674738 4744 4750 4726 4756 478820 4772 4793 4798 4893 4808 4812 4817~11amp 4783 shy

4821U 4826 4830 4834 48424838 48574846 4850 4854 486122 4864 4868 48874871 4875 4878 48904881 4884

23 4893 4896 4898 49164901 4913 4906 4909 49114904 24 4918 4922492Q 49364925shy 49294927 49344931 4932

4943493825 49524940 4941 4945 4946 49514948 4949 495326 4955 49644956 4957 4959 4960 4961 49634962 4965 497427 49674966 4968 4969 4970 49734971 4972 4974 498128 4975 4976 4977 4977 4978 49804979 4979

29 49864981 4982 4982 4983 4984 4984 49864985~49B5 30 4987 4987middot4987 4988 4988 4989 4990 49904989 4989 i

Source Abridged from Table I ofA Hald Statistical Table$ and Fonnulas (New YorkWiley) 1952 Reproduced by permission of A Hald

796 -~~ - -

o ~~~-TABLE VI Critical Values of t

f(t)

bull

Degrees of -Freedom tJWtlOO t050 tOO51010 tooi tooos

1 63143078 12706 31821 63657 31831 63662 2 1886 2920 4303 6965 9925 22326 31598 3 4541 1638 2353 3-182 5841 10213 12924 4 1533 2132 2776 3747 4604 7173 8610 5 1476 2015 2571 3365 4032 5893 6869 6 19431440 2447 3143 3707 5208 5959 7 1415 23651895 2998 3499 4785 5408

1397 1860 2306i 2896 3355 4501 5041 9 1383 1833 2262 2821 3250 4297 4781

10 18121372 2228 2764 3169 _ 4144 4587 U 1363 1796 2201 2718 3106 4025 4431 12 1356 1782 2179 shy 2681 3055 3930 4318

13~~ ~_ ~-i77113 i1reg 2650 3012 3852 4221 14 1345 21451761 2024 ~ 2977 41403787 15 1341 1753 2131 2602 29fl7 4073 16

373~ 1337 1746 2120 - 29212583 3686 4015

17 1333 1740 2110 2567 -2898 3646 3965 18 1330 1734 2101 t28782552 3610 3922 19 1328 t729 2093 28612539 38833579 - i

li~r~5I - 20 1725 2086 2528 2845 3850 21

3552 t 1323 2080 2518 1721 2831 3527 3819

22 20741321 1717 2508 2819 3505 3~792 23 1319 1714 2069 2500 3767 24

2807 3485 1318 1711 2064 2492 3467 3745

25 2middot797

1316 1708 2060 2485 3725 26

2787 3450 1315 1706 2056 2479 2779 3435 3707

27 1314 1703 2052 2473 2771 36903421 28 1313 170f 1 2048 2763 3674 29

~467 3408 1311 1699 2045 - 2462 2756 36593396

30 1310 1697 2D42 2457 - 2750 3646 40

3385 1303 1684 2021 2423 2704 3551

60 3307

1296 -16n 2000 2390 2660 3460 120

3232 1289 1658 1980 2358 2617 33733160

00 1645 1282 1960 2326 2576 3090 3291

Source 1his table is reproduced with the lcind permtsSioil of~e frustees of Biometrika from E S Pearson and R O Hartley (eds) The Biometrika Tables for Statisticians Vol 1 3d ed Biometrika 1966

r7~~

c

~ ApPENDIX A Tables 801

i~ 1lt TABLE IX Percentage Points of the F-distribution a = 05

~ liZ 1 2

1 1614 1995 2 1851 1900 3 1013 955 4 771 694

II

NUMERATOR DEGREES OF FREEDOM I11

3 4 9 I5 6 7 8

5

2157 2246 2302 2340 1916 1925 1930 1933

928 912 901 894 659 639 626 616

~ 661 579 541 519~ 505 495 6 599 514 476 453 439 428~7 559 474 435 412 397 387

5328 446 407 384 369 358 9 512 426 386 363 348 337~

10 -496 4100 371 348 333 322Q 11 484 7f~i 359398--~ 336 320 309

34938~middot12middot 475 326 300 ~~311~ 13 381467 341 318 303 292 0 fi

14 460 374 334 311 296 285 ~ 15 454 368 329 306 290foil 279 rr

449 363~ 16 324 301 285 274

~ 17 445 )20 296 281359 270 foil 18 441 3-16 293li))~5 277 266Q

438 352middot 313 290= 19 274 263 0 20 435 349 310 287 271 260 ~ 21 432 347 257307 284 268

i 22 430 344 305 282 266 255 303 0 23 428 342 280 264 253

Z 24 426 340 301 278 262 251foil Q 424middot25 339 299 276 260 249

26 423 337 298 274 259 247 27 421 335 296 273 257 246 28 420 334 295 271 245 256 29 418 333 293I 270 255 243 30 417 332 292 269 253 242 40 408 323 284 261 245 234 60 40q 315 276 253 237 225

120 307 392 268 245 229 217 00 384 300 260 237 221 210

2368 2389 1935 1937 889 885 609 604 488 482 421 415 379 373 350 344 329 323 314 307 301 295 291 285

283 277 271 1276 270 265 271 264 259 i266 259 254 1rII261 255 249

251 258 ~i-middot246 f 254 248 242 I

I 251 245 239 249 242 237 246 240 234 244 237 232 242 236 230 240 234 228 239 232 2]7 237 231 225 236 229 224 235 228 222 233 227 221

225 218 212 I

1217 210 204 209 202 196 201 194 188

2405 1938 881 600 477 410 368 339 318 302 290 280 ji

Source From M Merrington and eM Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 19433373-88

i (continued) I

I

I iii 11 iii

II I

I l

iINi II

802 Appendix A Tables

TABLE IX Continued

~1 NUMERATOR DEGREES OF FREEDOM Vl ~r--l-0----12----15-----r--2-0----24---r--30----40----60~--~- ~ OQ

1 2 3 4 S 6 7 8 9

~ 10 11

~ ~ ~ 14 o 15 rIJ (

Iiil 16 ~ 17 ~ 18 ~ 19 = 20

~ 21

i ii shy~ 24 ~ 2S

26 27 28 29 30

middot40 60

120 OQ

2419 1940 879 596 474 406 364 335 ~314

298 285 275 267 260 254 249 245 241 238 235 232 230 227 225 224 222 220 219 218 216

208 199 191 1~

2439 2459 2480 2491 1941 1943 1945 1945 874 870 866 864 591 586 580 577 468 462 456 453 400 394 387 384 357 351 344 341 328 322 315 312 307 301 294 290 291 285 277 274 279 272 265 261 269 262 254 251 260 253 246 242 253 246 239 235 248 240 233 229 242 235 228 224 238 231 223 219 234 227 219 215 231 223 216 211 iJi ~r~ 220 212 208 225 218 210 205 223 215 207 203 220 213 205 201 218 211 203 198 216 209 201 196

215 2Q7 199 195j 2i3 206 197 193 212 204 196 191 210 203 194 190 209 201 193 189

200 192184 179 192 184 175 170 183 i 175 166 161 U5 1m 1~ 1~

2501 1946 862 575 450 381 33S 308 286 270 257 247

238 231 225 219 215 211 207 204 201 198 196 194 192 190 188 187 185 184

174 165 155 146

2511 Imiddotmiddotmiddot

1947 859 572 446 377middot 334 304 283 266 253 243 234 227 220 215 210 206 203

~199 196 f 194 1~1 189 187 185 184 182 181 179

169 159 150 139

2522 1948 857 5~69

443 374 330 301 279 262 249 238 230 222

216 211 206 202 198 195 192 189 186 184 182 180 179 177 175 174

164 153 143 132

2533 2543 1949 1950 855 853 566 563 440 436 370 367 327 323 297 293 275 271 258 254 245 240 234 230 225 221 218 213 211 207 206 201 201 196 197 192 193 188 190 184 1~7 181 184 178

181 176 179 173 177 ~ 171

169 17~ 173 167 171 165

164170 162168

151158 147 139

125135 100122 i

ApPENDIX A Tables~c63 y

TABLE X Percentage Points of the F-distribution a = 025

f(F)

o FfJ25

- -NUMERATOR DEGREES OF FREEDOM ~ Jz 1 2 3 4 5 6 7 8 9

1 6478 7995 8642 8996 9218 9371 9482 9567 9633 2 3851 3900 3917 3925 3930 3933 3936 3937 3939 3 1744 1604 1544 1510 1488 1473 1462 1454 1447 4 1222 1065 998 960 936 920 907 898 890 5 1001 843 776 739 715 698 685 676 668 ( 881 726 660 623 599 582 570 560 552

589 7 807 654 552 529 512 499 490 482 8 757 606 542 505 482 465 453 443 436 9 721 571 508 472 448 432 420 410 403

gt1 10 - 694 483 447 424 407 395 385 378-4~~i 0 003f2~~ 11 672 463 428 404 88 376 366 359fIl

12- 655 510 447 412 389 373 351 344~61f l~ 13 641 497 435 400 377 360 348 339 331

0 ~ 14 630 486 424 389 366 329338-~$P 321 IIl 15 620 477 A15 380 358 34-1 329 320 312fIl

16 61_2 469 408 373 350 334 322 305 31~r-~ ~ 17 462- 4016Cl1) 366 344 328 316 298I j r~ 30(hmiddotfIl 45618 598 395 361 338 322 310 301 293~

592 451 390 356 333= 19 317 305 296 288

~ 20 587 446 386 351 329 313 301 291 284 21 583 442 382 348 325 309 297 287 280 22 - 579 438 378 344 322 305 293 284 276~

0 23 575 435 375 341 318 302 290 281 273 ~ 24 572 432 372 338 315 299 287 278 270 ~ 25 569 429 369 335 313 297 285 275 268

26 566 427 367 333 310 294 282 273 265 27 563 424 365 331 - 292 308 280 271 263 28 561 422 363 329 306 290 278 269 261 29 559 420 361 327 304 288 276 267 259 30 557 418 359 325 303 287 275 265 257

40 542 405 346 290 313 274 262 253 245 60 529 393 334 301 279 263 251 241 233

120 515 380 323 289 267 252 239 230 222 00 502 369 312 279 257 241 229 219 211

SourceFrom M Merrington and C M Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 1943337388

(continued)

4 804 Appendix A Tables

TABLE X Continued

~Vl NUMERATOR DEGREES OF FREEDOM V2~r---l-0--r--U--~-~~--~~20-----~~---~------~~middotmiddot----6-0-----U~O-----~---

9686 9767 9849 9931 9972 1001 1006 _1010 1~014-1 1018 2 3940 3941 3943 3945 3946 3946 3947 3948 394~ 3950 3 1442 1434 1425 1417 1412 1408 1404 1399 1395 1390

884 875 866 856 851 846 841 836 8314 826 662 652 643 633 628 623 618 612 6075 602 546 537 527 517 512 507 501 496 4906 485

7 476 467 457 447 442 436 431 425 420 414 8 430 420 410 400 395 389 384 378 373 367 9 396 387 377 367 361 356 351 345 339 333

372 362 352 342 337 331 326 320 314 308S 10 353 343 333 323 317 312 306 300 294Q 11 288 337 328 318 307 302 296 291 285 279 272 325 315 305 295 289 284 278 272 266 260E ~

)i 14 315 305 295 284 279 273 267 261 255 249 o 15 306 296 286 276 270 264 259 252 246 240rIl ~ 16 232299 289 279 268 263 257 251 245 238 ~ ~17 292 282 272 262 256 250 244 238 232 225

287 277 267 256 250 244 238 232 226~ 18 219 Q 19 282 272 262 251 245 239 233 227 220 21~=

277 268 257 246 241 235 229 222 216 209~ 20middot 273 bull =I~~~ 253 242 237 231 ~ 225 218 211lt 21 204 270 middot260 250 239 233 227 21 214 208 200~ 22 267 257 247 236 230 224 $ 218 211 204 197o 23

z 24 264 254 244 D3 227 221 ~ Ii 215 208 201 194 261 251 241 230 224 218 212 205 198 191~ 25

26 259 249 239 228 222 216 209 203 )95 I 188 27 257 247 236 225 219 213 207 200 i~~3 185 28 2d5tf i 245middot 234 223217 211 205 198 191 183 29 253 243 232 221 215 209 203 196 189 181 30 251 241 231 220 214 207 201 194 187 179

40 164239 229 218 207 201 194 188 180 172 60 227 217 206 194 188 182 174 167 158 148

120 216 205 194 182 176 169 161 153 143 131 I

QO 100205 194 I 183 171 I 164 157 148 139 127

Page 3: STA220_Final_2012S.pdf

Full Name Student Number

Fill in the blanks with the final answers

[1 m] (a) Find the range

[1 m] (b) Find the inter-quartile range IQR

~~ (c) If Lt~l Xi =1858 and Lt~l xl =235368 find the sample mean and sample standard deviation

s= ________X=------ shy

[1 m] (d) Find a 99 confidence interval for the mean

[1 m] (e) With 99 confidence the manufacturer can advertise that the mean lifetime of their printers is

at least characters

[1 m] (I) Suppose the popUlation standard deviation is known to be 21 million characters

Find a 99 confidence interval for the mean

[3 m] (g) What assumptions were required to use the CIs in (d) and (f) Assess whether or not these

conditions are met

Page 3 of 12

Full Name Student Number

[10 marks] 3 The table below shows results of a 1998 study on internet communication in particular flaming in chat rooms The investigators were interested in whether being personally criticized by 0 thers wou ld rnake one more lk1 ely to cntlclze 1 0 thers

Personally Criticized

Not Been Personally Criticized Total

Criticized Others 19 8 127 Has Not Criticized Others 23 143 166 Total 42 151 193

Now suppose a chat room user is randomly selected

[2m] (a) Label with letters and name the two events of interest

For each ofthe following write the events in question with proper notation and then estimate the probabilities Put theflnal answer on the blank where indicated

[1m] (b) Probability that the individual has criticized others

~~ (c) Probability of being criticized or criticizing others

[1m] (d) Probability of being criticized and criticizing others

[2 m] (e) The user has been personally criticized by someone else What is the probability that he will

criticize another individual

[3 m] (1) Are criticizing others and being personally criticized independent events Justify

Page 4 of 12

Full Name Student Number

[10 marks - 1 each blanksentence] 4 Religion Studies researchers were interested in making inferences about the true proportion of

Canadians who believe in some fonnmiddotofGod They randomly surveyed 100 Canadians and asked them whether or not they believed in God (Yes or No) Here is the Minitab output

Test and CI for One Proportion

Test of p = 05 vs p not = 05

Event = Yes

Sample X N Sample p 95 CI Z-Value P-Value 1 61 100 0610000 (0514403 0705597) 220 0028

Use the output to fill in the blanks

(a) people in the survey said they believed in God

(b) As given in the output the null hypothesis is and the alternative hypothesis is [use proper notation and symbols]

(c) The value of the test statistic is ________

(d) If Ho is true the distribution of the test statistic is _______

(e) The 95 confidence interval for the true proportion of Canadians Who do NOT believe in some fonn of God is _____________

(f) Using a confidence interval decide on whether or not you agree with this claim More than 34 ofCanadians believe in some form ofGod Fill in agree or disagree _______

(g) If the researchers are interested in answering the question Do the majority ofCanadians believe in some form ofGoel the appropriate p-value is At the 5 significance level the conclusion would be [reject or FTR Ho ]

(b) Give a one sentence conclusion in plain English to answer the question of interest from (g)

Page 5 of 12

Full Name Student Number

[20 marks] 5 An intersection in Mississauga has an average of 6 accidents per month

Before answering each part label the random variable ofinterest and identify its distribution with correct parameters Show your work and circle your final answers

[3 m] (a) What is the probability that there are exactly 2 accidents in September at this intersection

[3mr (b) What is the probability that in the next year 3 months will have exactly 2 accidents in them

each

[4 m] (c) Approximate the probability that in the next two years at least 10 months will have 2 accidents

in them each

Page 6 of 12

Full Name Student Number

[2 m] (d) On average how long would you have to wait for an accident to occur at this intersection

(Include the units in your answer)

[3 m] (e) An accident just occurred today What is the probability that the next accident is more than 12

a month from today

[5 m] (f) You take a random sample of 100 months and record the number of accidents that occur What

is the probability that the average number exceeds 6S accidents

Page 7 of 12

Full Name Student Number

lOmarksj 6 A new toothpaste has been put on the market that claims to reduce the number of teeth that have

problems (cavities sensitive teeth etc) In order to test the effect ofthe new toothpaste researchers randomly select participants and randomly assign each person to use either the new toothpaste or a standard toothpaste (a popular brand that has been used for several years) After 6 months of use each participant reports the number of teeth they have with problems It can be assumed that both populations are Normal Partial Minitab output is given

DESCRIPTIVE STATISTICS standard DESCRIPTIVE STATISTICS new

Variable N Mean StDev SE MeanVariable N Mean StDev SE Mean new 20 330 213 048standard 15 370 198 051

Test for Equal Variances

95 Bonferroni confidence intervals for standard deviations

Sample N Lower StDev Upper standard 15 138964 1 98 335600 new 20 156050 213 330000

F-Test (normal distribution) Test statistic = 086 p-value = 079

[2 m)

(a) Using a=O05 test for the equality of the variances of the standard and new toothpaste populations Include all steps requiredfor a Hypothesis Test (you may copy any relevant parts from the output without calculating them yourself)

Page 8 of 12

Full Name Student Number

[5 m] (b) Which type of hypothesis test would you use to test if the new toothpaste is effective as

claimed by the manufacturer using a005 Why Test the relevant hypothesis from scratch Include all steps requiredfor a Hypothesis Test

[3 m] (c) The manufacturer of the new toothpaste now decides to collect data on consumers ofthe new

toothpaste separately in order to improve their study How large of sample is required to estimate the mean number of teeth with problems within 05 and retain 95 confidence

Page 9 of 12

Full Name Student Number

flOmarksJ 7 Is insomnia related to education Researchers conducted a study to determine if insomnia is related to the education of Tennessee residents 575 adults in Tennessee were selected to participate in the study using a random-digit telephone procedure Thenumber of years of education and insomnia status (normal sleeper or chronic insomniac) were recorded for each participant The researchers discovered that the fewer the years of education the more likely the person is to have chronic insomnia

~Insomnia is a sleep disorder (problems falling asleepstaying asleep throughout the night) In a random-digit telephone procedure telephone numbers are generated at random and the people with those phone numbers are contacted to participate in the study

Fill in the blanks or give a one-sentence brief answer where appropriate to following

[1 m] (a) Identify the population of interest

[1 m] (b) Identify the sample

[4 m] (c) List the variables measured and whether they are categorical or quantitative

[1 m] (d) What inference(s) did the researchers make

[1 m] (e) In general conclusions about a sample can be generalized to the population if the sample is

___________ of the population

[2 m] (f) In this case do the selected participants form a random sample Explain why or why not

Page 10 of 12

Full Name Student Number

[10marksj 8 You and your friend playa dice-rolling game You each roll a fair die once and observe the

number that appears on the up face for each person Ifboth ofyou get the same number nobody wins or loses any money If you get different numbers whoever gets the lower number has to pay the other person $1 Let X represent your winnings (in $) from this game

[4 m] (a) Find the probability distribution for X and display it in table form

[3 m] (b) Calculate the mean ofX and interpret your answer in terms of the game Show your

calculations and then circle your final answer

[3 m] (c) Compute the variance ofX Show your work and circle thefinal answer

Page 11 of 12

Full Name Student Number

10 marks 9 A public school is considering adding to the curriculum a course that it believes will improve the reading comprehension skills of its students The school has decided to conduct an experiment to evaluate the course before it is offered to the general student body Six students are randomly selected and their reading scores are recorded before and after the course These scores are composite measurements ofboth speed and comprehension (where higher scores indicate higher

h )Th I dblcompre enSlOn e resu ts are summanze e ow Student Number Score After Course Score Before Course

1 221 211 2 231 216 3 203 191 4 216 224 5 207 201 6 203 178

The school will implement the new course if there is sufficient evidence at the 5 significance level that the course improves reading comprehension

[2 m) (a) Which type of hypothesis test is appropriate for this situation and why

[6m (b) Conduct the hypothesis test of interest Include all steps requiredfor a Hypothesis Test

[2 m (c) List the assumption(s) required to use this type ofhypothesis test

Page 12 of 12 --END OF EXAM--shy

794 Appendix A Tables

TABLE IV Normal Curve Areas

o 1

00 01 bull02z 03 OS04 06 07 OS 09

0 0000 0040 0080 0120 01990160 0239 0279 0319 03591 0398 0438 0478 0517 05960557 0636 0675 0714 0753

07932 0832 0871 0910 0948 0987 1026 1064 1103 1141 3 1179 1217 1255 1293 13681331 1406 1443 1480 1517

15544 1591 1628 1664 1700 1736 1772 1808 1844 1879

1915oS 1950 1985 2019 2054 2088 2123 2157 2190 2224

22576 2291 2324 2357 24222389 2454 2486 2517 2549

25807 2611 2642 2673 2704 2734 2764 2794 2823 2852

28818 2910 2939 2967 2995 3023 3051 3078 3106 3133 3212 9 y3159 3186 3238 3264 3289 3315 3340 3365 3389

10 3413 3438 3461 3485 35313508 3554 3577 3599 3621 372911 3643 3665 3686 3708 3749 3770 3790 - 3810 3830

3849 386912 3888 3907 3925 3944 3962 3980 3997 4015

4032 4049 13 bulliQ6timiddot 4082 41154099 4131 4147 4162 4177 42071~m224192L4 4236 4251 4265 43194179 4292 4306

4332 4345 4357 4370 4382 4394middot 4406 4418 4429lS 4441 ltIJ

4452L6 4463 4474 4484 4495 4505 4515 4525 4535 4345

4554L7 4564 4573 4582 4591 4599 4608 4616 4625 4633

464118 4656 46644P49 4671 4678 4686 4693 47064699

4713 4761 L9 4719 4732 ~ 47674738 4744 4750 4726 4756 478820 4772 4793 4798 4893 4808 4812 4817~11amp 4783 shy

4821U 4826 4830 4834 48424838 48574846 4850 4854 486122 4864 4868 48874871 4875 4878 48904881 4884

23 4893 4896 4898 49164901 4913 4906 4909 49114904 24 4918 4922492Q 49364925shy 49294927 49344931 4932

4943493825 49524940 4941 4945 4946 49514948 4949 495326 4955 49644956 4957 4959 4960 4961 49634962 4965 497427 49674966 4968 4969 4970 49734971 4972 4974 498128 4975 4976 4977 4977 4978 49804979 4979

29 49864981 4982 4982 4983 4984 4984 49864985~49B5 30 4987 4987middot4987 4988 4988 4989 4990 49904989 4989 i

Source Abridged from Table I ofA Hald Statistical Table$ and Fonnulas (New YorkWiley) 1952 Reproduced by permission of A Hald

796 -~~ - -

o ~~~-TABLE VI Critical Values of t

f(t)

bull

Degrees of -Freedom tJWtlOO t050 tOO51010 tooi tooos

1 63143078 12706 31821 63657 31831 63662 2 1886 2920 4303 6965 9925 22326 31598 3 4541 1638 2353 3-182 5841 10213 12924 4 1533 2132 2776 3747 4604 7173 8610 5 1476 2015 2571 3365 4032 5893 6869 6 19431440 2447 3143 3707 5208 5959 7 1415 23651895 2998 3499 4785 5408

1397 1860 2306i 2896 3355 4501 5041 9 1383 1833 2262 2821 3250 4297 4781

10 18121372 2228 2764 3169 _ 4144 4587 U 1363 1796 2201 2718 3106 4025 4431 12 1356 1782 2179 shy 2681 3055 3930 4318

13~~ ~_ ~-i77113 i1reg 2650 3012 3852 4221 14 1345 21451761 2024 ~ 2977 41403787 15 1341 1753 2131 2602 29fl7 4073 16

373~ 1337 1746 2120 - 29212583 3686 4015

17 1333 1740 2110 2567 -2898 3646 3965 18 1330 1734 2101 t28782552 3610 3922 19 1328 t729 2093 28612539 38833579 - i

li~r~5I - 20 1725 2086 2528 2845 3850 21

3552 t 1323 2080 2518 1721 2831 3527 3819

22 20741321 1717 2508 2819 3505 3~792 23 1319 1714 2069 2500 3767 24

2807 3485 1318 1711 2064 2492 3467 3745

25 2middot797

1316 1708 2060 2485 3725 26

2787 3450 1315 1706 2056 2479 2779 3435 3707

27 1314 1703 2052 2473 2771 36903421 28 1313 170f 1 2048 2763 3674 29

~467 3408 1311 1699 2045 - 2462 2756 36593396

30 1310 1697 2D42 2457 - 2750 3646 40

3385 1303 1684 2021 2423 2704 3551

60 3307

1296 -16n 2000 2390 2660 3460 120

3232 1289 1658 1980 2358 2617 33733160

00 1645 1282 1960 2326 2576 3090 3291

Source 1his table is reproduced with the lcind permtsSioil of~e frustees of Biometrika from E S Pearson and R O Hartley (eds) The Biometrika Tables for Statisticians Vol 1 3d ed Biometrika 1966

r7~~

c

~ ApPENDIX A Tables 801

i~ 1lt TABLE IX Percentage Points of the F-distribution a = 05

~ liZ 1 2

1 1614 1995 2 1851 1900 3 1013 955 4 771 694

II

NUMERATOR DEGREES OF FREEDOM I11

3 4 9 I5 6 7 8

5

2157 2246 2302 2340 1916 1925 1930 1933

928 912 901 894 659 639 626 616

~ 661 579 541 519~ 505 495 6 599 514 476 453 439 428~7 559 474 435 412 397 387

5328 446 407 384 369 358 9 512 426 386 363 348 337~

10 -496 4100 371 348 333 322Q 11 484 7f~i 359398--~ 336 320 309

34938~middot12middot 475 326 300 ~~311~ 13 381467 341 318 303 292 0 fi

14 460 374 334 311 296 285 ~ 15 454 368 329 306 290foil 279 rr

449 363~ 16 324 301 285 274

~ 17 445 )20 296 281359 270 foil 18 441 3-16 293li))~5 277 266Q

438 352middot 313 290= 19 274 263 0 20 435 349 310 287 271 260 ~ 21 432 347 257307 284 268

i 22 430 344 305 282 266 255 303 0 23 428 342 280 264 253

Z 24 426 340 301 278 262 251foil Q 424middot25 339 299 276 260 249

26 423 337 298 274 259 247 27 421 335 296 273 257 246 28 420 334 295 271 245 256 29 418 333 293I 270 255 243 30 417 332 292 269 253 242 40 408 323 284 261 245 234 60 40q 315 276 253 237 225

120 307 392 268 245 229 217 00 384 300 260 237 221 210

2368 2389 1935 1937 889 885 609 604 488 482 421 415 379 373 350 344 329 323 314 307 301 295 291 285

283 277 271 1276 270 265 271 264 259 i266 259 254 1rII261 255 249

251 258 ~i-middot246 f 254 248 242 I

I 251 245 239 249 242 237 246 240 234 244 237 232 242 236 230 240 234 228 239 232 2]7 237 231 225 236 229 224 235 228 222 233 227 221

225 218 212 I

1217 210 204 209 202 196 201 194 188

2405 1938 881 600 477 410 368 339 318 302 290 280 ji

Source From M Merrington and eM Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 19433373-88

i (continued) I

I

I iii 11 iii

II I

I l

iINi II

802 Appendix A Tables

TABLE IX Continued

~1 NUMERATOR DEGREES OF FREEDOM Vl ~r--l-0----12----15-----r--2-0----24---r--30----40----60~--~- ~ OQ

1 2 3 4 S 6 7 8 9

~ 10 11

~ ~ ~ 14 o 15 rIJ (

Iiil 16 ~ 17 ~ 18 ~ 19 = 20

~ 21

i ii shy~ 24 ~ 2S

26 27 28 29 30

middot40 60

120 OQ

2419 1940 879 596 474 406 364 335 ~314

298 285 275 267 260 254 249 245 241 238 235 232 230 227 225 224 222 220 219 218 216

208 199 191 1~

2439 2459 2480 2491 1941 1943 1945 1945 874 870 866 864 591 586 580 577 468 462 456 453 400 394 387 384 357 351 344 341 328 322 315 312 307 301 294 290 291 285 277 274 279 272 265 261 269 262 254 251 260 253 246 242 253 246 239 235 248 240 233 229 242 235 228 224 238 231 223 219 234 227 219 215 231 223 216 211 iJi ~r~ 220 212 208 225 218 210 205 223 215 207 203 220 213 205 201 218 211 203 198 216 209 201 196

215 2Q7 199 195j 2i3 206 197 193 212 204 196 191 210 203 194 190 209 201 193 189

200 192184 179 192 184 175 170 183 i 175 166 161 U5 1m 1~ 1~

2501 1946 862 575 450 381 33S 308 286 270 257 247

238 231 225 219 215 211 207 204 201 198 196 194 192 190 188 187 185 184

174 165 155 146

2511 Imiddotmiddotmiddot

1947 859 572 446 377middot 334 304 283 266 253 243 234 227 220 215 210 206 203

~199 196 f 194 1~1 189 187 185 184 182 181 179

169 159 150 139

2522 1948 857 5~69

443 374 330 301 279 262 249 238 230 222

216 211 206 202 198 195 192 189 186 184 182 180 179 177 175 174

164 153 143 132

2533 2543 1949 1950 855 853 566 563 440 436 370 367 327 323 297 293 275 271 258 254 245 240 234 230 225 221 218 213 211 207 206 201 201 196 197 192 193 188 190 184 1~7 181 184 178

181 176 179 173 177 ~ 171

169 17~ 173 167 171 165

164170 162168

151158 147 139

125135 100122 i

ApPENDIX A Tables~c63 y

TABLE X Percentage Points of the F-distribution a = 025

f(F)

o FfJ25

- -NUMERATOR DEGREES OF FREEDOM ~ Jz 1 2 3 4 5 6 7 8 9

1 6478 7995 8642 8996 9218 9371 9482 9567 9633 2 3851 3900 3917 3925 3930 3933 3936 3937 3939 3 1744 1604 1544 1510 1488 1473 1462 1454 1447 4 1222 1065 998 960 936 920 907 898 890 5 1001 843 776 739 715 698 685 676 668 ( 881 726 660 623 599 582 570 560 552

589 7 807 654 552 529 512 499 490 482 8 757 606 542 505 482 465 453 443 436 9 721 571 508 472 448 432 420 410 403

gt1 10 - 694 483 447 424 407 395 385 378-4~~i 0 003f2~~ 11 672 463 428 404 88 376 366 359fIl

12- 655 510 447 412 389 373 351 344~61f l~ 13 641 497 435 400 377 360 348 339 331

0 ~ 14 630 486 424 389 366 329338-~$P 321 IIl 15 620 477 A15 380 358 34-1 329 320 312fIl

16 61_2 469 408 373 350 334 322 305 31~r-~ ~ 17 462- 4016Cl1) 366 344 328 316 298I j r~ 30(hmiddotfIl 45618 598 395 361 338 322 310 301 293~

592 451 390 356 333= 19 317 305 296 288

~ 20 587 446 386 351 329 313 301 291 284 21 583 442 382 348 325 309 297 287 280 22 - 579 438 378 344 322 305 293 284 276~

0 23 575 435 375 341 318 302 290 281 273 ~ 24 572 432 372 338 315 299 287 278 270 ~ 25 569 429 369 335 313 297 285 275 268

26 566 427 367 333 310 294 282 273 265 27 563 424 365 331 - 292 308 280 271 263 28 561 422 363 329 306 290 278 269 261 29 559 420 361 327 304 288 276 267 259 30 557 418 359 325 303 287 275 265 257

40 542 405 346 290 313 274 262 253 245 60 529 393 334 301 279 263 251 241 233

120 515 380 323 289 267 252 239 230 222 00 502 369 312 279 257 241 229 219 211

SourceFrom M Merrington and C M Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 1943337388

(continued)

4 804 Appendix A Tables

TABLE X Continued

~Vl NUMERATOR DEGREES OF FREEDOM V2~r---l-0--r--U--~-~~--~~20-----~~---~------~~middotmiddot----6-0-----U~O-----~---

9686 9767 9849 9931 9972 1001 1006 _1010 1~014-1 1018 2 3940 3941 3943 3945 3946 3946 3947 3948 394~ 3950 3 1442 1434 1425 1417 1412 1408 1404 1399 1395 1390

884 875 866 856 851 846 841 836 8314 826 662 652 643 633 628 623 618 612 6075 602 546 537 527 517 512 507 501 496 4906 485

7 476 467 457 447 442 436 431 425 420 414 8 430 420 410 400 395 389 384 378 373 367 9 396 387 377 367 361 356 351 345 339 333

372 362 352 342 337 331 326 320 314 308S 10 353 343 333 323 317 312 306 300 294Q 11 288 337 328 318 307 302 296 291 285 279 272 325 315 305 295 289 284 278 272 266 260E ~

)i 14 315 305 295 284 279 273 267 261 255 249 o 15 306 296 286 276 270 264 259 252 246 240rIl ~ 16 232299 289 279 268 263 257 251 245 238 ~ ~17 292 282 272 262 256 250 244 238 232 225

287 277 267 256 250 244 238 232 226~ 18 219 Q 19 282 272 262 251 245 239 233 227 220 21~=

277 268 257 246 241 235 229 222 216 209~ 20middot 273 bull =I~~~ 253 242 237 231 ~ 225 218 211lt 21 204 270 middot260 250 239 233 227 21 214 208 200~ 22 267 257 247 236 230 224 $ 218 211 204 197o 23

z 24 264 254 244 D3 227 221 ~ Ii 215 208 201 194 261 251 241 230 224 218 212 205 198 191~ 25

26 259 249 239 228 222 216 209 203 )95 I 188 27 257 247 236 225 219 213 207 200 i~~3 185 28 2d5tf i 245middot 234 223217 211 205 198 191 183 29 253 243 232 221 215 209 203 196 189 181 30 251 241 231 220 214 207 201 194 187 179

40 164239 229 218 207 201 194 188 180 172 60 227 217 206 194 188 182 174 167 158 148

120 216 205 194 182 176 169 161 153 143 131 I

QO 100205 194 I 183 171 I 164 157 148 139 127

Page 4: STA220_Final_2012S.pdf

Full Name Student Number

[10 marks] 3 The table below shows results of a 1998 study on internet communication in particular flaming in chat rooms The investigators were interested in whether being personally criticized by 0 thers wou ld rnake one more lk1 ely to cntlclze 1 0 thers

Personally Criticized

Not Been Personally Criticized Total

Criticized Others 19 8 127 Has Not Criticized Others 23 143 166 Total 42 151 193

Now suppose a chat room user is randomly selected

[2m] (a) Label with letters and name the two events of interest

For each ofthe following write the events in question with proper notation and then estimate the probabilities Put theflnal answer on the blank where indicated

[1m] (b) Probability that the individual has criticized others

~~ (c) Probability of being criticized or criticizing others

[1m] (d) Probability of being criticized and criticizing others

[2 m] (e) The user has been personally criticized by someone else What is the probability that he will

criticize another individual

[3 m] (1) Are criticizing others and being personally criticized independent events Justify

Page 4 of 12

Full Name Student Number

[10 marks - 1 each blanksentence] 4 Religion Studies researchers were interested in making inferences about the true proportion of

Canadians who believe in some fonnmiddotofGod They randomly surveyed 100 Canadians and asked them whether or not they believed in God (Yes or No) Here is the Minitab output

Test and CI for One Proportion

Test of p = 05 vs p not = 05

Event = Yes

Sample X N Sample p 95 CI Z-Value P-Value 1 61 100 0610000 (0514403 0705597) 220 0028

Use the output to fill in the blanks

(a) people in the survey said they believed in God

(b) As given in the output the null hypothesis is and the alternative hypothesis is [use proper notation and symbols]

(c) The value of the test statistic is ________

(d) If Ho is true the distribution of the test statistic is _______

(e) The 95 confidence interval for the true proportion of Canadians Who do NOT believe in some fonn of God is _____________

(f) Using a confidence interval decide on whether or not you agree with this claim More than 34 ofCanadians believe in some form ofGod Fill in agree or disagree _______

(g) If the researchers are interested in answering the question Do the majority ofCanadians believe in some form ofGoel the appropriate p-value is At the 5 significance level the conclusion would be [reject or FTR Ho ]

(b) Give a one sentence conclusion in plain English to answer the question of interest from (g)

Page 5 of 12

Full Name Student Number

[20 marks] 5 An intersection in Mississauga has an average of 6 accidents per month

Before answering each part label the random variable ofinterest and identify its distribution with correct parameters Show your work and circle your final answers

[3 m] (a) What is the probability that there are exactly 2 accidents in September at this intersection

[3mr (b) What is the probability that in the next year 3 months will have exactly 2 accidents in them

each

[4 m] (c) Approximate the probability that in the next two years at least 10 months will have 2 accidents

in them each

Page 6 of 12

Full Name Student Number

[2 m] (d) On average how long would you have to wait for an accident to occur at this intersection

(Include the units in your answer)

[3 m] (e) An accident just occurred today What is the probability that the next accident is more than 12

a month from today

[5 m] (f) You take a random sample of 100 months and record the number of accidents that occur What

is the probability that the average number exceeds 6S accidents

Page 7 of 12

Full Name Student Number

lOmarksj 6 A new toothpaste has been put on the market that claims to reduce the number of teeth that have

problems (cavities sensitive teeth etc) In order to test the effect ofthe new toothpaste researchers randomly select participants and randomly assign each person to use either the new toothpaste or a standard toothpaste (a popular brand that has been used for several years) After 6 months of use each participant reports the number of teeth they have with problems It can be assumed that both populations are Normal Partial Minitab output is given

DESCRIPTIVE STATISTICS standard DESCRIPTIVE STATISTICS new

Variable N Mean StDev SE MeanVariable N Mean StDev SE Mean new 20 330 213 048standard 15 370 198 051

Test for Equal Variances

95 Bonferroni confidence intervals for standard deviations

Sample N Lower StDev Upper standard 15 138964 1 98 335600 new 20 156050 213 330000

F-Test (normal distribution) Test statistic = 086 p-value = 079

[2 m)

(a) Using a=O05 test for the equality of the variances of the standard and new toothpaste populations Include all steps requiredfor a Hypothesis Test (you may copy any relevant parts from the output without calculating them yourself)

Page 8 of 12

Full Name Student Number

[5 m] (b) Which type of hypothesis test would you use to test if the new toothpaste is effective as

claimed by the manufacturer using a005 Why Test the relevant hypothesis from scratch Include all steps requiredfor a Hypothesis Test

[3 m] (c) The manufacturer of the new toothpaste now decides to collect data on consumers ofthe new

toothpaste separately in order to improve their study How large of sample is required to estimate the mean number of teeth with problems within 05 and retain 95 confidence

Page 9 of 12

Full Name Student Number

flOmarksJ 7 Is insomnia related to education Researchers conducted a study to determine if insomnia is related to the education of Tennessee residents 575 adults in Tennessee were selected to participate in the study using a random-digit telephone procedure Thenumber of years of education and insomnia status (normal sleeper or chronic insomniac) were recorded for each participant The researchers discovered that the fewer the years of education the more likely the person is to have chronic insomnia

~Insomnia is a sleep disorder (problems falling asleepstaying asleep throughout the night) In a random-digit telephone procedure telephone numbers are generated at random and the people with those phone numbers are contacted to participate in the study

Fill in the blanks or give a one-sentence brief answer where appropriate to following

[1 m] (a) Identify the population of interest

[1 m] (b) Identify the sample

[4 m] (c) List the variables measured and whether they are categorical or quantitative

[1 m] (d) What inference(s) did the researchers make

[1 m] (e) In general conclusions about a sample can be generalized to the population if the sample is

___________ of the population

[2 m] (f) In this case do the selected participants form a random sample Explain why or why not

Page 10 of 12

Full Name Student Number

[10marksj 8 You and your friend playa dice-rolling game You each roll a fair die once and observe the

number that appears on the up face for each person Ifboth ofyou get the same number nobody wins or loses any money If you get different numbers whoever gets the lower number has to pay the other person $1 Let X represent your winnings (in $) from this game

[4 m] (a) Find the probability distribution for X and display it in table form

[3 m] (b) Calculate the mean ofX and interpret your answer in terms of the game Show your

calculations and then circle your final answer

[3 m] (c) Compute the variance ofX Show your work and circle thefinal answer

Page 11 of 12

Full Name Student Number

10 marks 9 A public school is considering adding to the curriculum a course that it believes will improve the reading comprehension skills of its students The school has decided to conduct an experiment to evaluate the course before it is offered to the general student body Six students are randomly selected and their reading scores are recorded before and after the course These scores are composite measurements ofboth speed and comprehension (where higher scores indicate higher

h )Th I dblcompre enSlOn e resu ts are summanze e ow Student Number Score After Course Score Before Course

1 221 211 2 231 216 3 203 191 4 216 224 5 207 201 6 203 178

The school will implement the new course if there is sufficient evidence at the 5 significance level that the course improves reading comprehension

[2 m) (a) Which type of hypothesis test is appropriate for this situation and why

[6m (b) Conduct the hypothesis test of interest Include all steps requiredfor a Hypothesis Test

[2 m (c) List the assumption(s) required to use this type ofhypothesis test

Page 12 of 12 --END OF EXAM--shy

794 Appendix A Tables

TABLE IV Normal Curve Areas

o 1

00 01 bull02z 03 OS04 06 07 OS 09

0 0000 0040 0080 0120 01990160 0239 0279 0319 03591 0398 0438 0478 0517 05960557 0636 0675 0714 0753

07932 0832 0871 0910 0948 0987 1026 1064 1103 1141 3 1179 1217 1255 1293 13681331 1406 1443 1480 1517

15544 1591 1628 1664 1700 1736 1772 1808 1844 1879

1915oS 1950 1985 2019 2054 2088 2123 2157 2190 2224

22576 2291 2324 2357 24222389 2454 2486 2517 2549

25807 2611 2642 2673 2704 2734 2764 2794 2823 2852

28818 2910 2939 2967 2995 3023 3051 3078 3106 3133 3212 9 y3159 3186 3238 3264 3289 3315 3340 3365 3389

10 3413 3438 3461 3485 35313508 3554 3577 3599 3621 372911 3643 3665 3686 3708 3749 3770 3790 - 3810 3830

3849 386912 3888 3907 3925 3944 3962 3980 3997 4015

4032 4049 13 bulliQ6timiddot 4082 41154099 4131 4147 4162 4177 42071~m224192L4 4236 4251 4265 43194179 4292 4306

4332 4345 4357 4370 4382 4394middot 4406 4418 4429lS 4441 ltIJ

4452L6 4463 4474 4484 4495 4505 4515 4525 4535 4345

4554L7 4564 4573 4582 4591 4599 4608 4616 4625 4633

464118 4656 46644P49 4671 4678 4686 4693 47064699

4713 4761 L9 4719 4732 ~ 47674738 4744 4750 4726 4756 478820 4772 4793 4798 4893 4808 4812 4817~11amp 4783 shy

4821U 4826 4830 4834 48424838 48574846 4850 4854 486122 4864 4868 48874871 4875 4878 48904881 4884

23 4893 4896 4898 49164901 4913 4906 4909 49114904 24 4918 4922492Q 49364925shy 49294927 49344931 4932

4943493825 49524940 4941 4945 4946 49514948 4949 495326 4955 49644956 4957 4959 4960 4961 49634962 4965 497427 49674966 4968 4969 4970 49734971 4972 4974 498128 4975 4976 4977 4977 4978 49804979 4979

29 49864981 4982 4982 4983 4984 4984 49864985~49B5 30 4987 4987middot4987 4988 4988 4989 4990 49904989 4989 i

Source Abridged from Table I ofA Hald Statistical Table$ and Fonnulas (New YorkWiley) 1952 Reproduced by permission of A Hald

796 -~~ - -

o ~~~-TABLE VI Critical Values of t

f(t)

bull

Degrees of -Freedom tJWtlOO t050 tOO51010 tooi tooos

1 63143078 12706 31821 63657 31831 63662 2 1886 2920 4303 6965 9925 22326 31598 3 4541 1638 2353 3-182 5841 10213 12924 4 1533 2132 2776 3747 4604 7173 8610 5 1476 2015 2571 3365 4032 5893 6869 6 19431440 2447 3143 3707 5208 5959 7 1415 23651895 2998 3499 4785 5408

1397 1860 2306i 2896 3355 4501 5041 9 1383 1833 2262 2821 3250 4297 4781

10 18121372 2228 2764 3169 _ 4144 4587 U 1363 1796 2201 2718 3106 4025 4431 12 1356 1782 2179 shy 2681 3055 3930 4318

13~~ ~_ ~-i77113 i1reg 2650 3012 3852 4221 14 1345 21451761 2024 ~ 2977 41403787 15 1341 1753 2131 2602 29fl7 4073 16

373~ 1337 1746 2120 - 29212583 3686 4015

17 1333 1740 2110 2567 -2898 3646 3965 18 1330 1734 2101 t28782552 3610 3922 19 1328 t729 2093 28612539 38833579 - i

li~r~5I - 20 1725 2086 2528 2845 3850 21

3552 t 1323 2080 2518 1721 2831 3527 3819

22 20741321 1717 2508 2819 3505 3~792 23 1319 1714 2069 2500 3767 24

2807 3485 1318 1711 2064 2492 3467 3745

25 2middot797

1316 1708 2060 2485 3725 26

2787 3450 1315 1706 2056 2479 2779 3435 3707

27 1314 1703 2052 2473 2771 36903421 28 1313 170f 1 2048 2763 3674 29

~467 3408 1311 1699 2045 - 2462 2756 36593396

30 1310 1697 2D42 2457 - 2750 3646 40

3385 1303 1684 2021 2423 2704 3551

60 3307

1296 -16n 2000 2390 2660 3460 120

3232 1289 1658 1980 2358 2617 33733160

00 1645 1282 1960 2326 2576 3090 3291

Source 1his table is reproduced with the lcind permtsSioil of~e frustees of Biometrika from E S Pearson and R O Hartley (eds) The Biometrika Tables for Statisticians Vol 1 3d ed Biometrika 1966

r7~~

c

~ ApPENDIX A Tables 801

i~ 1lt TABLE IX Percentage Points of the F-distribution a = 05

~ liZ 1 2

1 1614 1995 2 1851 1900 3 1013 955 4 771 694

II

NUMERATOR DEGREES OF FREEDOM I11

3 4 9 I5 6 7 8

5

2157 2246 2302 2340 1916 1925 1930 1933

928 912 901 894 659 639 626 616

~ 661 579 541 519~ 505 495 6 599 514 476 453 439 428~7 559 474 435 412 397 387

5328 446 407 384 369 358 9 512 426 386 363 348 337~

10 -496 4100 371 348 333 322Q 11 484 7f~i 359398--~ 336 320 309

34938~middot12middot 475 326 300 ~~311~ 13 381467 341 318 303 292 0 fi

14 460 374 334 311 296 285 ~ 15 454 368 329 306 290foil 279 rr

449 363~ 16 324 301 285 274

~ 17 445 )20 296 281359 270 foil 18 441 3-16 293li))~5 277 266Q

438 352middot 313 290= 19 274 263 0 20 435 349 310 287 271 260 ~ 21 432 347 257307 284 268

i 22 430 344 305 282 266 255 303 0 23 428 342 280 264 253

Z 24 426 340 301 278 262 251foil Q 424middot25 339 299 276 260 249

26 423 337 298 274 259 247 27 421 335 296 273 257 246 28 420 334 295 271 245 256 29 418 333 293I 270 255 243 30 417 332 292 269 253 242 40 408 323 284 261 245 234 60 40q 315 276 253 237 225

120 307 392 268 245 229 217 00 384 300 260 237 221 210

2368 2389 1935 1937 889 885 609 604 488 482 421 415 379 373 350 344 329 323 314 307 301 295 291 285

283 277 271 1276 270 265 271 264 259 i266 259 254 1rII261 255 249

251 258 ~i-middot246 f 254 248 242 I

I 251 245 239 249 242 237 246 240 234 244 237 232 242 236 230 240 234 228 239 232 2]7 237 231 225 236 229 224 235 228 222 233 227 221

225 218 212 I

1217 210 204 209 202 196 201 194 188

2405 1938 881 600 477 410 368 339 318 302 290 280 ji

Source From M Merrington and eM Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 19433373-88

i (continued) I

I

I iii 11 iii

II I

I l

iINi II

802 Appendix A Tables

TABLE IX Continued

~1 NUMERATOR DEGREES OF FREEDOM Vl ~r--l-0----12----15-----r--2-0----24---r--30----40----60~--~- ~ OQ

1 2 3 4 S 6 7 8 9

~ 10 11

~ ~ ~ 14 o 15 rIJ (

Iiil 16 ~ 17 ~ 18 ~ 19 = 20

~ 21

i ii shy~ 24 ~ 2S

26 27 28 29 30

middot40 60

120 OQ

2419 1940 879 596 474 406 364 335 ~314

298 285 275 267 260 254 249 245 241 238 235 232 230 227 225 224 222 220 219 218 216

208 199 191 1~

2439 2459 2480 2491 1941 1943 1945 1945 874 870 866 864 591 586 580 577 468 462 456 453 400 394 387 384 357 351 344 341 328 322 315 312 307 301 294 290 291 285 277 274 279 272 265 261 269 262 254 251 260 253 246 242 253 246 239 235 248 240 233 229 242 235 228 224 238 231 223 219 234 227 219 215 231 223 216 211 iJi ~r~ 220 212 208 225 218 210 205 223 215 207 203 220 213 205 201 218 211 203 198 216 209 201 196

215 2Q7 199 195j 2i3 206 197 193 212 204 196 191 210 203 194 190 209 201 193 189

200 192184 179 192 184 175 170 183 i 175 166 161 U5 1m 1~ 1~

2501 1946 862 575 450 381 33S 308 286 270 257 247

238 231 225 219 215 211 207 204 201 198 196 194 192 190 188 187 185 184

174 165 155 146

2511 Imiddotmiddotmiddot

1947 859 572 446 377middot 334 304 283 266 253 243 234 227 220 215 210 206 203

~199 196 f 194 1~1 189 187 185 184 182 181 179

169 159 150 139

2522 1948 857 5~69

443 374 330 301 279 262 249 238 230 222

216 211 206 202 198 195 192 189 186 184 182 180 179 177 175 174

164 153 143 132

2533 2543 1949 1950 855 853 566 563 440 436 370 367 327 323 297 293 275 271 258 254 245 240 234 230 225 221 218 213 211 207 206 201 201 196 197 192 193 188 190 184 1~7 181 184 178

181 176 179 173 177 ~ 171

169 17~ 173 167 171 165

164170 162168

151158 147 139

125135 100122 i

ApPENDIX A Tables~c63 y

TABLE X Percentage Points of the F-distribution a = 025

f(F)

o FfJ25

- -NUMERATOR DEGREES OF FREEDOM ~ Jz 1 2 3 4 5 6 7 8 9

1 6478 7995 8642 8996 9218 9371 9482 9567 9633 2 3851 3900 3917 3925 3930 3933 3936 3937 3939 3 1744 1604 1544 1510 1488 1473 1462 1454 1447 4 1222 1065 998 960 936 920 907 898 890 5 1001 843 776 739 715 698 685 676 668 ( 881 726 660 623 599 582 570 560 552

589 7 807 654 552 529 512 499 490 482 8 757 606 542 505 482 465 453 443 436 9 721 571 508 472 448 432 420 410 403

gt1 10 - 694 483 447 424 407 395 385 378-4~~i 0 003f2~~ 11 672 463 428 404 88 376 366 359fIl

12- 655 510 447 412 389 373 351 344~61f l~ 13 641 497 435 400 377 360 348 339 331

0 ~ 14 630 486 424 389 366 329338-~$P 321 IIl 15 620 477 A15 380 358 34-1 329 320 312fIl

16 61_2 469 408 373 350 334 322 305 31~r-~ ~ 17 462- 4016Cl1) 366 344 328 316 298I j r~ 30(hmiddotfIl 45618 598 395 361 338 322 310 301 293~

592 451 390 356 333= 19 317 305 296 288

~ 20 587 446 386 351 329 313 301 291 284 21 583 442 382 348 325 309 297 287 280 22 - 579 438 378 344 322 305 293 284 276~

0 23 575 435 375 341 318 302 290 281 273 ~ 24 572 432 372 338 315 299 287 278 270 ~ 25 569 429 369 335 313 297 285 275 268

26 566 427 367 333 310 294 282 273 265 27 563 424 365 331 - 292 308 280 271 263 28 561 422 363 329 306 290 278 269 261 29 559 420 361 327 304 288 276 267 259 30 557 418 359 325 303 287 275 265 257

40 542 405 346 290 313 274 262 253 245 60 529 393 334 301 279 263 251 241 233

120 515 380 323 289 267 252 239 230 222 00 502 369 312 279 257 241 229 219 211

SourceFrom M Merrington and C M Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 1943337388

(continued)

4 804 Appendix A Tables

TABLE X Continued

~Vl NUMERATOR DEGREES OF FREEDOM V2~r---l-0--r--U--~-~~--~~20-----~~---~------~~middotmiddot----6-0-----U~O-----~---

9686 9767 9849 9931 9972 1001 1006 _1010 1~014-1 1018 2 3940 3941 3943 3945 3946 3946 3947 3948 394~ 3950 3 1442 1434 1425 1417 1412 1408 1404 1399 1395 1390

884 875 866 856 851 846 841 836 8314 826 662 652 643 633 628 623 618 612 6075 602 546 537 527 517 512 507 501 496 4906 485

7 476 467 457 447 442 436 431 425 420 414 8 430 420 410 400 395 389 384 378 373 367 9 396 387 377 367 361 356 351 345 339 333

372 362 352 342 337 331 326 320 314 308S 10 353 343 333 323 317 312 306 300 294Q 11 288 337 328 318 307 302 296 291 285 279 272 325 315 305 295 289 284 278 272 266 260E ~

)i 14 315 305 295 284 279 273 267 261 255 249 o 15 306 296 286 276 270 264 259 252 246 240rIl ~ 16 232299 289 279 268 263 257 251 245 238 ~ ~17 292 282 272 262 256 250 244 238 232 225

287 277 267 256 250 244 238 232 226~ 18 219 Q 19 282 272 262 251 245 239 233 227 220 21~=

277 268 257 246 241 235 229 222 216 209~ 20middot 273 bull =I~~~ 253 242 237 231 ~ 225 218 211lt 21 204 270 middot260 250 239 233 227 21 214 208 200~ 22 267 257 247 236 230 224 $ 218 211 204 197o 23

z 24 264 254 244 D3 227 221 ~ Ii 215 208 201 194 261 251 241 230 224 218 212 205 198 191~ 25

26 259 249 239 228 222 216 209 203 )95 I 188 27 257 247 236 225 219 213 207 200 i~~3 185 28 2d5tf i 245middot 234 223217 211 205 198 191 183 29 253 243 232 221 215 209 203 196 189 181 30 251 241 231 220 214 207 201 194 187 179

40 164239 229 218 207 201 194 188 180 172 60 227 217 206 194 188 182 174 167 158 148

120 216 205 194 182 176 169 161 153 143 131 I

QO 100205 194 I 183 171 I 164 157 148 139 127

Page 5: STA220_Final_2012S.pdf

Full Name Student Number

[10 marks - 1 each blanksentence] 4 Religion Studies researchers were interested in making inferences about the true proportion of

Canadians who believe in some fonnmiddotofGod They randomly surveyed 100 Canadians and asked them whether or not they believed in God (Yes or No) Here is the Minitab output

Test and CI for One Proportion

Test of p = 05 vs p not = 05

Event = Yes

Sample X N Sample p 95 CI Z-Value P-Value 1 61 100 0610000 (0514403 0705597) 220 0028

Use the output to fill in the blanks

(a) people in the survey said they believed in God

(b) As given in the output the null hypothesis is and the alternative hypothesis is [use proper notation and symbols]

(c) The value of the test statistic is ________

(d) If Ho is true the distribution of the test statistic is _______

(e) The 95 confidence interval for the true proportion of Canadians Who do NOT believe in some fonn of God is _____________

(f) Using a confidence interval decide on whether or not you agree with this claim More than 34 ofCanadians believe in some form ofGod Fill in agree or disagree _______

(g) If the researchers are interested in answering the question Do the majority ofCanadians believe in some form ofGoel the appropriate p-value is At the 5 significance level the conclusion would be [reject or FTR Ho ]

(b) Give a one sentence conclusion in plain English to answer the question of interest from (g)

Page 5 of 12

Full Name Student Number

[20 marks] 5 An intersection in Mississauga has an average of 6 accidents per month

Before answering each part label the random variable ofinterest and identify its distribution with correct parameters Show your work and circle your final answers

[3 m] (a) What is the probability that there are exactly 2 accidents in September at this intersection

[3mr (b) What is the probability that in the next year 3 months will have exactly 2 accidents in them

each

[4 m] (c) Approximate the probability that in the next two years at least 10 months will have 2 accidents

in them each

Page 6 of 12

Full Name Student Number

[2 m] (d) On average how long would you have to wait for an accident to occur at this intersection

(Include the units in your answer)

[3 m] (e) An accident just occurred today What is the probability that the next accident is more than 12

a month from today

[5 m] (f) You take a random sample of 100 months and record the number of accidents that occur What

is the probability that the average number exceeds 6S accidents

Page 7 of 12

Full Name Student Number

lOmarksj 6 A new toothpaste has been put on the market that claims to reduce the number of teeth that have

problems (cavities sensitive teeth etc) In order to test the effect ofthe new toothpaste researchers randomly select participants and randomly assign each person to use either the new toothpaste or a standard toothpaste (a popular brand that has been used for several years) After 6 months of use each participant reports the number of teeth they have with problems It can be assumed that both populations are Normal Partial Minitab output is given

DESCRIPTIVE STATISTICS standard DESCRIPTIVE STATISTICS new

Variable N Mean StDev SE MeanVariable N Mean StDev SE Mean new 20 330 213 048standard 15 370 198 051

Test for Equal Variances

95 Bonferroni confidence intervals for standard deviations

Sample N Lower StDev Upper standard 15 138964 1 98 335600 new 20 156050 213 330000

F-Test (normal distribution) Test statistic = 086 p-value = 079

[2 m)

(a) Using a=O05 test for the equality of the variances of the standard and new toothpaste populations Include all steps requiredfor a Hypothesis Test (you may copy any relevant parts from the output without calculating them yourself)

Page 8 of 12

Full Name Student Number

[5 m] (b) Which type of hypothesis test would you use to test if the new toothpaste is effective as

claimed by the manufacturer using a005 Why Test the relevant hypothesis from scratch Include all steps requiredfor a Hypothesis Test

[3 m] (c) The manufacturer of the new toothpaste now decides to collect data on consumers ofthe new

toothpaste separately in order to improve their study How large of sample is required to estimate the mean number of teeth with problems within 05 and retain 95 confidence

Page 9 of 12

Full Name Student Number

flOmarksJ 7 Is insomnia related to education Researchers conducted a study to determine if insomnia is related to the education of Tennessee residents 575 adults in Tennessee were selected to participate in the study using a random-digit telephone procedure Thenumber of years of education and insomnia status (normal sleeper or chronic insomniac) were recorded for each participant The researchers discovered that the fewer the years of education the more likely the person is to have chronic insomnia

~Insomnia is a sleep disorder (problems falling asleepstaying asleep throughout the night) In a random-digit telephone procedure telephone numbers are generated at random and the people with those phone numbers are contacted to participate in the study

Fill in the blanks or give a one-sentence brief answer where appropriate to following

[1 m] (a) Identify the population of interest

[1 m] (b) Identify the sample

[4 m] (c) List the variables measured and whether they are categorical or quantitative

[1 m] (d) What inference(s) did the researchers make

[1 m] (e) In general conclusions about a sample can be generalized to the population if the sample is

___________ of the population

[2 m] (f) In this case do the selected participants form a random sample Explain why or why not

Page 10 of 12

Full Name Student Number

[10marksj 8 You and your friend playa dice-rolling game You each roll a fair die once and observe the

number that appears on the up face for each person Ifboth ofyou get the same number nobody wins or loses any money If you get different numbers whoever gets the lower number has to pay the other person $1 Let X represent your winnings (in $) from this game

[4 m] (a) Find the probability distribution for X and display it in table form

[3 m] (b) Calculate the mean ofX and interpret your answer in terms of the game Show your

calculations and then circle your final answer

[3 m] (c) Compute the variance ofX Show your work and circle thefinal answer

Page 11 of 12

Full Name Student Number

10 marks 9 A public school is considering adding to the curriculum a course that it believes will improve the reading comprehension skills of its students The school has decided to conduct an experiment to evaluate the course before it is offered to the general student body Six students are randomly selected and their reading scores are recorded before and after the course These scores are composite measurements ofboth speed and comprehension (where higher scores indicate higher

h )Th I dblcompre enSlOn e resu ts are summanze e ow Student Number Score After Course Score Before Course

1 221 211 2 231 216 3 203 191 4 216 224 5 207 201 6 203 178

The school will implement the new course if there is sufficient evidence at the 5 significance level that the course improves reading comprehension

[2 m) (a) Which type of hypothesis test is appropriate for this situation and why

[6m (b) Conduct the hypothesis test of interest Include all steps requiredfor a Hypothesis Test

[2 m (c) List the assumption(s) required to use this type ofhypothesis test

Page 12 of 12 --END OF EXAM--shy

794 Appendix A Tables

TABLE IV Normal Curve Areas

o 1

00 01 bull02z 03 OS04 06 07 OS 09

0 0000 0040 0080 0120 01990160 0239 0279 0319 03591 0398 0438 0478 0517 05960557 0636 0675 0714 0753

07932 0832 0871 0910 0948 0987 1026 1064 1103 1141 3 1179 1217 1255 1293 13681331 1406 1443 1480 1517

15544 1591 1628 1664 1700 1736 1772 1808 1844 1879

1915oS 1950 1985 2019 2054 2088 2123 2157 2190 2224

22576 2291 2324 2357 24222389 2454 2486 2517 2549

25807 2611 2642 2673 2704 2734 2764 2794 2823 2852

28818 2910 2939 2967 2995 3023 3051 3078 3106 3133 3212 9 y3159 3186 3238 3264 3289 3315 3340 3365 3389

10 3413 3438 3461 3485 35313508 3554 3577 3599 3621 372911 3643 3665 3686 3708 3749 3770 3790 - 3810 3830

3849 386912 3888 3907 3925 3944 3962 3980 3997 4015

4032 4049 13 bulliQ6timiddot 4082 41154099 4131 4147 4162 4177 42071~m224192L4 4236 4251 4265 43194179 4292 4306

4332 4345 4357 4370 4382 4394middot 4406 4418 4429lS 4441 ltIJ

4452L6 4463 4474 4484 4495 4505 4515 4525 4535 4345

4554L7 4564 4573 4582 4591 4599 4608 4616 4625 4633

464118 4656 46644P49 4671 4678 4686 4693 47064699

4713 4761 L9 4719 4732 ~ 47674738 4744 4750 4726 4756 478820 4772 4793 4798 4893 4808 4812 4817~11amp 4783 shy

4821U 4826 4830 4834 48424838 48574846 4850 4854 486122 4864 4868 48874871 4875 4878 48904881 4884

23 4893 4896 4898 49164901 4913 4906 4909 49114904 24 4918 4922492Q 49364925shy 49294927 49344931 4932

4943493825 49524940 4941 4945 4946 49514948 4949 495326 4955 49644956 4957 4959 4960 4961 49634962 4965 497427 49674966 4968 4969 4970 49734971 4972 4974 498128 4975 4976 4977 4977 4978 49804979 4979

29 49864981 4982 4982 4983 4984 4984 49864985~49B5 30 4987 4987middot4987 4988 4988 4989 4990 49904989 4989 i

Source Abridged from Table I ofA Hald Statistical Table$ and Fonnulas (New YorkWiley) 1952 Reproduced by permission of A Hald

796 -~~ - -

o ~~~-TABLE VI Critical Values of t

f(t)

bull

Degrees of -Freedom tJWtlOO t050 tOO51010 tooi tooos

1 63143078 12706 31821 63657 31831 63662 2 1886 2920 4303 6965 9925 22326 31598 3 4541 1638 2353 3-182 5841 10213 12924 4 1533 2132 2776 3747 4604 7173 8610 5 1476 2015 2571 3365 4032 5893 6869 6 19431440 2447 3143 3707 5208 5959 7 1415 23651895 2998 3499 4785 5408

1397 1860 2306i 2896 3355 4501 5041 9 1383 1833 2262 2821 3250 4297 4781

10 18121372 2228 2764 3169 _ 4144 4587 U 1363 1796 2201 2718 3106 4025 4431 12 1356 1782 2179 shy 2681 3055 3930 4318

13~~ ~_ ~-i77113 i1reg 2650 3012 3852 4221 14 1345 21451761 2024 ~ 2977 41403787 15 1341 1753 2131 2602 29fl7 4073 16

373~ 1337 1746 2120 - 29212583 3686 4015

17 1333 1740 2110 2567 -2898 3646 3965 18 1330 1734 2101 t28782552 3610 3922 19 1328 t729 2093 28612539 38833579 - i

li~r~5I - 20 1725 2086 2528 2845 3850 21

3552 t 1323 2080 2518 1721 2831 3527 3819

22 20741321 1717 2508 2819 3505 3~792 23 1319 1714 2069 2500 3767 24

2807 3485 1318 1711 2064 2492 3467 3745

25 2middot797

1316 1708 2060 2485 3725 26

2787 3450 1315 1706 2056 2479 2779 3435 3707

27 1314 1703 2052 2473 2771 36903421 28 1313 170f 1 2048 2763 3674 29

~467 3408 1311 1699 2045 - 2462 2756 36593396

30 1310 1697 2D42 2457 - 2750 3646 40

3385 1303 1684 2021 2423 2704 3551

60 3307

1296 -16n 2000 2390 2660 3460 120

3232 1289 1658 1980 2358 2617 33733160

00 1645 1282 1960 2326 2576 3090 3291

Source 1his table is reproduced with the lcind permtsSioil of~e frustees of Biometrika from E S Pearson and R O Hartley (eds) The Biometrika Tables for Statisticians Vol 1 3d ed Biometrika 1966

r7~~

c

~ ApPENDIX A Tables 801

i~ 1lt TABLE IX Percentage Points of the F-distribution a = 05

~ liZ 1 2

1 1614 1995 2 1851 1900 3 1013 955 4 771 694

II

NUMERATOR DEGREES OF FREEDOM I11

3 4 9 I5 6 7 8

5

2157 2246 2302 2340 1916 1925 1930 1933

928 912 901 894 659 639 626 616

~ 661 579 541 519~ 505 495 6 599 514 476 453 439 428~7 559 474 435 412 397 387

5328 446 407 384 369 358 9 512 426 386 363 348 337~

10 -496 4100 371 348 333 322Q 11 484 7f~i 359398--~ 336 320 309

34938~middot12middot 475 326 300 ~~311~ 13 381467 341 318 303 292 0 fi

14 460 374 334 311 296 285 ~ 15 454 368 329 306 290foil 279 rr

449 363~ 16 324 301 285 274

~ 17 445 )20 296 281359 270 foil 18 441 3-16 293li))~5 277 266Q

438 352middot 313 290= 19 274 263 0 20 435 349 310 287 271 260 ~ 21 432 347 257307 284 268

i 22 430 344 305 282 266 255 303 0 23 428 342 280 264 253

Z 24 426 340 301 278 262 251foil Q 424middot25 339 299 276 260 249

26 423 337 298 274 259 247 27 421 335 296 273 257 246 28 420 334 295 271 245 256 29 418 333 293I 270 255 243 30 417 332 292 269 253 242 40 408 323 284 261 245 234 60 40q 315 276 253 237 225

120 307 392 268 245 229 217 00 384 300 260 237 221 210

2368 2389 1935 1937 889 885 609 604 488 482 421 415 379 373 350 344 329 323 314 307 301 295 291 285

283 277 271 1276 270 265 271 264 259 i266 259 254 1rII261 255 249

251 258 ~i-middot246 f 254 248 242 I

I 251 245 239 249 242 237 246 240 234 244 237 232 242 236 230 240 234 228 239 232 2]7 237 231 225 236 229 224 235 228 222 233 227 221

225 218 212 I

1217 210 204 209 202 196 201 194 188

2405 1938 881 600 477 410 368 339 318 302 290 280 ji

Source From M Merrington and eM Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 19433373-88

i (continued) I

I

I iii 11 iii

II I

I l

iINi II

802 Appendix A Tables

TABLE IX Continued

~1 NUMERATOR DEGREES OF FREEDOM Vl ~r--l-0----12----15-----r--2-0----24---r--30----40----60~--~- ~ OQ

1 2 3 4 S 6 7 8 9

~ 10 11

~ ~ ~ 14 o 15 rIJ (

Iiil 16 ~ 17 ~ 18 ~ 19 = 20

~ 21

i ii shy~ 24 ~ 2S

26 27 28 29 30

middot40 60

120 OQ

2419 1940 879 596 474 406 364 335 ~314

298 285 275 267 260 254 249 245 241 238 235 232 230 227 225 224 222 220 219 218 216

208 199 191 1~

2439 2459 2480 2491 1941 1943 1945 1945 874 870 866 864 591 586 580 577 468 462 456 453 400 394 387 384 357 351 344 341 328 322 315 312 307 301 294 290 291 285 277 274 279 272 265 261 269 262 254 251 260 253 246 242 253 246 239 235 248 240 233 229 242 235 228 224 238 231 223 219 234 227 219 215 231 223 216 211 iJi ~r~ 220 212 208 225 218 210 205 223 215 207 203 220 213 205 201 218 211 203 198 216 209 201 196

215 2Q7 199 195j 2i3 206 197 193 212 204 196 191 210 203 194 190 209 201 193 189

200 192184 179 192 184 175 170 183 i 175 166 161 U5 1m 1~ 1~

2501 1946 862 575 450 381 33S 308 286 270 257 247

238 231 225 219 215 211 207 204 201 198 196 194 192 190 188 187 185 184

174 165 155 146

2511 Imiddotmiddotmiddot

1947 859 572 446 377middot 334 304 283 266 253 243 234 227 220 215 210 206 203

~199 196 f 194 1~1 189 187 185 184 182 181 179

169 159 150 139

2522 1948 857 5~69

443 374 330 301 279 262 249 238 230 222

216 211 206 202 198 195 192 189 186 184 182 180 179 177 175 174

164 153 143 132

2533 2543 1949 1950 855 853 566 563 440 436 370 367 327 323 297 293 275 271 258 254 245 240 234 230 225 221 218 213 211 207 206 201 201 196 197 192 193 188 190 184 1~7 181 184 178

181 176 179 173 177 ~ 171

169 17~ 173 167 171 165

164170 162168

151158 147 139

125135 100122 i

ApPENDIX A Tables~c63 y

TABLE X Percentage Points of the F-distribution a = 025

f(F)

o FfJ25

- -NUMERATOR DEGREES OF FREEDOM ~ Jz 1 2 3 4 5 6 7 8 9

1 6478 7995 8642 8996 9218 9371 9482 9567 9633 2 3851 3900 3917 3925 3930 3933 3936 3937 3939 3 1744 1604 1544 1510 1488 1473 1462 1454 1447 4 1222 1065 998 960 936 920 907 898 890 5 1001 843 776 739 715 698 685 676 668 ( 881 726 660 623 599 582 570 560 552

589 7 807 654 552 529 512 499 490 482 8 757 606 542 505 482 465 453 443 436 9 721 571 508 472 448 432 420 410 403

gt1 10 - 694 483 447 424 407 395 385 378-4~~i 0 003f2~~ 11 672 463 428 404 88 376 366 359fIl

12- 655 510 447 412 389 373 351 344~61f l~ 13 641 497 435 400 377 360 348 339 331

0 ~ 14 630 486 424 389 366 329338-~$P 321 IIl 15 620 477 A15 380 358 34-1 329 320 312fIl

16 61_2 469 408 373 350 334 322 305 31~r-~ ~ 17 462- 4016Cl1) 366 344 328 316 298I j r~ 30(hmiddotfIl 45618 598 395 361 338 322 310 301 293~

592 451 390 356 333= 19 317 305 296 288

~ 20 587 446 386 351 329 313 301 291 284 21 583 442 382 348 325 309 297 287 280 22 - 579 438 378 344 322 305 293 284 276~

0 23 575 435 375 341 318 302 290 281 273 ~ 24 572 432 372 338 315 299 287 278 270 ~ 25 569 429 369 335 313 297 285 275 268

26 566 427 367 333 310 294 282 273 265 27 563 424 365 331 - 292 308 280 271 263 28 561 422 363 329 306 290 278 269 261 29 559 420 361 327 304 288 276 267 259 30 557 418 359 325 303 287 275 265 257

40 542 405 346 290 313 274 262 253 245 60 529 393 334 301 279 263 251 241 233

120 515 380 323 289 267 252 239 230 222 00 502 369 312 279 257 241 229 219 211

SourceFrom M Merrington and C M Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 1943337388

(continued)

4 804 Appendix A Tables

TABLE X Continued

~Vl NUMERATOR DEGREES OF FREEDOM V2~r---l-0--r--U--~-~~--~~20-----~~---~------~~middotmiddot----6-0-----U~O-----~---

9686 9767 9849 9931 9972 1001 1006 _1010 1~014-1 1018 2 3940 3941 3943 3945 3946 3946 3947 3948 394~ 3950 3 1442 1434 1425 1417 1412 1408 1404 1399 1395 1390

884 875 866 856 851 846 841 836 8314 826 662 652 643 633 628 623 618 612 6075 602 546 537 527 517 512 507 501 496 4906 485

7 476 467 457 447 442 436 431 425 420 414 8 430 420 410 400 395 389 384 378 373 367 9 396 387 377 367 361 356 351 345 339 333

372 362 352 342 337 331 326 320 314 308S 10 353 343 333 323 317 312 306 300 294Q 11 288 337 328 318 307 302 296 291 285 279 272 325 315 305 295 289 284 278 272 266 260E ~

)i 14 315 305 295 284 279 273 267 261 255 249 o 15 306 296 286 276 270 264 259 252 246 240rIl ~ 16 232299 289 279 268 263 257 251 245 238 ~ ~17 292 282 272 262 256 250 244 238 232 225

287 277 267 256 250 244 238 232 226~ 18 219 Q 19 282 272 262 251 245 239 233 227 220 21~=

277 268 257 246 241 235 229 222 216 209~ 20middot 273 bull =I~~~ 253 242 237 231 ~ 225 218 211lt 21 204 270 middot260 250 239 233 227 21 214 208 200~ 22 267 257 247 236 230 224 $ 218 211 204 197o 23

z 24 264 254 244 D3 227 221 ~ Ii 215 208 201 194 261 251 241 230 224 218 212 205 198 191~ 25

26 259 249 239 228 222 216 209 203 )95 I 188 27 257 247 236 225 219 213 207 200 i~~3 185 28 2d5tf i 245middot 234 223217 211 205 198 191 183 29 253 243 232 221 215 209 203 196 189 181 30 251 241 231 220 214 207 201 194 187 179

40 164239 229 218 207 201 194 188 180 172 60 227 217 206 194 188 182 174 167 158 148

120 216 205 194 182 176 169 161 153 143 131 I

QO 100205 194 I 183 171 I 164 157 148 139 127

Page 6: STA220_Final_2012S.pdf

Full Name Student Number

[20 marks] 5 An intersection in Mississauga has an average of 6 accidents per month

Before answering each part label the random variable ofinterest and identify its distribution with correct parameters Show your work and circle your final answers

[3 m] (a) What is the probability that there are exactly 2 accidents in September at this intersection

[3mr (b) What is the probability that in the next year 3 months will have exactly 2 accidents in them

each

[4 m] (c) Approximate the probability that in the next two years at least 10 months will have 2 accidents

in them each

Page 6 of 12

Full Name Student Number

[2 m] (d) On average how long would you have to wait for an accident to occur at this intersection

(Include the units in your answer)

[3 m] (e) An accident just occurred today What is the probability that the next accident is more than 12

a month from today

[5 m] (f) You take a random sample of 100 months and record the number of accidents that occur What

is the probability that the average number exceeds 6S accidents

Page 7 of 12

Full Name Student Number

lOmarksj 6 A new toothpaste has been put on the market that claims to reduce the number of teeth that have

problems (cavities sensitive teeth etc) In order to test the effect ofthe new toothpaste researchers randomly select participants and randomly assign each person to use either the new toothpaste or a standard toothpaste (a popular brand that has been used for several years) After 6 months of use each participant reports the number of teeth they have with problems It can be assumed that both populations are Normal Partial Minitab output is given

DESCRIPTIVE STATISTICS standard DESCRIPTIVE STATISTICS new

Variable N Mean StDev SE MeanVariable N Mean StDev SE Mean new 20 330 213 048standard 15 370 198 051

Test for Equal Variances

95 Bonferroni confidence intervals for standard deviations

Sample N Lower StDev Upper standard 15 138964 1 98 335600 new 20 156050 213 330000

F-Test (normal distribution) Test statistic = 086 p-value = 079

[2 m)

(a) Using a=O05 test for the equality of the variances of the standard and new toothpaste populations Include all steps requiredfor a Hypothesis Test (you may copy any relevant parts from the output without calculating them yourself)

Page 8 of 12

Full Name Student Number

[5 m] (b) Which type of hypothesis test would you use to test if the new toothpaste is effective as

claimed by the manufacturer using a005 Why Test the relevant hypothesis from scratch Include all steps requiredfor a Hypothesis Test

[3 m] (c) The manufacturer of the new toothpaste now decides to collect data on consumers ofthe new

toothpaste separately in order to improve their study How large of sample is required to estimate the mean number of teeth with problems within 05 and retain 95 confidence

Page 9 of 12

Full Name Student Number

flOmarksJ 7 Is insomnia related to education Researchers conducted a study to determine if insomnia is related to the education of Tennessee residents 575 adults in Tennessee were selected to participate in the study using a random-digit telephone procedure Thenumber of years of education and insomnia status (normal sleeper or chronic insomniac) were recorded for each participant The researchers discovered that the fewer the years of education the more likely the person is to have chronic insomnia

~Insomnia is a sleep disorder (problems falling asleepstaying asleep throughout the night) In a random-digit telephone procedure telephone numbers are generated at random and the people with those phone numbers are contacted to participate in the study

Fill in the blanks or give a one-sentence brief answer where appropriate to following

[1 m] (a) Identify the population of interest

[1 m] (b) Identify the sample

[4 m] (c) List the variables measured and whether they are categorical or quantitative

[1 m] (d) What inference(s) did the researchers make

[1 m] (e) In general conclusions about a sample can be generalized to the population if the sample is

___________ of the population

[2 m] (f) In this case do the selected participants form a random sample Explain why or why not

Page 10 of 12

Full Name Student Number

[10marksj 8 You and your friend playa dice-rolling game You each roll a fair die once and observe the

number that appears on the up face for each person Ifboth ofyou get the same number nobody wins or loses any money If you get different numbers whoever gets the lower number has to pay the other person $1 Let X represent your winnings (in $) from this game

[4 m] (a) Find the probability distribution for X and display it in table form

[3 m] (b) Calculate the mean ofX and interpret your answer in terms of the game Show your

calculations and then circle your final answer

[3 m] (c) Compute the variance ofX Show your work and circle thefinal answer

Page 11 of 12

Full Name Student Number

10 marks 9 A public school is considering adding to the curriculum a course that it believes will improve the reading comprehension skills of its students The school has decided to conduct an experiment to evaluate the course before it is offered to the general student body Six students are randomly selected and their reading scores are recorded before and after the course These scores are composite measurements ofboth speed and comprehension (where higher scores indicate higher

h )Th I dblcompre enSlOn e resu ts are summanze e ow Student Number Score After Course Score Before Course

1 221 211 2 231 216 3 203 191 4 216 224 5 207 201 6 203 178

The school will implement the new course if there is sufficient evidence at the 5 significance level that the course improves reading comprehension

[2 m) (a) Which type of hypothesis test is appropriate for this situation and why

[6m (b) Conduct the hypothesis test of interest Include all steps requiredfor a Hypothesis Test

[2 m (c) List the assumption(s) required to use this type ofhypothesis test

Page 12 of 12 --END OF EXAM--shy

794 Appendix A Tables

TABLE IV Normal Curve Areas

o 1

00 01 bull02z 03 OS04 06 07 OS 09

0 0000 0040 0080 0120 01990160 0239 0279 0319 03591 0398 0438 0478 0517 05960557 0636 0675 0714 0753

07932 0832 0871 0910 0948 0987 1026 1064 1103 1141 3 1179 1217 1255 1293 13681331 1406 1443 1480 1517

15544 1591 1628 1664 1700 1736 1772 1808 1844 1879

1915oS 1950 1985 2019 2054 2088 2123 2157 2190 2224

22576 2291 2324 2357 24222389 2454 2486 2517 2549

25807 2611 2642 2673 2704 2734 2764 2794 2823 2852

28818 2910 2939 2967 2995 3023 3051 3078 3106 3133 3212 9 y3159 3186 3238 3264 3289 3315 3340 3365 3389

10 3413 3438 3461 3485 35313508 3554 3577 3599 3621 372911 3643 3665 3686 3708 3749 3770 3790 - 3810 3830

3849 386912 3888 3907 3925 3944 3962 3980 3997 4015

4032 4049 13 bulliQ6timiddot 4082 41154099 4131 4147 4162 4177 42071~m224192L4 4236 4251 4265 43194179 4292 4306

4332 4345 4357 4370 4382 4394middot 4406 4418 4429lS 4441 ltIJ

4452L6 4463 4474 4484 4495 4505 4515 4525 4535 4345

4554L7 4564 4573 4582 4591 4599 4608 4616 4625 4633

464118 4656 46644P49 4671 4678 4686 4693 47064699

4713 4761 L9 4719 4732 ~ 47674738 4744 4750 4726 4756 478820 4772 4793 4798 4893 4808 4812 4817~11amp 4783 shy

4821U 4826 4830 4834 48424838 48574846 4850 4854 486122 4864 4868 48874871 4875 4878 48904881 4884

23 4893 4896 4898 49164901 4913 4906 4909 49114904 24 4918 4922492Q 49364925shy 49294927 49344931 4932

4943493825 49524940 4941 4945 4946 49514948 4949 495326 4955 49644956 4957 4959 4960 4961 49634962 4965 497427 49674966 4968 4969 4970 49734971 4972 4974 498128 4975 4976 4977 4977 4978 49804979 4979

29 49864981 4982 4982 4983 4984 4984 49864985~49B5 30 4987 4987middot4987 4988 4988 4989 4990 49904989 4989 i

Source Abridged from Table I ofA Hald Statistical Table$ and Fonnulas (New YorkWiley) 1952 Reproduced by permission of A Hald

796 -~~ - -

o ~~~-TABLE VI Critical Values of t

f(t)

bull

Degrees of -Freedom tJWtlOO t050 tOO51010 tooi tooos

1 63143078 12706 31821 63657 31831 63662 2 1886 2920 4303 6965 9925 22326 31598 3 4541 1638 2353 3-182 5841 10213 12924 4 1533 2132 2776 3747 4604 7173 8610 5 1476 2015 2571 3365 4032 5893 6869 6 19431440 2447 3143 3707 5208 5959 7 1415 23651895 2998 3499 4785 5408

1397 1860 2306i 2896 3355 4501 5041 9 1383 1833 2262 2821 3250 4297 4781

10 18121372 2228 2764 3169 _ 4144 4587 U 1363 1796 2201 2718 3106 4025 4431 12 1356 1782 2179 shy 2681 3055 3930 4318

13~~ ~_ ~-i77113 i1reg 2650 3012 3852 4221 14 1345 21451761 2024 ~ 2977 41403787 15 1341 1753 2131 2602 29fl7 4073 16

373~ 1337 1746 2120 - 29212583 3686 4015

17 1333 1740 2110 2567 -2898 3646 3965 18 1330 1734 2101 t28782552 3610 3922 19 1328 t729 2093 28612539 38833579 - i

li~r~5I - 20 1725 2086 2528 2845 3850 21

3552 t 1323 2080 2518 1721 2831 3527 3819

22 20741321 1717 2508 2819 3505 3~792 23 1319 1714 2069 2500 3767 24

2807 3485 1318 1711 2064 2492 3467 3745

25 2middot797

1316 1708 2060 2485 3725 26

2787 3450 1315 1706 2056 2479 2779 3435 3707

27 1314 1703 2052 2473 2771 36903421 28 1313 170f 1 2048 2763 3674 29

~467 3408 1311 1699 2045 - 2462 2756 36593396

30 1310 1697 2D42 2457 - 2750 3646 40

3385 1303 1684 2021 2423 2704 3551

60 3307

1296 -16n 2000 2390 2660 3460 120

3232 1289 1658 1980 2358 2617 33733160

00 1645 1282 1960 2326 2576 3090 3291

Source 1his table is reproduced with the lcind permtsSioil of~e frustees of Biometrika from E S Pearson and R O Hartley (eds) The Biometrika Tables for Statisticians Vol 1 3d ed Biometrika 1966

r7~~

c

~ ApPENDIX A Tables 801

i~ 1lt TABLE IX Percentage Points of the F-distribution a = 05

~ liZ 1 2

1 1614 1995 2 1851 1900 3 1013 955 4 771 694

II

NUMERATOR DEGREES OF FREEDOM I11

3 4 9 I5 6 7 8

5

2157 2246 2302 2340 1916 1925 1930 1933

928 912 901 894 659 639 626 616

~ 661 579 541 519~ 505 495 6 599 514 476 453 439 428~7 559 474 435 412 397 387

5328 446 407 384 369 358 9 512 426 386 363 348 337~

10 -496 4100 371 348 333 322Q 11 484 7f~i 359398--~ 336 320 309

34938~middot12middot 475 326 300 ~~311~ 13 381467 341 318 303 292 0 fi

14 460 374 334 311 296 285 ~ 15 454 368 329 306 290foil 279 rr

449 363~ 16 324 301 285 274

~ 17 445 )20 296 281359 270 foil 18 441 3-16 293li))~5 277 266Q

438 352middot 313 290= 19 274 263 0 20 435 349 310 287 271 260 ~ 21 432 347 257307 284 268

i 22 430 344 305 282 266 255 303 0 23 428 342 280 264 253

Z 24 426 340 301 278 262 251foil Q 424middot25 339 299 276 260 249

26 423 337 298 274 259 247 27 421 335 296 273 257 246 28 420 334 295 271 245 256 29 418 333 293I 270 255 243 30 417 332 292 269 253 242 40 408 323 284 261 245 234 60 40q 315 276 253 237 225

120 307 392 268 245 229 217 00 384 300 260 237 221 210

2368 2389 1935 1937 889 885 609 604 488 482 421 415 379 373 350 344 329 323 314 307 301 295 291 285

283 277 271 1276 270 265 271 264 259 i266 259 254 1rII261 255 249

251 258 ~i-middot246 f 254 248 242 I

I 251 245 239 249 242 237 246 240 234 244 237 232 242 236 230 240 234 228 239 232 2]7 237 231 225 236 229 224 235 228 222 233 227 221

225 218 212 I

1217 210 204 209 202 196 201 194 188

2405 1938 881 600 477 410 368 339 318 302 290 280 ji

Source From M Merrington and eM Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 19433373-88

i (continued) I

I

I iii 11 iii

II I

I l

iINi II

802 Appendix A Tables

TABLE IX Continued

~1 NUMERATOR DEGREES OF FREEDOM Vl ~r--l-0----12----15-----r--2-0----24---r--30----40----60~--~- ~ OQ

1 2 3 4 S 6 7 8 9

~ 10 11

~ ~ ~ 14 o 15 rIJ (

Iiil 16 ~ 17 ~ 18 ~ 19 = 20

~ 21

i ii shy~ 24 ~ 2S

26 27 28 29 30

middot40 60

120 OQ

2419 1940 879 596 474 406 364 335 ~314

298 285 275 267 260 254 249 245 241 238 235 232 230 227 225 224 222 220 219 218 216

208 199 191 1~

2439 2459 2480 2491 1941 1943 1945 1945 874 870 866 864 591 586 580 577 468 462 456 453 400 394 387 384 357 351 344 341 328 322 315 312 307 301 294 290 291 285 277 274 279 272 265 261 269 262 254 251 260 253 246 242 253 246 239 235 248 240 233 229 242 235 228 224 238 231 223 219 234 227 219 215 231 223 216 211 iJi ~r~ 220 212 208 225 218 210 205 223 215 207 203 220 213 205 201 218 211 203 198 216 209 201 196

215 2Q7 199 195j 2i3 206 197 193 212 204 196 191 210 203 194 190 209 201 193 189

200 192184 179 192 184 175 170 183 i 175 166 161 U5 1m 1~ 1~

2501 1946 862 575 450 381 33S 308 286 270 257 247

238 231 225 219 215 211 207 204 201 198 196 194 192 190 188 187 185 184

174 165 155 146

2511 Imiddotmiddotmiddot

1947 859 572 446 377middot 334 304 283 266 253 243 234 227 220 215 210 206 203

~199 196 f 194 1~1 189 187 185 184 182 181 179

169 159 150 139

2522 1948 857 5~69

443 374 330 301 279 262 249 238 230 222

216 211 206 202 198 195 192 189 186 184 182 180 179 177 175 174

164 153 143 132

2533 2543 1949 1950 855 853 566 563 440 436 370 367 327 323 297 293 275 271 258 254 245 240 234 230 225 221 218 213 211 207 206 201 201 196 197 192 193 188 190 184 1~7 181 184 178

181 176 179 173 177 ~ 171

169 17~ 173 167 171 165

164170 162168

151158 147 139

125135 100122 i

ApPENDIX A Tables~c63 y

TABLE X Percentage Points of the F-distribution a = 025

f(F)

o FfJ25

- -NUMERATOR DEGREES OF FREEDOM ~ Jz 1 2 3 4 5 6 7 8 9

1 6478 7995 8642 8996 9218 9371 9482 9567 9633 2 3851 3900 3917 3925 3930 3933 3936 3937 3939 3 1744 1604 1544 1510 1488 1473 1462 1454 1447 4 1222 1065 998 960 936 920 907 898 890 5 1001 843 776 739 715 698 685 676 668 ( 881 726 660 623 599 582 570 560 552

589 7 807 654 552 529 512 499 490 482 8 757 606 542 505 482 465 453 443 436 9 721 571 508 472 448 432 420 410 403

gt1 10 - 694 483 447 424 407 395 385 378-4~~i 0 003f2~~ 11 672 463 428 404 88 376 366 359fIl

12- 655 510 447 412 389 373 351 344~61f l~ 13 641 497 435 400 377 360 348 339 331

0 ~ 14 630 486 424 389 366 329338-~$P 321 IIl 15 620 477 A15 380 358 34-1 329 320 312fIl

16 61_2 469 408 373 350 334 322 305 31~r-~ ~ 17 462- 4016Cl1) 366 344 328 316 298I j r~ 30(hmiddotfIl 45618 598 395 361 338 322 310 301 293~

592 451 390 356 333= 19 317 305 296 288

~ 20 587 446 386 351 329 313 301 291 284 21 583 442 382 348 325 309 297 287 280 22 - 579 438 378 344 322 305 293 284 276~

0 23 575 435 375 341 318 302 290 281 273 ~ 24 572 432 372 338 315 299 287 278 270 ~ 25 569 429 369 335 313 297 285 275 268

26 566 427 367 333 310 294 282 273 265 27 563 424 365 331 - 292 308 280 271 263 28 561 422 363 329 306 290 278 269 261 29 559 420 361 327 304 288 276 267 259 30 557 418 359 325 303 287 275 265 257

40 542 405 346 290 313 274 262 253 245 60 529 393 334 301 279 263 251 241 233

120 515 380 323 289 267 252 239 230 222 00 502 369 312 279 257 241 229 219 211

SourceFrom M Merrington and C M Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 1943337388

(continued)

4 804 Appendix A Tables

TABLE X Continued

~Vl NUMERATOR DEGREES OF FREEDOM V2~r---l-0--r--U--~-~~--~~20-----~~---~------~~middotmiddot----6-0-----U~O-----~---

9686 9767 9849 9931 9972 1001 1006 _1010 1~014-1 1018 2 3940 3941 3943 3945 3946 3946 3947 3948 394~ 3950 3 1442 1434 1425 1417 1412 1408 1404 1399 1395 1390

884 875 866 856 851 846 841 836 8314 826 662 652 643 633 628 623 618 612 6075 602 546 537 527 517 512 507 501 496 4906 485

7 476 467 457 447 442 436 431 425 420 414 8 430 420 410 400 395 389 384 378 373 367 9 396 387 377 367 361 356 351 345 339 333

372 362 352 342 337 331 326 320 314 308S 10 353 343 333 323 317 312 306 300 294Q 11 288 337 328 318 307 302 296 291 285 279 272 325 315 305 295 289 284 278 272 266 260E ~

)i 14 315 305 295 284 279 273 267 261 255 249 o 15 306 296 286 276 270 264 259 252 246 240rIl ~ 16 232299 289 279 268 263 257 251 245 238 ~ ~17 292 282 272 262 256 250 244 238 232 225

287 277 267 256 250 244 238 232 226~ 18 219 Q 19 282 272 262 251 245 239 233 227 220 21~=

277 268 257 246 241 235 229 222 216 209~ 20middot 273 bull =I~~~ 253 242 237 231 ~ 225 218 211lt 21 204 270 middot260 250 239 233 227 21 214 208 200~ 22 267 257 247 236 230 224 $ 218 211 204 197o 23

z 24 264 254 244 D3 227 221 ~ Ii 215 208 201 194 261 251 241 230 224 218 212 205 198 191~ 25

26 259 249 239 228 222 216 209 203 )95 I 188 27 257 247 236 225 219 213 207 200 i~~3 185 28 2d5tf i 245middot 234 223217 211 205 198 191 183 29 253 243 232 221 215 209 203 196 189 181 30 251 241 231 220 214 207 201 194 187 179

40 164239 229 218 207 201 194 188 180 172 60 227 217 206 194 188 182 174 167 158 148

120 216 205 194 182 176 169 161 153 143 131 I

QO 100205 194 I 183 171 I 164 157 148 139 127

Page 7: STA220_Final_2012S.pdf

Full Name Student Number

[2 m] (d) On average how long would you have to wait for an accident to occur at this intersection

(Include the units in your answer)

[3 m] (e) An accident just occurred today What is the probability that the next accident is more than 12

a month from today

[5 m] (f) You take a random sample of 100 months and record the number of accidents that occur What

is the probability that the average number exceeds 6S accidents

Page 7 of 12

Full Name Student Number

lOmarksj 6 A new toothpaste has been put on the market that claims to reduce the number of teeth that have

problems (cavities sensitive teeth etc) In order to test the effect ofthe new toothpaste researchers randomly select participants and randomly assign each person to use either the new toothpaste or a standard toothpaste (a popular brand that has been used for several years) After 6 months of use each participant reports the number of teeth they have with problems It can be assumed that both populations are Normal Partial Minitab output is given

DESCRIPTIVE STATISTICS standard DESCRIPTIVE STATISTICS new

Variable N Mean StDev SE MeanVariable N Mean StDev SE Mean new 20 330 213 048standard 15 370 198 051

Test for Equal Variances

95 Bonferroni confidence intervals for standard deviations

Sample N Lower StDev Upper standard 15 138964 1 98 335600 new 20 156050 213 330000

F-Test (normal distribution) Test statistic = 086 p-value = 079

[2 m)

(a) Using a=O05 test for the equality of the variances of the standard and new toothpaste populations Include all steps requiredfor a Hypothesis Test (you may copy any relevant parts from the output without calculating them yourself)

Page 8 of 12

Full Name Student Number

[5 m] (b) Which type of hypothesis test would you use to test if the new toothpaste is effective as

claimed by the manufacturer using a005 Why Test the relevant hypothesis from scratch Include all steps requiredfor a Hypothesis Test

[3 m] (c) The manufacturer of the new toothpaste now decides to collect data on consumers ofthe new

toothpaste separately in order to improve their study How large of sample is required to estimate the mean number of teeth with problems within 05 and retain 95 confidence

Page 9 of 12

Full Name Student Number

flOmarksJ 7 Is insomnia related to education Researchers conducted a study to determine if insomnia is related to the education of Tennessee residents 575 adults in Tennessee were selected to participate in the study using a random-digit telephone procedure Thenumber of years of education and insomnia status (normal sleeper or chronic insomniac) were recorded for each participant The researchers discovered that the fewer the years of education the more likely the person is to have chronic insomnia

~Insomnia is a sleep disorder (problems falling asleepstaying asleep throughout the night) In a random-digit telephone procedure telephone numbers are generated at random and the people with those phone numbers are contacted to participate in the study

Fill in the blanks or give a one-sentence brief answer where appropriate to following

[1 m] (a) Identify the population of interest

[1 m] (b) Identify the sample

[4 m] (c) List the variables measured and whether they are categorical or quantitative

[1 m] (d) What inference(s) did the researchers make

[1 m] (e) In general conclusions about a sample can be generalized to the population if the sample is

___________ of the population

[2 m] (f) In this case do the selected participants form a random sample Explain why or why not

Page 10 of 12

Full Name Student Number

[10marksj 8 You and your friend playa dice-rolling game You each roll a fair die once and observe the

number that appears on the up face for each person Ifboth ofyou get the same number nobody wins or loses any money If you get different numbers whoever gets the lower number has to pay the other person $1 Let X represent your winnings (in $) from this game

[4 m] (a) Find the probability distribution for X and display it in table form

[3 m] (b) Calculate the mean ofX and interpret your answer in terms of the game Show your

calculations and then circle your final answer

[3 m] (c) Compute the variance ofX Show your work and circle thefinal answer

Page 11 of 12

Full Name Student Number

10 marks 9 A public school is considering adding to the curriculum a course that it believes will improve the reading comprehension skills of its students The school has decided to conduct an experiment to evaluate the course before it is offered to the general student body Six students are randomly selected and their reading scores are recorded before and after the course These scores are composite measurements ofboth speed and comprehension (where higher scores indicate higher

h )Th I dblcompre enSlOn e resu ts are summanze e ow Student Number Score After Course Score Before Course

1 221 211 2 231 216 3 203 191 4 216 224 5 207 201 6 203 178

The school will implement the new course if there is sufficient evidence at the 5 significance level that the course improves reading comprehension

[2 m) (a) Which type of hypothesis test is appropriate for this situation and why

[6m (b) Conduct the hypothesis test of interest Include all steps requiredfor a Hypothesis Test

[2 m (c) List the assumption(s) required to use this type ofhypothesis test

Page 12 of 12 --END OF EXAM--shy

794 Appendix A Tables

TABLE IV Normal Curve Areas

o 1

00 01 bull02z 03 OS04 06 07 OS 09

0 0000 0040 0080 0120 01990160 0239 0279 0319 03591 0398 0438 0478 0517 05960557 0636 0675 0714 0753

07932 0832 0871 0910 0948 0987 1026 1064 1103 1141 3 1179 1217 1255 1293 13681331 1406 1443 1480 1517

15544 1591 1628 1664 1700 1736 1772 1808 1844 1879

1915oS 1950 1985 2019 2054 2088 2123 2157 2190 2224

22576 2291 2324 2357 24222389 2454 2486 2517 2549

25807 2611 2642 2673 2704 2734 2764 2794 2823 2852

28818 2910 2939 2967 2995 3023 3051 3078 3106 3133 3212 9 y3159 3186 3238 3264 3289 3315 3340 3365 3389

10 3413 3438 3461 3485 35313508 3554 3577 3599 3621 372911 3643 3665 3686 3708 3749 3770 3790 - 3810 3830

3849 386912 3888 3907 3925 3944 3962 3980 3997 4015

4032 4049 13 bulliQ6timiddot 4082 41154099 4131 4147 4162 4177 42071~m224192L4 4236 4251 4265 43194179 4292 4306

4332 4345 4357 4370 4382 4394middot 4406 4418 4429lS 4441 ltIJ

4452L6 4463 4474 4484 4495 4505 4515 4525 4535 4345

4554L7 4564 4573 4582 4591 4599 4608 4616 4625 4633

464118 4656 46644P49 4671 4678 4686 4693 47064699

4713 4761 L9 4719 4732 ~ 47674738 4744 4750 4726 4756 478820 4772 4793 4798 4893 4808 4812 4817~11amp 4783 shy

4821U 4826 4830 4834 48424838 48574846 4850 4854 486122 4864 4868 48874871 4875 4878 48904881 4884

23 4893 4896 4898 49164901 4913 4906 4909 49114904 24 4918 4922492Q 49364925shy 49294927 49344931 4932

4943493825 49524940 4941 4945 4946 49514948 4949 495326 4955 49644956 4957 4959 4960 4961 49634962 4965 497427 49674966 4968 4969 4970 49734971 4972 4974 498128 4975 4976 4977 4977 4978 49804979 4979

29 49864981 4982 4982 4983 4984 4984 49864985~49B5 30 4987 4987middot4987 4988 4988 4989 4990 49904989 4989 i

Source Abridged from Table I ofA Hald Statistical Table$ and Fonnulas (New YorkWiley) 1952 Reproduced by permission of A Hald

796 -~~ - -

o ~~~-TABLE VI Critical Values of t

f(t)

bull

Degrees of -Freedom tJWtlOO t050 tOO51010 tooi tooos

1 63143078 12706 31821 63657 31831 63662 2 1886 2920 4303 6965 9925 22326 31598 3 4541 1638 2353 3-182 5841 10213 12924 4 1533 2132 2776 3747 4604 7173 8610 5 1476 2015 2571 3365 4032 5893 6869 6 19431440 2447 3143 3707 5208 5959 7 1415 23651895 2998 3499 4785 5408

1397 1860 2306i 2896 3355 4501 5041 9 1383 1833 2262 2821 3250 4297 4781

10 18121372 2228 2764 3169 _ 4144 4587 U 1363 1796 2201 2718 3106 4025 4431 12 1356 1782 2179 shy 2681 3055 3930 4318

13~~ ~_ ~-i77113 i1reg 2650 3012 3852 4221 14 1345 21451761 2024 ~ 2977 41403787 15 1341 1753 2131 2602 29fl7 4073 16

373~ 1337 1746 2120 - 29212583 3686 4015

17 1333 1740 2110 2567 -2898 3646 3965 18 1330 1734 2101 t28782552 3610 3922 19 1328 t729 2093 28612539 38833579 - i

li~r~5I - 20 1725 2086 2528 2845 3850 21

3552 t 1323 2080 2518 1721 2831 3527 3819

22 20741321 1717 2508 2819 3505 3~792 23 1319 1714 2069 2500 3767 24

2807 3485 1318 1711 2064 2492 3467 3745

25 2middot797

1316 1708 2060 2485 3725 26

2787 3450 1315 1706 2056 2479 2779 3435 3707

27 1314 1703 2052 2473 2771 36903421 28 1313 170f 1 2048 2763 3674 29

~467 3408 1311 1699 2045 - 2462 2756 36593396

30 1310 1697 2D42 2457 - 2750 3646 40

3385 1303 1684 2021 2423 2704 3551

60 3307

1296 -16n 2000 2390 2660 3460 120

3232 1289 1658 1980 2358 2617 33733160

00 1645 1282 1960 2326 2576 3090 3291

Source 1his table is reproduced with the lcind permtsSioil of~e frustees of Biometrika from E S Pearson and R O Hartley (eds) The Biometrika Tables for Statisticians Vol 1 3d ed Biometrika 1966

r7~~

c

~ ApPENDIX A Tables 801

i~ 1lt TABLE IX Percentage Points of the F-distribution a = 05

~ liZ 1 2

1 1614 1995 2 1851 1900 3 1013 955 4 771 694

II

NUMERATOR DEGREES OF FREEDOM I11

3 4 9 I5 6 7 8

5

2157 2246 2302 2340 1916 1925 1930 1933

928 912 901 894 659 639 626 616

~ 661 579 541 519~ 505 495 6 599 514 476 453 439 428~7 559 474 435 412 397 387

5328 446 407 384 369 358 9 512 426 386 363 348 337~

10 -496 4100 371 348 333 322Q 11 484 7f~i 359398--~ 336 320 309

34938~middot12middot 475 326 300 ~~311~ 13 381467 341 318 303 292 0 fi

14 460 374 334 311 296 285 ~ 15 454 368 329 306 290foil 279 rr

449 363~ 16 324 301 285 274

~ 17 445 )20 296 281359 270 foil 18 441 3-16 293li))~5 277 266Q

438 352middot 313 290= 19 274 263 0 20 435 349 310 287 271 260 ~ 21 432 347 257307 284 268

i 22 430 344 305 282 266 255 303 0 23 428 342 280 264 253

Z 24 426 340 301 278 262 251foil Q 424middot25 339 299 276 260 249

26 423 337 298 274 259 247 27 421 335 296 273 257 246 28 420 334 295 271 245 256 29 418 333 293I 270 255 243 30 417 332 292 269 253 242 40 408 323 284 261 245 234 60 40q 315 276 253 237 225

120 307 392 268 245 229 217 00 384 300 260 237 221 210

2368 2389 1935 1937 889 885 609 604 488 482 421 415 379 373 350 344 329 323 314 307 301 295 291 285

283 277 271 1276 270 265 271 264 259 i266 259 254 1rII261 255 249

251 258 ~i-middot246 f 254 248 242 I

I 251 245 239 249 242 237 246 240 234 244 237 232 242 236 230 240 234 228 239 232 2]7 237 231 225 236 229 224 235 228 222 233 227 221

225 218 212 I

1217 210 204 209 202 196 201 194 188

2405 1938 881 600 477 410 368 339 318 302 290 280 ji

Source From M Merrington and eM Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 19433373-88

i (continued) I

I

I iii 11 iii

II I

I l

iINi II

802 Appendix A Tables

TABLE IX Continued

~1 NUMERATOR DEGREES OF FREEDOM Vl ~r--l-0----12----15-----r--2-0----24---r--30----40----60~--~- ~ OQ

1 2 3 4 S 6 7 8 9

~ 10 11

~ ~ ~ 14 o 15 rIJ (

Iiil 16 ~ 17 ~ 18 ~ 19 = 20

~ 21

i ii shy~ 24 ~ 2S

26 27 28 29 30

middot40 60

120 OQ

2419 1940 879 596 474 406 364 335 ~314

298 285 275 267 260 254 249 245 241 238 235 232 230 227 225 224 222 220 219 218 216

208 199 191 1~

2439 2459 2480 2491 1941 1943 1945 1945 874 870 866 864 591 586 580 577 468 462 456 453 400 394 387 384 357 351 344 341 328 322 315 312 307 301 294 290 291 285 277 274 279 272 265 261 269 262 254 251 260 253 246 242 253 246 239 235 248 240 233 229 242 235 228 224 238 231 223 219 234 227 219 215 231 223 216 211 iJi ~r~ 220 212 208 225 218 210 205 223 215 207 203 220 213 205 201 218 211 203 198 216 209 201 196

215 2Q7 199 195j 2i3 206 197 193 212 204 196 191 210 203 194 190 209 201 193 189

200 192184 179 192 184 175 170 183 i 175 166 161 U5 1m 1~ 1~

2501 1946 862 575 450 381 33S 308 286 270 257 247

238 231 225 219 215 211 207 204 201 198 196 194 192 190 188 187 185 184

174 165 155 146

2511 Imiddotmiddotmiddot

1947 859 572 446 377middot 334 304 283 266 253 243 234 227 220 215 210 206 203

~199 196 f 194 1~1 189 187 185 184 182 181 179

169 159 150 139

2522 1948 857 5~69

443 374 330 301 279 262 249 238 230 222

216 211 206 202 198 195 192 189 186 184 182 180 179 177 175 174

164 153 143 132

2533 2543 1949 1950 855 853 566 563 440 436 370 367 327 323 297 293 275 271 258 254 245 240 234 230 225 221 218 213 211 207 206 201 201 196 197 192 193 188 190 184 1~7 181 184 178

181 176 179 173 177 ~ 171

169 17~ 173 167 171 165

164170 162168

151158 147 139

125135 100122 i

ApPENDIX A Tables~c63 y

TABLE X Percentage Points of the F-distribution a = 025

f(F)

o FfJ25

- -NUMERATOR DEGREES OF FREEDOM ~ Jz 1 2 3 4 5 6 7 8 9

1 6478 7995 8642 8996 9218 9371 9482 9567 9633 2 3851 3900 3917 3925 3930 3933 3936 3937 3939 3 1744 1604 1544 1510 1488 1473 1462 1454 1447 4 1222 1065 998 960 936 920 907 898 890 5 1001 843 776 739 715 698 685 676 668 ( 881 726 660 623 599 582 570 560 552

589 7 807 654 552 529 512 499 490 482 8 757 606 542 505 482 465 453 443 436 9 721 571 508 472 448 432 420 410 403

gt1 10 - 694 483 447 424 407 395 385 378-4~~i 0 003f2~~ 11 672 463 428 404 88 376 366 359fIl

12- 655 510 447 412 389 373 351 344~61f l~ 13 641 497 435 400 377 360 348 339 331

0 ~ 14 630 486 424 389 366 329338-~$P 321 IIl 15 620 477 A15 380 358 34-1 329 320 312fIl

16 61_2 469 408 373 350 334 322 305 31~r-~ ~ 17 462- 4016Cl1) 366 344 328 316 298I j r~ 30(hmiddotfIl 45618 598 395 361 338 322 310 301 293~

592 451 390 356 333= 19 317 305 296 288

~ 20 587 446 386 351 329 313 301 291 284 21 583 442 382 348 325 309 297 287 280 22 - 579 438 378 344 322 305 293 284 276~

0 23 575 435 375 341 318 302 290 281 273 ~ 24 572 432 372 338 315 299 287 278 270 ~ 25 569 429 369 335 313 297 285 275 268

26 566 427 367 333 310 294 282 273 265 27 563 424 365 331 - 292 308 280 271 263 28 561 422 363 329 306 290 278 269 261 29 559 420 361 327 304 288 276 267 259 30 557 418 359 325 303 287 275 265 257

40 542 405 346 290 313 274 262 253 245 60 529 393 334 301 279 263 251 241 233

120 515 380 323 289 267 252 239 230 222 00 502 369 312 279 257 241 229 219 211

SourceFrom M Merrington and C M Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 1943337388

(continued)

4 804 Appendix A Tables

TABLE X Continued

~Vl NUMERATOR DEGREES OF FREEDOM V2~r---l-0--r--U--~-~~--~~20-----~~---~------~~middotmiddot----6-0-----U~O-----~---

9686 9767 9849 9931 9972 1001 1006 _1010 1~014-1 1018 2 3940 3941 3943 3945 3946 3946 3947 3948 394~ 3950 3 1442 1434 1425 1417 1412 1408 1404 1399 1395 1390

884 875 866 856 851 846 841 836 8314 826 662 652 643 633 628 623 618 612 6075 602 546 537 527 517 512 507 501 496 4906 485

7 476 467 457 447 442 436 431 425 420 414 8 430 420 410 400 395 389 384 378 373 367 9 396 387 377 367 361 356 351 345 339 333

372 362 352 342 337 331 326 320 314 308S 10 353 343 333 323 317 312 306 300 294Q 11 288 337 328 318 307 302 296 291 285 279 272 325 315 305 295 289 284 278 272 266 260E ~

)i 14 315 305 295 284 279 273 267 261 255 249 o 15 306 296 286 276 270 264 259 252 246 240rIl ~ 16 232299 289 279 268 263 257 251 245 238 ~ ~17 292 282 272 262 256 250 244 238 232 225

287 277 267 256 250 244 238 232 226~ 18 219 Q 19 282 272 262 251 245 239 233 227 220 21~=

277 268 257 246 241 235 229 222 216 209~ 20middot 273 bull =I~~~ 253 242 237 231 ~ 225 218 211lt 21 204 270 middot260 250 239 233 227 21 214 208 200~ 22 267 257 247 236 230 224 $ 218 211 204 197o 23

z 24 264 254 244 D3 227 221 ~ Ii 215 208 201 194 261 251 241 230 224 218 212 205 198 191~ 25

26 259 249 239 228 222 216 209 203 )95 I 188 27 257 247 236 225 219 213 207 200 i~~3 185 28 2d5tf i 245middot 234 223217 211 205 198 191 183 29 253 243 232 221 215 209 203 196 189 181 30 251 241 231 220 214 207 201 194 187 179

40 164239 229 218 207 201 194 188 180 172 60 227 217 206 194 188 182 174 167 158 148

120 216 205 194 182 176 169 161 153 143 131 I

QO 100205 194 I 183 171 I 164 157 148 139 127

Page 8: STA220_Final_2012S.pdf

Full Name Student Number

lOmarksj 6 A new toothpaste has been put on the market that claims to reduce the number of teeth that have

problems (cavities sensitive teeth etc) In order to test the effect ofthe new toothpaste researchers randomly select participants and randomly assign each person to use either the new toothpaste or a standard toothpaste (a popular brand that has been used for several years) After 6 months of use each participant reports the number of teeth they have with problems It can be assumed that both populations are Normal Partial Minitab output is given

DESCRIPTIVE STATISTICS standard DESCRIPTIVE STATISTICS new

Variable N Mean StDev SE MeanVariable N Mean StDev SE Mean new 20 330 213 048standard 15 370 198 051

Test for Equal Variances

95 Bonferroni confidence intervals for standard deviations

Sample N Lower StDev Upper standard 15 138964 1 98 335600 new 20 156050 213 330000

F-Test (normal distribution) Test statistic = 086 p-value = 079

[2 m)

(a) Using a=O05 test for the equality of the variances of the standard and new toothpaste populations Include all steps requiredfor a Hypothesis Test (you may copy any relevant parts from the output without calculating them yourself)

Page 8 of 12

Full Name Student Number

[5 m] (b) Which type of hypothesis test would you use to test if the new toothpaste is effective as

claimed by the manufacturer using a005 Why Test the relevant hypothesis from scratch Include all steps requiredfor a Hypothesis Test

[3 m] (c) The manufacturer of the new toothpaste now decides to collect data on consumers ofthe new

toothpaste separately in order to improve their study How large of sample is required to estimate the mean number of teeth with problems within 05 and retain 95 confidence

Page 9 of 12

Full Name Student Number

flOmarksJ 7 Is insomnia related to education Researchers conducted a study to determine if insomnia is related to the education of Tennessee residents 575 adults in Tennessee were selected to participate in the study using a random-digit telephone procedure Thenumber of years of education and insomnia status (normal sleeper or chronic insomniac) were recorded for each participant The researchers discovered that the fewer the years of education the more likely the person is to have chronic insomnia

~Insomnia is a sleep disorder (problems falling asleepstaying asleep throughout the night) In a random-digit telephone procedure telephone numbers are generated at random and the people with those phone numbers are contacted to participate in the study

Fill in the blanks or give a one-sentence brief answer where appropriate to following

[1 m] (a) Identify the population of interest

[1 m] (b) Identify the sample

[4 m] (c) List the variables measured and whether they are categorical or quantitative

[1 m] (d) What inference(s) did the researchers make

[1 m] (e) In general conclusions about a sample can be generalized to the population if the sample is

___________ of the population

[2 m] (f) In this case do the selected participants form a random sample Explain why or why not

Page 10 of 12

Full Name Student Number

[10marksj 8 You and your friend playa dice-rolling game You each roll a fair die once and observe the

number that appears on the up face for each person Ifboth ofyou get the same number nobody wins or loses any money If you get different numbers whoever gets the lower number has to pay the other person $1 Let X represent your winnings (in $) from this game

[4 m] (a) Find the probability distribution for X and display it in table form

[3 m] (b) Calculate the mean ofX and interpret your answer in terms of the game Show your

calculations and then circle your final answer

[3 m] (c) Compute the variance ofX Show your work and circle thefinal answer

Page 11 of 12

Full Name Student Number

10 marks 9 A public school is considering adding to the curriculum a course that it believes will improve the reading comprehension skills of its students The school has decided to conduct an experiment to evaluate the course before it is offered to the general student body Six students are randomly selected and their reading scores are recorded before and after the course These scores are composite measurements ofboth speed and comprehension (where higher scores indicate higher

h )Th I dblcompre enSlOn e resu ts are summanze e ow Student Number Score After Course Score Before Course

1 221 211 2 231 216 3 203 191 4 216 224 5 207 201 6 203 178

The school will implement the new course if there is sufficient evidence at the 5 significance level that the course improves reading comprehension

[2 m) (a) Which type of hypothesis test is appropriate for this situation and why

[6m (b) Conduct the hypothesis test of interest Include all steps requiredfor a Hypothesis Test

[2 m (c) List the assumption(s) required to use this type ofhypothesis test

Page 12 of 12 --END OF EXAM--shy

794 Appendix A Tables

TABLE IV Normal Curve Areas

o 1

00 01 bull02z 03 OS04 06 07 OS 09

0 0000 0040 0080 0120 01990160 0239 0279 0319 03591 0398 0438 0478 0517 05960557 0636 0675 0714 0753

07932 0832 0871 0910 0948 0987 1026 1064 1103 1141 3 1179 1217 1255 1293 13681331 1406 1443 1480 1517

15544 1591 1628 1664 1700 1736 1772 1808 1844 1879

1915oS 1950 1985 2019 2054 2088 2123 2157 2190 2224

22576 2291 2324 2357 24222389 2454 2486 2517 2549

25807 2611 2642 2673 2704 2734 2764 2794 2823 2852

28818 2910 2939 2967 2995 3023 3051 3078 3106 3133 3212 9 y3159 3186 3238 3264 3289 3315 3340 3365 3389

10 3413 3438 3461 3485 35313508 3554 3577 3599 3621 372911 3643 3665 3686 3708 3749 3770 3790 - 3810 3830

3849 386912 3888 3907 3925 3944 3962 3980 3997 4015

4032 4049 13 bulliQ6timiddot 4082 41154099 4131 4147 4162 4177 42071~m224192L4 4236 4251 4265 43194179 4292 4306

4332 4345 4357 4370 4382 4394middot 4406 4418 4429lS 4441 ltIJ

4452L6 4463 4474 4484 4495 4505 4515 4525 4535 4345

4554L7 4564 4573 4582 4591 4599 4608 4616 4625 4633

464118 4656 46644P49 4671 4678 4686 4693 47064699

4713 4761 L9 4719 4732 ~ 47674738 4744 4750 4726 4756 478820 4772 4793 4798 4893 4808 4812 4817~11amp 4783 shy

4821U 4826 4830 4834 48424838 48574846 4850 4854 486122 4864 4868 48874871 4875 4878 48904881 4884

23 4893 4896 4898 49164901 4913 4906 4909 49114904 24 4918 4922492Q 49364925shy 49294927 49344931 4932

4943493825 49524940 4941 4945 4946 49514948 4949 495326 4955 49644956 4957 4959 4960 4961 49634962 4965 497427 49674966 4968 4969 4970 49734971 4972 4974 498128 4975 4976 4977 4977 4978 49804979 4979

29 49864981 4982 4982 4983 4984 4984 49864985~49B5 30 4987 4987middot4987 4988 4988 4989 4990 49904989 4989 i

Source Abridged from Table I ofA Hald Statistical Table$ and Fonnulas (New YorkWiley) 1952 Reproduced by permission of A Hald

796 -~~ - -

o ~~~-TABLE VI Critical Values of t

f(t)

bull

Degrees of -Freedom tJWtlOO t050 tOO51010 tooi tooos

1 63143078 12706 31821 63657 31831 63662 2 1886 2920 4303 6965 9925 22326 31598 3 4541 1638 2353 3-182 5841 10213 12924 4 1533 2132 2776 3747 4604 7173 8610 5 1476 2015 2571 3365 4032 5893 6869 6 19431440 2447 3143 3707 5208 5959 7 1415 23651895 2998 3499 4785 5408

1397 1860 2306i 2896 3355 4501 5041 9 1383 1833 2262 2821 3250 4297 4781

10 18121372 2228 2764 3169 _ 4144 4587 U 1363 1796 2201 2718 3106 4025 4431 12 1356 1782 2179 shy 2681 3055 3930 4318

13~~ ~_ ~-i77113 i1reg 2650 3012 3852 4221 14 1345 21451761 2024 ~ 2977 41403787 15 1341 1753 2131 2602 29fl7 4073 16

373~ 1337 1746 2120 - 29212583 3686 4015

17 1333 1740 2110 2567 -2898 3646 3965 18 1330 1734 2101 t28782552 3610 3922 19 1328 t729 2093 28612539 38833579 - i

li~r~5I - 20 1725 2086 2528 2845 3850 21

3552 t 1323 2080 2518 1721 2831 3527 3819

22 20741321 1717 2508 2819 3505 3~792 23 1319 1714 2069 2500 3767 24

2807 3485 1318 1711 2064 2492 3467 3745

25 2middot797

1316 1708 2060 2485 3725 26

2787 3450 1315 1706 2056 2479 2779 3435 3707

27 1314 1703 2052 2473 2771 36903421 28 1313 170f 1 2048 2763 3674 29

~467 3408 1311 1699 2045 - 2462 2756 36593396

30 1310 1697 2D42 2457 - 2750 3646 40

3385 1303 1684 2021 2423 2704 3551

60 3307

1296 -16n 2000 2390 2660 3460 120

3232 1289 1658 1980 2358 2617 33733160

00 1645 1282 1960 2326 2576 3090 3291

Source 1his table is reproduced with the lcind permtsSioil of~e frustees of Biometrika from E S Pearson and R O Hartley (eds) The Biometrika Tables for Statisticians Vol 1 3d ed Biometrika 1966

r7~~

c

~ ApPENDIX A Tables 801

i~ 1lt TABLE IX Percentage Points of the F-distribution a = 05

~ liZ 1 2

1 1614 1995 2 1851 1900 3 1013 955 4 771 694

II

NUMERATOR DEGREES OF FREEDOM I11

3 4 9 I5 6 7 8

5

2157 2246 2302 2340 1916 1925 1930 1933

928 912 901 894 659 639 626 616

~ 661 579 541 519~ 505 495 6 599 514 476 453 439 428~7 559 474 435 412 397 387

5328 446 407 384 369 358 9 512 426 386 363 348 337~

10 -496 4100 371 348 333 322Q 11 484 7f~i 359398--~ 336 320 309

34938~middot12middot 475 326 300 ~~311~ 13 381467 341 318 303 292 0 fi

14 460 374 334 311 296 285 ~ 15 454 368 329 306 290foil 279 rr

449 363~ 16 324 301 285 274

~ 17 445 )20 296 281359 270 foil 18 441 3-16 293li))~5 277 266Q

438 352middot 313 290= 19 274 263 0 20 435 349 310 287 271 260 ~ 21 432 347 257307 284 268

i 22 430 344 305 282 266 255 303 0 23 428 342 280 264 253

Z 24 426 340 301 278 262 251foil Q 424middot25 339 299 276 260 249

26 423 337 298 274 259 247 27 421 335 296 273 257 246 28 420 334 295 271 245 256 29 418 333 293I 270 255 243 30 417 332 292 269 253 242 40 408 323 284 261 245 234 60 40q 315 276 253 237 225

120 307 392 268 245 229 217 00 384 300 260 237 221 210

2368 2389 1935 1937 889 885 609 604 488 482 421 415 379 373 350 344 329 323 314 307 301 295 291 285

283 277 271 1276 270 265 271 264 259 i266 259 254 1rII261 255 249

251 258 ~i-middot246 f 254 248 242 I

I 251 245 239 249 242 237 246 240 234 244 237 232 242 236 230 240 234 228 239 232 2]7 237 231 225 236 229 224 235 228 222 233 227 221

225 218 212 I

1217 210 204 209 202 196 201 194 188

2405 1938 881 600 477 410 368 339 318 302 290 280 ji

Source From M Merrington and eM Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 19433373-88

i (continued) I

I

I iii 11 iii

II I

I l

iINi II

802 Appendix A Tables

TABLE IX Continued

~1 NUMERATOR DEGREES OF FREEDOM Vl ~r--l-0----12----15-----r--2-0----24---r--30----40----60~--~- ~ OQ

1 2 3 4 S 6 7 8 9

~ 10 11

~ ~ ~ 14 o 15 rIJ (

Iiil 16 ~ 17 ~ 18 ~ 19 = 20

~ 21

i ii shy~ 24 ~ 2S

26 27 28 29 30

middot40 60

120 OQ

2419 1940 879 596 474 406 364 335 ~314

298 285 275 267 260 254 249 245 241 238 235 232 230 227 225 224 222 220 219 218 216

208 199 191 1~

2439 2459 2480 2491 1941 1943 1945 1945 874 870 866 864 591 586 580 577 468 462 456 453 400 394 387 384 357 351 344 341 328 322 315 312 307 301 294 290 291 285 277 274 279 272 265 261 269 262 254 251 260 253 246 242 253 246 239 235 248 240 233 229 242 235 228 224 238 231 223 219 234 227 219 215 231 223 216 211 iJi ~r~ 220 212 208 225 218 210 205 223 215 207 203 220 213 205 201 218 211 203 198 216 209 201 196

215 2Q7 199 195j 2i3 206 197 193 212 204 196 191 210 203 194 190 209 201 193 189

200 192184 179 192 184 175 170 183 i 175 166 161 U5 1m 1~ 1~

2501 1946 862 575 450 381 33S 308 286 270 257 247

238 231 225 219 215 211 207 204 201 198 196 194 192 190 188 187 185 184

174 165 155 146

2511 Imiddotmiddotmiddot

1947 859 572 446 377middot 334 304 283 266 253 243 234 227 220 215 210 206 203

~199 196 f 194 1~1 189 187 185 184 182 181 179

169 159 150 139

2522 1948 857 5~69

443 374 330 301 279 262 249 238 230 222

216 211 206 202 198 195 192 189 186 184 182 180 179 177 175 174

164 153 143 132

2533 2543 1949 1950 855 853 566 563 440 436 370 367 327 323 297 293 275 271 258 254 245 240 234 230 225 221 218 213 211 207 206 201 201 196 197 192 193 188 190 184 1~7 181 184 178

181 176 179 173 177 ~ 171

169 17~ 173 167 171 165

164170 162168

151158 147 139

125135 100122 i

ApPENDIX A Tables~c63 y

TABLE X Percentage Points of the F-distribution a = 025

f(F)

o FfJ25

- -NUMERATOR DEGREES OF FREEDOM ~ Jz 1 2 3 4 5 6 7 8 9

1 6478 7995 8642 8996 9218 9371 9482 9567 9633 2 3851 3900 3917 3925 3930 3933 3936 3937 3939 3 1744 1604 1544 1510 1488 1473 1462 1454 1447 4 1222 1065 998 960 936 920 907 898 890 5 1001 843 776 739 715 698 685 676 668 ( 881 726 660 623 599 582 570 560 552

589 7 807 654 552 529 512 499 490 482 8 757 606 542 505 482 465 453 443 436 9 721 571 508 472 448 432 420 410 403

gt1 10 - 694 483 447 424 407 395 385 378-4~~i 0 003f2~~ 11 672 463 428 404 88 376 366 359fIl

12- 655 510 447 412 389 373 351 344~61f l~ 13 641 497 435 400 377 360 348 339 331

0 ~ 14 630 486 424 389 366 329338-~$P 321 IIl 15 620 477 A15 380 358 34-1 329 320 312fIl

16 61_2 469 408 373 350 334 322 305 31~r-~ ~ 17 462- 4016Cl1) 366 344 328 316 298I j r~ 30(hmiddotfIl 45618 598 395 361 338 322 310 301 293~

592 451 390 356 333= 19 317 305 296 288

~ 20 587 446 386 351 329 313 301 291 284 21 583 442 382 348 325 309 297 287 280 22 - 579 438 378 344 322 305 293 284 276~

0 23 575 435 375 341 318 302 290 281 273 ~ 24 572 432 372 338 315 299 287 278 270 ~ 25 569 429 369 335 313 297 285 275 268

26 566 427 367 333 310 294 282 273 265 27 563 424 365 331 - 292 308 280 271 263 28 561 422 363 329 306 290 278 269 261 29 559 420 361 327 304 288 276 267 259 30 557 418 359 325 303 287 275 265 257

40 542 405 346 290 313 274 262 253 245 60 529 393 334 301 279 263 251 241 233

120 515 380 323 289 267 252 239 230 222 00 502 369 312 279 257 241 229 219 211

SourceFrom M Merrington and C M Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 1943337388

(continued)

4 804 Appendix A Tables

TABLE X Continued

~Vl NUMERATOR DEGREES OF FREEDOM V2~r---l-0--r--U--~-~~--~~20-----~~---~------~~middotmiddot----6-0-----U~O-----~---

9686 9767 9849 9931 9972 1001 1006 _1010 1~014-1 1018 2 3940 3941 3943 3945 3946 3946 3947 3948 394~ 3950 3 1442 1434 1425 1417 1412 1408 1404 1399 1395 1390

884 875 866 856 851 846 841 836 8314 826 662 652 643 633 628 623 618 612 6075 602 546 537 527 517 512 507 501 496 4906 485

7 476 467 457 447 442 436 431 425 420 414 8 430 420 410 400 395 389 384 378 373 367 9 396 387 377 367 361 356 351 345 339 333

372 362 352 342 337 331 326 320 314 308S 10 353 343 333 323 317 312 306 300 294Q 11 288 337 328 318 307 302 296 291 285 279 272 325 315 305 295 289 284 278 272 266 260E ~

)i 14 315 305 295 284 279 273 267 261 255 249 o 15 306 296 286 276 270 264 259 252 246 240rIl ~ 16 232299 289 279 268 263 257 251 245 238 ~ ~17 292 282 272 262 256 250 244 238 232 225

287 277 267 256 250 244 238 232 226~ 18 219 Q 19 282 272 262 251 245 239 233 227 220 21~=

277 268 257 246 241 235 229 222 216 209~ 20middot 273 bull =I~~~ 253 242 237 231 ~ 225 218 211lt 21 204 270 middot260 250 239 233 227 21 214 208 200~ 22 267 257 247 236 230 224 $ 218 211 204 197o 23

z 24 264 254 244 D3 227 221 ~ Ii 215 208 201 194 261 251 241 230 224 218 212 205 198 191~ 25

26 259 249 239 228 222 216 209 203 )95 I 188 27 257 247 236 225 219 213 207 200 i~~3 185 28 2d5tf i 245middot 234 223217 211 205 198 191 183 29 253 243 232 221 215 209 203 196 189 181 30 251 241 231 220 214 207 201 194 187 179

40 164239 229 218 207 201 194 188 180 172 60 227 217 206 194 188 182 174 167 158 148

120 216 205 194 182 176 169 161 153 143 131 I

QO 100205 194 I 183 171 I 164 157 148 139 127

Page 9: STA220_Final_2012S.pdf

Full Name Student Number

[5 m] (b) Which type of hypothesis test would you use to test if the new toothpaste is effective as

claimed by the manufacturer using a005 Why Test the relevant hypothesis from scratch Include all steps requiredfor a Hypothesis Test

[3 m] (c) The manufacturer of the new toothpaste now decides to collect data on consumers ofthe new

toothpaste separately in order to improve their study How large of sample is required to estimate the mean number of teeth with problems within 05 and retain 95 confidence

Page 9 of 12

Full Name Student Number

flOmarksJ 7 Is insomnia related to education Researchers conducted a study to determine if insomnia is related to the education of Tennessee residents 575 adults in Tennessee were selected to participate in the study using a random-digit telephone procedure Thenumber of years of education and insomnia status (normal sleeper or chronic insomniac) were recorded for each participant The researchers discovered that the fewer the years of education the more likely the person is to have chronic insomnia

~Insomnia is a sleep disorder (problems falling asleepstaying asleep throughout the night) In a random-digit telephone procedure telephone numbers are generated at random and the people with those phone numbers are contacted to participate in the study

Fill in the blanks or give a one-sentence brief answer where appropriate to following

[1 m] (a) Identify the population of interest

[1 m] (b) Identify the sample

[4 m] (c) List the variables measured and whether they are categorical or quantitative

[1 m] (d) What inference(s) did the researchers make

[1 m] (e) In general conclusions about a sample can be generalized to the population if the sample is

___________ of the population

[2 m] (f) In this case do the selected participants form a random sample Explain why or why not

Page 10 of 12

Full Name Student Number

[10marksj 8 You and your friend playa dice-rolling game You each roll a fair die once and observe the

number that appears on the up face for each person Ifboth ofyou get the same number nobody wins or loses any money If you get different numbers whoever gets the lower number has to pay the other person $1 Let X represent your winnings (in $) from this game

[4 m] (a) Find the probability distribution for X and display it in table form

[3 m] (b) Calculate the mean ofX and interpret your answer in terms of the game Show your

calculations and then circle your final answer

[3 m] (c) Compute the variance ofX Show your work and circle thefinal answer

Page 11 of 12

Full Name Student Number

10 marks 9 A public school is considering adding to the curriculum a course that it believes will improve the reading comprehension skills of its students The school has decided to conduct an experiment to evaluate the course before it is offered to the general student body Six students are randomly selected and their reading scores are recorded before and after the course These scores are composite measurements ofboth speed and comprehension (where higher scores indicate higher

h )Th I dblcompre enSlOn e resu ts are summanze e ow Student Number Score After Course Score Before Course

1 221 211 2 231 216 3 203 191 4 216 224 5 207 201 6 203 178

The school will implement the new course if there is sufficient evidence at the 5 significance level that the course improves reading comprehension

[2 m) (a) Which type of hypothesis test is appropriate for this situation and why

[6m (b) Conduct the hypothesis test of interest Include all steps requiredfor a Hypothesis Test

[2 m (c) List the assumption(s) required to use this type ofhypothesis test

Page 12 of 12 --END OF EXAM--shy

794 Appendix A Tables

TABLE IV Normal Curve Areas

o 1

00 01 bull02z 03 OS04 06 07 OS 09

0 0000 0040 0080 0120 01990160 0239 0279 0319 03591 0398 0438 0478 0517 05960557 0636 0675 0714 0753

07932 0832 0871 0910 0948 0987 1026 1064 1103 1141 3 1179 1217 1255 1293 13681331 1406 1443 1480 1517

15544 1591 1628 1664 1700 1736 1772 1808 1844 1879

1915oS 1950 1985 2019 2054 2088 2123 2157 2190 2224

22576 2291 2324 2357 24222389 2454 2486 2517 2549

25807 2611 2642 2673 2704 2734 2764 2794 2823 2852

28818 2910 2939 2967 2995 3023 3051 3078 3106 3133 3212 9 y3159 3186 3238 3264 3289 3315 3340 3365 3389

10 3413 3438 3461 3485 35313508 3554 3577 3599 3621 372911 3643 3665 3686 3708 3749 3770 3790 - 3810 3830

3849 386912 3888 3907 3925 3944 3962 3980 3997 4015

4032 4049 13 bulliQ6timiddot 4082 41154099 4131 4147 4162 4177 42071~m224192L4 4236 4251 4265 43194179 4292 4306

4332 4345 4357 4370 4382 4394middot 4406 4418 4429lS 4441 ltIJ

4452L6 4463 4474 4484 4495 4505 4515 4525 4535 4345

4554L7 4564 4573 4582 4591 4599 4608 4616 4625 4633

464118 4656 46644P49 4671 4678 4686 4693 47064699

4713 4761 L9 4719 4732 ~ 47674738 4744 4750 4726 4756 478820 4772 4793 4798 4893 4808 4812 4817~11amp 4783 shy

4821U 4826 4830 4834 48424838 48574846 4850 4854 486122 4864 4868 48874871 4875 4878 48904881 4884

23 4893 4896 4898 49164901 4913 4906 4909 49114904 24 4918 4922492Q 49364925shy 49294927 49344931 4932

4943493825 49524940 4941 4945 4946 49514948 4949 495326 4955 49644956 4957 4959 4960 4961 49634962 4965 497427 49674966 4968 4969 4970 49734971 4972 4974 498128 4975 4976 4977 4977 4978 49804979 4979

29 49864981 4982 4982 4983 4984 4984 49864985~49B5 30 4987 4987middot4987 4988 4988 4989 4990 49904989 4989 i

Source Abridged from Table I ofA Hald Statistical Table$ and Fonnulas (New YorkWiley) 1952 Reproduced by permission of A Hald

796 -~~ - -

o ~~~-TABLE VI Critical Values of t

f(t)

bull

Degrees of -Freedom tJWtlOO t050 tOO51010 tooi tooos

1 63143078 12706 31821 63657 31831 63662 2 1886 2920 4303 6965 9925 22326 31598 3 4541 1638 2353 3-182 5841 10213 12924 4 1533 2132 2776 3747 4604 7173 8610 5 1476 2015 2571 3365 4032 5893 6869 6 19431440 2447 3143 3707 5208 5959 7 1415 23651895 2998 3499 4785 5408

1397 1860 2306i 2896 3355 4501 5041 9 1383 1833 2262 2821 3250 4297 4781

10 18121372 2228 2764 3169 _ 4144 4587 U 1363 1796 2201 2718 3106 4025 4431 12 1356 1782 2179 shy 2681 3055 3930 4318

13~~ ~_ ~-i77113 i1reg 2650 3012 3852 4221 14 1345 21451761 2024 ~ 2977 41403787 15 1341 1753 2131 2602 29fl7 4073 16

373~ 1337 1746 2120 - 29212583 3686 4015

17 1333 1740 2110 2567 -2898 3646 3965 18 1330 1734 2101 t28782552 3610 3922 19 1328 t729 2093 28612539 38833579 - i

li~r~5I - 20 1725 2086 2528 2845 3850 21

3552 t 1323 2080 2518 1721 2831 3527 3819

22 20741321 1717 2508 2819 3505 3~792 23 1319 1714 2069 2500 3767 24

2807 3485 1318 1711 2064 2492 3467 3745

25 2middot797

1316 1708 2060 2485 3725 26

2787 3450 1315 1706 2056 2479 2779 3435 3707

27 1314 1703 2052 2473 2771 36903421 28 1313 170f 1 2048 2763 3674 29

~467 3408 1311 1699 2045 - 2462 2756 36593396

30 1310 1697 2D42 2457 - 2750 3646 40

3385 1303 1684 2021 2423 2704 3551

60 3307

1296 -16n 2000 2390 2660 3460 120

3232 1289 1658 1980 2358 2617 33733160

00 1645 1282 1960 2326 2576 3090 3291

Source 1his table is reproduced with the lcind permtsSioil of~e frustees of Biometrika from E S Pearson and R O Hartley (eds) The Biometrika Tables for Statisticians Vol 1 3d ed Biometrika 1966

r7~~

c

~ ApPENDIX A Tables 801

i~ 1lt TABLE IX Percentage Points of the F-distribution a = 05

~ liZ 1 2

1 1614 1995 2 1851 1900 3 1013 955 4 771 694

II

NUMERATOR DEGREES OF FREEDOM I11

3 4 9 I5 6 7 8

5

2157 2246 2302 2340 1916 1925 1930 1933

928 912 901 894 659 639 626 616

~ 661 579 541 519~ 505 495 6 599 514 476 453 439 428~7 559 474 435 412 397 387

5328 446 407 384 369 358 9 512 426 386 363 348 337~

10 -496 4100 371 348 333 322Q 11 484 7f~i 359398--~ 336 320 309

34938~middot12middot 475 326 300 ~~311~ 13 381467 341 318 303 292 0 fi

14 460 374 334 311 296 285 ~ 15 454 368 329 306 290foil 279 rr

449 363~ 16 324 301 285 274

~ 17 445 )20 296 281359 270 foil 18 441 3-16 293li))~5 277 266Q

438 352middot 313 290= 19 274 263 0 20 435 349 310 287 271 260 ~ 21 432 347 257307 284 268

i 22 430 344 305 282 266 255 303 0 23 428 342 280 264 253

Z 24 426 340 301 278 262 251foil Q 424middot25 339 299 276 260 249

26 423 337 298 274 259 247 27 421 335 296 273 257 246 28 420 334 295 271 245 256 29 418 333 293I 270 255 243 30 417 332 292 269 253 242 40 408 323 284 261 245 234 60 40q 315 276 253 237 225

120 307 392 268 245 229 217 00 384 300 260 237 221 210

2368 2389 1935 1937 889 885 609 604 488 482 421 415 379 373 350 344 329 323 314 307 301 295 291 285

283 277 271 1276 270 265 271 264 259 i266 259 254 1rII261 255 249

251 258 ~i-middot246 f 254 248 242 I

I 251 245 239 249 242 237 246 240 234 244 237 232 242 236 230 240 234 228 239 232 2]7 237 231 225 236 229 224 235 228 222 233 227 221

225 218 212 I

1217 210 204 209 202 196 201 194 188

2405 1938 881 600 477 410 368 339 318 302 290 280 ji

Source From M Merrington and eM Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 19433373-88

i (continued) I

I

I iii 11 iii

II I

I l

iINi II

802 Appendix A Tables

TABLE IX Continued

~1 NUMERATOR DEGREES OF FREEDOM Vl ~r--l-0----12----15-----r--2-0----24---r--30----40----60~--~- ~ OQ

1 2 3 4 S 6 7 8 9

~ 10 11

~ ~ ~ 14 o 15 rIJ (

Iiil 16 ~ 17 ~ 18 ~ 19 = 20

~ 21

i ii shy~ 24 ~ 2S

26 27 28 29 30

middot40 60

120 OQ

2419 1940 879 596 474 406 364 335 ~314

298 285 275 267 260 254 249 245 241 238 235 232 230 227 225 224 222 220 219 218 216

208 199 191 1~

2439 2459 2480 2491 1941 1943 1945 1945 874 870 866 864 591 586 580 577 468 462 456 453 400 394 387 384 357 351 344 341 328 322 315 312 307 301 294 290 291 285 277 274 279 272 265 261 269 262 254 251 260 253 246 242 253 246 239 235 248 240 233 229 242 235 228 224 238 231 223 219 234 227 219 215 231 223 216 211 iJi ~r~ 220 212 208 225 218 210 205 223 215 207 203 220 213 205 201 218 211 203 198 216 209 201 196

215 2Q7 199 195j 2i3 206 197 193 212 204 196 191 210 203 194 190 209 201 193 189

200 192184 179 192 184 175 170 183 i 175 166 161 U5 1m 1~ 1~

2501 1946 862 575 450 381 33S 308 286 270 257 247

238 231 225 219 215 211 207 204 201 198 196 194 192 190 188 187 185 184

174 165 155 146

2511 Imiddotmiddotmiddot

1947 859 572 446 377middot 334 304 283 266 253 243 234 227 220 215 210 206 203

~199 196 f 194 1~1 189 187 185 184 182 181 179

169 159 150 139

2522 1948 857 5~69

443 374 330 301 279 262 249 238 230 222

216 211 206 202 198 195 192 189 186 184 182 180 179 177 175 174

164 153 143 132

2533 2543 1949 1950 855 853 566 563 440 436 370 367 327 323 297 293 275 271 258 254 245 240 234 230 225 221 218 213 211 207 206 201 201 196 197 192 193 188 190 184 1~7 181 184 178

181 176 179 173 177 ~ 171

169 17~ 173 167 171 165

164170 162168

151158 147 139

125135 100122 i

ApPENDIX A Tables~c63 y

TABLE X Percentage Points of the F-distribution a = 025

f(F)

o FfJ25

- -NUMERATOR DEGREES OF FREEDOM ~ Jz 1 2 3 4 5 6 7 8 9

1 6478 7995 8642 8996 9218 9371 9482 9567 9633 2 3851 3900 3917 3925 3930 3933 3936 3937 3939 3 1744 1604 1544 1510 1488 1473 1462 1454 1447 4 1222 1065 998 960 936 920 907 898 890 5 1001 843 776 739 715 698 685 676 668 ( 881 726 660 623 599 582 570 560 552

589 7 807 654 552 529 512 499 490 482 8 757 606 542 505 482 465 453 443 436 9 721 571 508 472 448 432 420 410 403

gt1 10 - 694 483 447 424 407 395 385 378-4~~i 0 003f2~~ 11 672 463 428 404 88 376 366 359fIl

12- 655 510 447 412 389 373 351 344~61f l~ 13 641 497 435 400 377 360 348 339 331

0 ~ 14 630 486 424 389 366 329338-~$P 321 IIl 15 620 477 A15 380 358 34-1 329 320 312fIl

16 61_2 469 408 373 350 334 322 305 31~r-~ ~ 17 462- 4016Cl1) 366 344 328 316 298I j r~ 30(hmiddotfIl 45618 598 395 361 338 322 310 301 293~

592 451 390 356 333= 19 317 305 296 288

~ 20 587 446 386 351 329 313 301 291 284 21 583 442 382 348 325 309 297 287 280 22 - 579 438 378 344 322 305 293 284 276~

0 23 575 435 375 341 318 302 290 281 273 ~ 24 572 432 372 338 315 299 287 278 270 ~ 25 569 429 369 335 313 297 285 275 268

26 566 427 367 333 310 294 282 273 265 27 563 424 365 331 - 292 308 280 271 263 28 561 422 363 329 306 290 278 269 261 29 559 420 361 327 304 288 276 267 259 30 557 418 359 325 303 287 275 265 257

40 542 405 346 290 313 274 262 253 245 60 529 393 334 301 279 263 251 241 233

120 515 380 323 289 267 252 239 230 222 00 502 369 312 279 257 241 229 219 211

SourceFrom M Merrington and C M Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 1943337388

(continued)

4 804 Appendix A Tables

TABLE X Continued

~Vl NUMERATOR DEGREES OF FREEDOM V2~r---l-0--r--U--~-~~--~~20-----~~---~------~~middotmiddot----6-0-----U~O-----~---

9686 9767 9849 9931 9972 1001 1006 _1010 1~014-1 1018 2 3940 3941 3943 3945 3946 3946 3947 3948 394~ 3950 3 1442 1434 1425 1417 1412 1408 1404 1399 1395 1390

884 875 866 856 851 846 841 836 8314 826 662 652 643 633 628 623 618 612 6075 602 546 537 527 517 512 507 501 496 4906 485

7 476 467 457 447 442 436 431 425 420 414 8 430 420 410 400 395 389 384 378 373 367 9 396 387 377 367 361 356 351 345 339 333

372 362 352 342 337 331 326 320 314 308S 10 353 343 333 323 317 312 306 300 294Q 11 288 337 328 318 307 302 296 291 285 279 272 325 315 305 295 289 284 278 272 266 260E ~

)i 14 315 305 295 284 279 273 267 261 255 249 o 15 306 296 286 276 270 264 259 252 246 240rIl ~ 16 232299 289 279 268 263 257 251 245 238 ~ ~17 292 282 272 262 256 250 244 238 232 225

287 277 267 256 250 244 238 232 226~ 18 219 Q 19 282 272 262 251 245 239 233 227 220 21~=

277 268 257 246 241 235 229 222 216 209~ 20middot 273 bull =I~~~ 253 242 237 231 ~ 225 218 211lt 21 204 270 middot260 250 239 233 227 21 214 208 200~ 22 267 257 247 236 230 224 $ 218 211 204 197o 23

z 24 264 254 244 D3 227 221 ~ Ii 215 208 201 194 261 251 241 230 224 218 212 205 198 191~ 25

26 259 249 239 228 222 216 209 203 )95 I 188 27 257 247 236 225 219 213 207 200 i~~3 185 28 2d5tf i 245middot 234 223217 211 205 198 191 183 29 253 243 232 221 215 209 203 196 189 181 30 251 241 231 220 214 207 201 194 187 179

40 164239 229 218 207 201 194 188 180 172 60 227 217 206 194 188 182 174 167 158 148

120 216 205 194 182 176 169 161 153 143 131 I

QO 100205 194 I 183 171 I 164 157 148 139 127

Page 10: STA220_Final_2012S.pdf

Full Name Student Number

flOmarksJ 7 Is insomnia related to education Researchers conducted a study to determine if insomnia is related to the education of Tennessee residents 575 adults in Tennessee were selected to participate in the study using a random-digit telephone procedure Thenumber of years of education and insomnia status (normal sleeper or chronic insomniac) were recorded for each participant The researchers discovered that the fewer the years of education the more likely the person is to have chronic insomnia

~Insomnia is a sleep disorder (problems falling asleepstaying asleep throughout the night) In a random-digit telephone procedure telephone numbers are generated at random and the people with those phone numbers are contacted to participate in the study

Fill in the blanks or give a one-sentence brief answer where appropriate to following

[1 m] (a) Identify the population of interest

[1 m] (b) Identify the sample

[4 m] (c) List the variables measured and whether they are categorical or quantitative

[1 m] (d) What inference(s) did the researchers make

[1 m] (e) In general conclusions about a sample can be generalized to the population if the sample is

___________ of the population

[2 m] (f) In this case do the selected participants form a random sample Explain why or why not

Page 10 of 12

Full Name Student Number

[10marksj 8 You and your friend playa dice-rolling game You each roll a fair die once and observe the

number that appears on the up face for each person Ifboth ofyou get the same number nobody wins or loses any money If you get different numbers whoever gets the lower number has to pay the other person $1 Let X represent your winnings (in $) from this game

[4 m] (a) Find the probability distribution for X and display it in table form

[3 m] (b) Calculate the mean ofX and interpret your answer in terms of the game Show your

calculations and then circle your final answer

[3 m] (c) Compute the variance ofX Show your work and circle thefinal answer

Page 11 of 12

Full Name Student Number

10 marks 9 A public school is considering adding to the curriculum a course that it believes will improve the reading comprehension skills of its students The school has decided to conduct an experiment to evaluate the course before it is offered to the general student body Six students are randomly selected and their reading scores are recorded before and after the course These scores are composite measurements ofboth speed and comprehension (where higher scores indicate higher

h )Th I dblcompre enSlOn e resu ts are summanze e ow Student Number Score After Course Score Before Course

1 221 211 2 231 216 3 203 191 4 216 224 5 207 201 6 203 178

The school will implement the new course if there is sufficient evidence at the 5 significance level that the course improves reading comprehension

[2 m) (a) Which type of hypothesis test is appropriate for this situation and why

[6m (b) Conduct the hypothesis test of interest Include all steps requiredfor a Hypothesis Test

[2 m (c) List the assumption(s) required to use this type ofhypothesis test

Page 12 of 12 --END OF EXAM--shy

794 Appendix A Tables

TABLE IV Normal Curve Areas

o 1

00 01 bull02z 03 OS04 06 07 OS 09

0 0000 0040 0080 0120 01990160 0239 0279 0319 03591 0398 0438 0478 0517 05960557 0636 0675 0714 0753

07932 0832 0871 0910 0948 0987 1026 1064 1103 1141 3 1179 1217 1255 1293 13681331 1406 1443 1480 1517

15544 1591 1628 1664 1700 1736 1772 1808 1844 1879

1915oS 1950 1985 2019 2054 2088 2123 2157 2190 2224

22576 2291 2324 2357 24222389 2454 2486 2517 2549

25807 2611 2642 2673 2704 2734 2764 2794 2823 2852

28818 2910 2939 2967 2995 3023 3051 3078 3106 3133 3212 9 y3159 3186 3238 3264 3289 3315 3340 3365 3389

10 3413 3438 3461 3485 35313508 3554 3577 3599 3621 372911 3643 3665 3686 3708 3749 3770 3790 - 3810 3830

3849 386912 3888 3907 3925 3944 3962 3980 3997 4015

4032 4049 13 bulliQ6timiddot 4082 41154099 4131 4147 4162 4177 42071~m224192L4 4236 4251 4265 43194179 4292 4306

4332 4345 4357 4370 4382 4394middot 4406 4418 4429lS 4441 ltIJ

4452L6 4463 4474 4484 4495 4505 4515 4525 4535 4345

4554L7 4564 4573 4582 4591 4599 4608 4616 4625 4633

464118 4656 46644P49 4671 4678 4686 4693 47064699

4713 4761 L9 4719 4732 ~ 47674738 4744 4750 4726 4756 478820 4772 4793 4798 4893 4808 4812 4817~11amp 4783 shy

4821U 4826 4830 4834 48424838 48574846 4850 4854 486122 4864 4868 48874871 4875 4878 48904881 4884

23 4893 4896 4898 49164901 4913 4906 4909 49114904 24 4918 4922492Q 49364925shy 49294927 49344931 4932

4943493825 49524940 4941 4945 4946 49514948 4949 495326 4955 49644956 4957 4959 4960 4961 49634962 4965 497427 49674966 4968 4969 4970 49734971 4972 4974 498128 4975 4976 4977 4977 4978 49804979 4979

29 49864981 4982 4982 4983 4984 4984 49864985~49B5 30 4987 4987middot4987 4988 4988 4989 4990 49904989 4989 i

Source Abridged from Table I ofA Hald Statistical Table$ and Fonnulas (New YorkWiley) 1952 Reproduced by permission of A Hald

796 -~~ - -

o ~~~-TABLE VI Critical Values of t

f(t)

bull

Degrees of -Freedom tJWtlOO t050 tOO51010 tooi tooos

1 63143078 12706 31821 63657 31831 63662 2 1886 2920 4303 6965 9925 22326 31598 3 4541 1638 2353 3-182 5841 10213 12924 4 1533 2132 2776 3747 4604 7173 8610 5 1476 2015 2571 3365 4032 5893 6869 6 19431440 2447 3143 3707 5208 5959 7 1415 23651895 2998 3499 4785 5408

1397 1860 2306i 2896 3355 4501 5041 9 1383 1833 2262 2821 3250 4297 4781

10 18121372 2228 2764 3169 _ 4144 4587 U 1363 1796 2201 2718 3106 4025 4431 12 1356 1782 2179 shy 2681 3055 3930 4318

13~~ ~_ ~-i77113 i1reg 2650 3012 3852 4221 14 1345 21451761 2024 ~ 2977 41403787 15 1341 1753 2131 2602 29fl7 4073 16

373~ 1337 1746 2120 - 29212583 3686 4015

17 1333 1740 2110 2567 -2898 3646 3965 18 1330 1734 2101 t28782552 3610 3922 19 1328 t729 2093 28612539 38833579 - i

li~r~5I - 20 1725 2086 2528 2845 3850 21

3552 t 1323 2080 2518 1721 2831 3527 3819

22 20741321 1717 2508 2819 3505 3~792 23 1319 1714 2069 2500 3767 24

2807 3485 1318 1711 2064 2492 3467 3745

25 2middot797

1316 1708 2060 2485 3725 26

2787 3450 1315 1706 2056 2479 2779 3435 3707

27 1314 1703 2052 2473 2771 36903421 28 1313 170f 1 2048 2763 3674 29

~467 3408 1311 1699 2045 - 2462 2756 36593396

30 1310 1697 2D42 2457 - 2750 3646 40

3385 1303 1684 2021 2423 2704 3551

60 3307

1296 -16n 2000 2390 2660 3460 120

3232 1289 1658 1980 2358 2617 33733160

00 1645 1282 1960 2326 2576 3090 3291

Source 1his table is reproduced with the lcind permtsSioil of~e frustees of Biometrika from E S Pearson and R O Hartley (eds) The Biometrika Tables for Statisticians Vol 1 3d ed Biometrika 1966

r7~~

c

~ ApPENDIX A Tables 801

i~ 1lt TABLE IX Percentage Points of the F-distribution a = 05

~ liZ 1 2

1 1614 1995 2 1851 1900 3 1013 955 4 771 694

II

NUMERATOR DEGREES OF FREEDOM I11

3 4 9 I5 6 7 8

5

2157 2246 2302 2340 1916 1925 1930 1933

928 912 901 894 659 639 626 616

~ 661 579 541 519~ 505 495 6 599 514 476 453 439 428~7 559 474 435 412 397 387

5328 446 407 384 369 358 9 512 426 386 363 348 337~

10 -496 4100 371 348 333 322Q 11 484 7f~i 359398--~ 336 320 309

34938~middot12middot 475 326 300 ~~311~ 13 381467 341 318 303 292 0 fi

14 460 374 334 311 296 285 ~ 15 454 368 329 306 290foil 279 rr

449 363~ 16 324 301 285 274

~ 17 445 )20 296 281359 270 foil 18 441 3-16 293li))~5 277 266Q

438 352middot 313 290= 19 274 263 0 20 435 349 310 287 271 260 ~ 21 432 347 257307 284 268

i 22 430 344 305 282 266 255 303 0 23 428 342 280 264 253

Z 24 426 340 301 278 262 251foil Q 424middot25 339 299 276 260 249

26 423 337 298 274 259 247 27 421 335 296 273 257 246 28 420 334 295 271 245 256 29 418 333 293I 270 255 243 30 417 332 292 269 253 242 40 408 323 284 261 245 234 60 40q 315 276 253 237 225

120 307 392 268 245 229 217 00 384 300 260 237 221 210

2368 2389 1935 1937 889 885 609 604 488 482 421 415 379 373 350 344 329 323 314 307 301 295 291 285

283 277 271 1276 270 265 271 264 259 i266 259 254 1rII261 255 249

251 258 ~i-middot246 f 254 248 242 I

I 251 245 239 249 242 237 246 240 234 244 237 232 242 236 230 240 234 228 239 232 2]7 237 231 225 236 229 224 235 228 222 233 227 221

225 218 212 I

1217 210 204 209 202 196 201 194 188

2405 1938 881 600 477 410 368 339 318 302 290 280 ji

Source From M Merrington and eM Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 19433373-88

i (continued) I

I

I iii 11 iii

II I

I l

iINi II

802 Appendix A Tables

TABLE IX Continued

~1 NUMERATOR DEGREES OF FREEDOM Vl ~r--l-0----12----15-----r--2-0----24---r--30----40----60~--~- ~ OQ

1 2 3 4 S 6 7 8 9

~ 10 11

~ ~ ~ 14 o 15 rIJ (

Iiil 16 ~ 17 ~ 18 ~ 19 = 20

~ 21

i ii shy~ 24 ~ 2S

26 27 28 29 30

middot40 60

120 OQ

2419 1940 879 596 474 406 364 335 ~314

298 285 275 267 260 254 249 245 241 238 235 232 230 227 225 224 222 220 219 218 216

208 199 191 1~

2439 2459 2480 2491 1941 1943 1945 1945 874 870 866 864 591 586 580 577 468 462 456 453 400 394 387 384 357 351 344 341 328 322 315 312 307 301 294 290 291 285 277 274 279 272 265 261 269 262 254 251 260 253 246 242 253 246 239 235 248 240 233 229 242 235 228 224 238 231 223 219 234 227 219 215 231 223 216 211 iJi ~r~ 220 212 208 225 218 210 205 223 215 207 203 220 213 205 201 218 211 203 198 216 209 201 196

215 2Q7 199 195j 2i3 206 197 193 212 204 196 191 210 203 194 190 209 201 193 189

200 192184 179 192 184 175 170 183 i 175 166 161 U5 1m 1~ 1~

2501 1946 862 575 450 381 33S 308 286 270 257 247

238 231 225 219 215 211 207 204 201 198 196 194 192 190 188 187 185 184

174 165 155 146

2511 Imiddotmiddotmiddot

1947 859 572 446 377middot 334 304 283 266 253 243 234 227 220 215 210 206 203

~199 196 f 194 1~1 189 187 185 184 182 181 179

169 159 150 139

2522 1948 857 5~69

443 374 330 301 279 262 249 238 230 222

216 211 206 202 198 195 192 189 186 184 182 180 179 177 175 174

164 153 143 132

2533 2543 1949 1950 855 853 566 563 440 436 370 367 327 323 297 293 275 271 258 254 245 240 234 230 225 221 218 213 211 207 206 201 201 196 197 192 193 188 190 184 1~7 181 184 178

181 176 179 173 177 ~ 171

169 17~ 173 167 171 165

164170 162168

151158 147 139

125135 100122 i

ApPENDIX A Tables~c63 y

TABLE X Percentage Points of the F-distribution a = 025

f(F)

o FfJ25

- -NUMERATOR DEGREES OF FREEDOM ~ Jz 1 2 3 4 5 6 7 8 9

1 6478 7995 8642 8996 9218 9371 9482 9567 9633 2 3851 3900 3917 3925 3930 3933 3936 3937 3939 3 1744 1604 1544 1510 1488 1473 1462 1454 1447 4 1222 1065 998 960 936 920 907 898 890 5 1001 843 776 739 715 698 685 676 668 ( 881 726 660 623 599 582 570 560 552

589 7 807 654 552 529 512 499 490 482 8 757 606 542 505 482 465 453 443 436 9 721 571 508 472 448 432 420 410 403

gt1 10 - 694 483 447 424 407 395 385 378-4~~i 0 003f2~~ 11 672 463 428 404 88 376 366 359fIl

12- 655 510 447 412 389 373 351 344~61f l~ 13 641 497 435 400 377 360 348 339 331

0 ~ 14 630 486 424 389 366 329338-~$P 321 IIl 15 620 477 A15 380 358 34-1 329 320 312fIl

16 61_2 469 408 373 350 334 322 305 31~r-~ ~ 17 462- 4016Cl1) 366 344 328 316 298I j r~ 30(hmiddotfIl 45618 598 395 361 338 322 310 301 293~

592 451 390 356 333= 19 317 305 296 288

~ 20 587 446 386 351 329 313 301 291 284 21 583 442 382 348 325 309 297 287 280 22 - 579 438 378 344 322 305 293 284 276~

0 23 575 435 375 341 318 302 290 281 273 ~ 24 572 432 372 338 315 299 287 278 270 ~ 25 569 429 369 335 313 297 285 275 268

26 566 427 367 333 310 294 282 273 265 27 563 424 365 331 - 292 308 280 271 263 28 561 422 363 329 306 290 278 269 261 29 559 420 361 327 304 288 276 267 259 30 557 418 359 325 303 287 275 265 257

40 542 405 346 290 313 274 262 253 245 60 529 393 334 301 279 263 251 241 233

120 515 380 323 289 267 252 239 230 222 00 502 369 312 279 257 241 229 219 211

SourceFrom M Merrington and C M Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 1943337388

(continued)

4 804 Appendix A Tables

TABLE X Continued

~Vl NUMERATOR DEGREES OF FREEDOM V2~r---l-0--r--U--~-~~--~~20-----~~---~------~~middotmiddot----6-0-----U~O-----~---

9686 9767 9849 9931 9972 1001 1006 _1010 1~014-1 1018 2 3940 3941 3943 3945 3946 3946 3947 3948 394~ 3950 3 1442 1434 1425 1417 1412 1408 1404 1399 1395 1390

884 875 866 856 851 846 841 836 8314 826 662 652 643 633 628 623 618 612 6075 602 546 537 527 517 512 507 501 496 4906 485

7 476 467 457 447 442 436 431 425 420 414 8 430 420 410 400 395 389 384 378 373 367 9 396 387 377 367 361 356 351 345 339 333

372 362 352 342 337 331 326 320 314 308S 10 353 343 333 323 317 312 306 300 294Q 11 288 337 328 318 307 302 296 291 285 279 272 325 315 305 295 289 284 278 272 266 260E ~

)i 14 315 305 295 284 279 273 267 261 255 249 o 15 306 296 286 276 270 264 259 252 246 240rIl ~ 16 232299 289 279 268 263 257 251 245 238 ~ ~17 292 282 272 262 256 250 244 238 232 225

287 277 267 256 250 244 238 232 226~ 18 219 Q 19 282 272 262 251 245 239 233 227 220 21~=

277 268 257 246 241 235 229 222 216 209~ 20middot 273 bull =I~~~ 253 242 237 231 ~ 225 218 211lt 21 204 270 middot260 250 239 233 227 21 214 208 200~ 22 267 257 247 236 230 224 $ 218 211 204 197o 23

z 24 264 254 244 D3 227 221 ~ Ii 215 208 201 194 261 251 241 230 224 218 212 205 198 191~ 25

26 259 249 239 228 222 216 209 203 )95 I 188 27 257 247 236 225 219 213 207 200 i~~3 185 28 2d5tf i 245middot 234 223217 211 205 198 191 183 29 253 243 232 221 215 209 203 196 189 181 30 251 241 231 220 214 207 201 194 187 179

40 164239 229 218 207 201 194 188 180 172 60 227 217 206 194 188 182 174 167 158 148

120 216 205 194 182 176 169 161 153 143 131 I

QO 100205 194 I 183 171 I 164 157 148 139 127

Page 11: STA220_Final_2012S.pdf

Full Name Student Number

[10marksj 8 You and your friend playa dice-rolling game You each roll a fair die once and observe the

number that appears on the up face for each person Ifboth ofyou get the same number nobody wins or loses any money If you get different numbers whoever gets the lower number has to pay the other person $1 Let X represent your winnings (in $) from this game

[4 m] (a) Find the probability distribution for X and display it in table form

[3 m] (b) Calculate the mean ofX and interpret your answer in terms of the game Show your

calculations and then circle your final answer

[3 m] (c) Compute the variance ofX Show your work and circle thefinal answer

Page 11 of 12

Full Name Student Number

10 marks 9 A public school is considering adding to the curriculum a course that it believes will improve the reading comprehension skills of its students The school has decided to conduct an experiment to evaluate the course before it is offered to the general student body Six students are randomly selected and their reading scores are recorded before and after the course These scores are composite measurements ofboth speed and comprehension (where higher scores indicate higher

h )Th I dblcompre enSlOn e resu ts are summanze e ow Student Number Score After Course Score Before Course

1 221 211 2 231 216 3 203 191 4 216 224 5 207 201 6 203 178

The school will implement the new course if there is sufficient evidence at the 5 significance level that the course improves reading comprehension

[2 m) (a) Which type of hypothesis test is appropriate for this situation and why

[6m (b) Conduct the hypothesis test of interest Include all steps requiredfor a Hypothesis Test

[2 m (c) List the assumption(s) required to use this type ofhypothesis test

Page 12 of 12 --END OF EXAM--shy

794 Appendix A Tables

TABLE IV Normal Curve Areas

o 1

00 01 bull02z 03 OS04 06 07 OS 09

0 0000 0040 0080 0120 01990160 0239 0279 0319 03591 0398 0438 0478 0517 05960557 0636 0675 0714 0753

07932 0832 0871 0910 0948 0987 1026 1064 1103 1141 3 1179 1217 1255 1293 13681331 1406 1443 1480 1517

15544 1591 1628 1664 1700 1736 1772 1808 1844 1879

1915oS 1950 1985 2019 2054 2088 2123 2157 2190 2224

22576 2291 2324 2357 24222389 2454 2486 2517 2549

25807 2611 2642 2673 2704 2734 2764 2794 2823 2852

28818 2910 2939 2967 2995 3023 3051 3078 3106 3133 3212 9 y3159 3186 3238 3264 3289 3315 3340 3365 3389

10 3413 3438 3461 3485 35313508 3554 3577 3599 3621 372911 3643 3665 3686 3708 3749 3770 3790 - 3810 3830

3849 386912 3888 3907 3925 3944 3962 3980 3997 4015

4032 4049 13 bulliQ6timiddot 4082 41154099 4131 4147 4162 4177 42071~m224192L4 4236 4251 4265 43194179 4292 4306

4332 4345 4357 4370 4382 4394middot 4406 4418 4429lS 4441 ltIJ

4452L6 4463 4474 4484 4495 4505 4515 4525 4535 4345

4554L7 4564 4573 4582 4591 4599 4608 4616 4625 4633

464118 4656 46644P49 4671 4678 4686 4693 47064699

4713 4761 L9 4719 4732 ~ 47674738 4744 4750 4726 4756 478820 4772 4793 4798 4893 4808 4812 4817~11amp 4783 shy

4821U 4826 4830 4834 48424838 48574846 4850 4854 486122 4864 4868 48874871 4875 4878 48904881 4884

23 4893 4896 4898 49164901 4913 4906 4909 49114904 24 4918 4922492Q 49364925shy 49294927 49344931 4932

4943493825 49524940 4941 4945 4946 49514948 4949 495326 4955 49644956 4957 4959 4960 4961 49634962 4965 497427 49674966 4968 4969 4970 49734971 4972 4974 498128 4975 4976 4977 4977 4978 49804979 4979

29 49864981 4982 4982 4983 4984 4984 49864985~49B5 30 4987 4987middot4987 4988 4988 4989 4990 49904989 4989 i

Source Abridged from Table I ofA Hald Statistical Table$ and Fonnulas (New YorkWiley) 1952 Reproduced by permission of A Hald

796 -~~ - -

o ~~~-TABLE VI Critical Values of t

f(t)

bull

Degrees of -Freedom tJWtlOO t050 tOO51010 tooi tooos

1 63143078 12706 31821 63657 31831 63662 2 1886 2920 4303 6965 9925 22326 31598 3 4541 1638 2353 3-182 5841 10213 12924 4 1533 2132 2776 3747 4604 7173 8610 5 1476 2015 2571 3365 4032 5893 6869 6 19431440 2447 3143 3707 5208 5959 7 1415 23651895 2998 3499 4785 5408

1397 1860 2306i 2896 3355 4501 5041 9 1383 1833 2262 2821 3250 4297 4781

10 18121372 2228 2764 3169 _ 4144 4587 U 1363 1796 2201 2718 3106 4025 4431 12 1356 1782 2179 shy 2681 3055 3930 4318

13~~ ~_ ~-i77113 i1reg 2650 3012 3852 4221 14 1345 21451761 2024 ~ 2977 41403787 15 1341 1753 2131 2602 29fl7 4073 16

373~ 1337 1746 2120 - 29212583 3686 4015

17 1333 1740 2110 2567 -2898 3646 3965 18 1330 1734 2101 t28782552 3610 3922 19 1328 t729 2093 28612539 38833579 - i

li~r~5I - 20 1725 2086 2528 2845 3850 21

3552 t 1323 2080 2518 1721 2831 3527 3819

22 20741321 1717 2508 2819 3505 3~792 23 1319 1714 2069 2500 3767 24

2807 3485 1318 1711 2064 2492 3467 3745

25 2middot797

1316 1708 2060 2485 3725 26

2787 3450 1315 1706 2056 2479 2779 3435 3707

27 1314 1703 2052 2473 2771 36903421 28 1313 170f 1 2048 2763 3674 29

~467 3408 1311 1699 2045 - 2462 2756 36593396

30 1310 1697 2D42 2457 - 2750 3646 40

3385 1303 1684 2021 2423 2704 3551

60 3307

1296 -16n 2000 2390 2660 3460 120

3232 1289 1658 1980 2358 2617 33733160

00 1645 1282 1960 2326 2576 3090 3291

Source 1his table is reproduced with the lcind permtsSioil of~e frustees of Biometrika from E S Pearson and R O Hartley (eds) The Biometrika Tables for Statisticians Vol 1 3d ed Biometrika 1966

r7~~

c

~ ApPENDIX A Tables 801

i~ 1lt TABLE IX Percentage Points of the F-distribution a = 05

~ liZ 1 2

1 1614 1995 2 1851 1900 3 1013 955 4 771 694

II

NUMERATOR DEGREES OF FREEDOM I11

3 4 9 I5 6 7 8

5

2157 2246 2302 2340 1916 1925 1930 1933

928 912 901 894 659 639 626 616

~ 661 579 541 519~ 505 495 6 599 514 476 453 439 428~7 559 474 435 412 397 387

5328 446 407 384 369 358 9 512 426 386 363 348 337~

10 -496 4100 371 348 333 322Q 11 484 7f~i 359398--~ 336 320 309

34938~middot12middot 475 326 300 ~~311~ 13 381467 341 318 303 292 0 fi

14 460 374 334 311 296 285 ~ 15 454 368 329 306 290foil 279 rr

449 363~ 16 324 301 285 274

~ 17 445 )20 296 281359 270 foil 18 441 3-16 293li))~5 277 266Q

438 352middot 313 290= 19 274 263 0 20 435 349 310 287 271 260 ~ 21 432 347 257307 284 268

i 22 430 344 305 282 266 255 303 0 23 428 342 280 264 253

Z 24 426 340 301 278 262 251foil Q 424middot25 339 299 276 260 249

26 423 337 298 274 259 247 27 421 335 296 273 257 246 28 420 334 295 271 245 256 29 418 333 293I 270 255 243 30 417 332 292 269 253 242 40 408 323 284 261 245 234 60 40q 315 276 253 237 225

120 307 392 268 245 229 217 00 384 300 260 237 221 210

2368 2389 1935 1937 889 885 609 604 488 482 421 415 379 373 350 344 329 323 314 307 301 295 291 285

283 277 271 1276 270 265 271 264 259 i266 259 254 1rII261 255 249

251 258 ~i-middot246 f 254 248 242 I

I 251 245 239 249 242 237 246 240 234 244 237 232 242 236 230 240 234 228 239 232 2]7 237 231 225 236 229 224 235 228 222 233 227 221

225 218 212 I

1217 210 204 209 202 196 201 194 188

2405 1938 881 600 477 410 368 339 318 302 290 280 ji

Source From M Merrington and eM Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 19433373-88

i (continued) I

I

I iii 11 iii

II I

I l

iINi II

802 Appendix A Tables

TABLE IX Continued

~1 NUMERATOR DEGREES OF FREEDOM Vl ~r--l-0----12----15-----r--2-0----24---r--30----40----60~--~- ~ OQ

1 2 3 4 S 6 7 8 9

~ 10 11

~ ~ ~ 14 o 15 rIJ (

Iiil 16 ~ 17 ~ 18 ~ 19 = 20

~ 21

i ii shy~ 24 ~ 2S

26 27 28 29 30

middot40 60

120 OQ

2419 1940 879 596 474 406 364 335 ~314

298 285 275 267 260 254 249 245 241 238 235 232 230 227 225 224 222 220 219 218 216

208 199 191 1~

2439 2459 2480 2491 1941 1943 1945 1945 874 870 866 864 591 586 580 577 468 462 456 453 400 394 387 384 357 351 344 341 328 322 315 312 307 301 294 290 291 285 277 274 279 272 265 261 269 262 254 251 260 253 246 242 253 246 239 235 248 240 233 229 242 235 228 224 238 231 223 219 234 227 219 215 231 223 216 211 iJi ~r~ 220 212 208 225 218 210 205 223 215 207 203 220 213 205 201 218 211 203 198 216 209 201 196

215 2Q7 199 195j 2i3 206 197 193 212 204 196 191 210 203 194 190 209 201 193 189

200 192184 179 192 184 175 170 183 i 175 166 161 U5 1m 1~ 1~

2501 1946 862 575 450 381 33S 308 286 270 257 247

238 231 225 219 215 211 207 204 201 198 196 194 192 190 188 187 185 184

174 165 155 146

2511 Imiddotmiddotmiddot

1947 859 572 446 377middot 334 304 283 266 253 243 234 227 220 215 210 206 203

~199 196 f 194 1~1 189 187 185 184 182 181 179

169 159 150 139

2522 1948 857 5~69

443 374 330 301 279 262 249 238 230 222

216 211 206 202 198 195 192 189 186 184 182 180 179 177 175 174

164 153 143 132

2533 2543 1949 1950 855 853 566 563 440 436 370 367 327 323 297 293 275 271 258 254 245 240 234 230 225 221 218 213 211 207 206 201 201 196 197 192 193 188 190 184 1~7 181 184 178

181 176 179 173 177 ~ 171

169 17~ 173 167 171 165

164170 162168

151158 147 139

125135 100122 i

ApPENDIX A Tables~c63 y

TABLE X Percentage Points of the F-distribution a = 025

f(F)

o FfJ25

- -NUMERATOR DEGREES OF FREEDOM ~ Jz 1 2 3 4 5 6 7 8 9

1 6478 7995 8642 8996 9218 9371 9482 9567 9633 2 3851 3900 3917 3925 3930 3933 3936 3937 3939 3 1744 1604 1544 1510 1488 1473 1462 1454 1447 4 1222 1065 998 960 936 920 907 898 890 5 1001 843 776 739 715 698 685 676 668 ( 881 726 660 623 599 582 570 560 552

589 7 807 654 552 529 512 499 490 482 8 757 606 542 505 482 465 453 443 436 9 721 571 508 472 448 432 420 410 403

gt1 10 - 694 483 447 424 407 395 385 378-4~~i 0 003f2~~ 11 672 463 428 404 88 376 366 359fIl

12- 655 510 447 412 389 373 351 344~61f l~ 13 641 497 435 400 377 360 348 339 331

0 ~ 14 630 486 424 389 366 329338-~$P 321 IIl 15 620 477 A15 380 358 34-1 329 320 312fIl

16 61_2 469 408 373 350 334 322 305 31~r-~ ~ 17 462- 4016Cl1) 366 344 328 316 298I j r~ 30(hmiddotfIl 45618 598 395 361 338 322 310 301 293~

592 451 390 356 333= 19 317 305 296 288

~ 20 587 446 386 351 329 313 301 291 284 21 583 442 382 348 325 309 297 287 280 22 - 579 438 378 344 322 305 293 284 276~

0 23 575 435 375 341 318 302 290 281 273 ~ 24 572 432 372 338 315 299 287 278 270 ~ 25 569 429 369 335 313 297 285 275 268

26 566 427 367 333 310 294 282 273 265 27 563 424 365 331 - 292 308 280 271 263 28 561 422 363 329 306 290 278 269 261 29 559 420 361 327 304 288 276 267 259 30 557 418 359 325 303 287 275 265 257

40 542 405 346 290 313 274 262 253 245 60 529 393 334 301 279 263 251 241 233

120 515 380 323 289 267 252 239 230 222 00 502 369 312 279 257 241 229 219 211

SourceFrom M Merrington and C M Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 1943337388

(continued)

4 804 Appendix A Tables

TABLE X Continued

~Vl NUMERATOR DEGREES OF FREEDOM V2~r---l-0--r--U--~-~~--~~20-----~~---~------~~middotmiddot----6-0-----U~O-----~---

9686 9767 9849 9931 9972 1001 1006 _1010 1~014-1 1018 2 3940 3941 3943 3945 3946 3946 3947 3948 394~ 3950 3 1442 1434 1425 1417 1412 1408 1404 1399 1395 1390

884 875 866 856 851 846 841 836 8314 826 662 652 643 633 628 623 618 612 6075 602 546 537 527 517 512 507 501 496 4906 485

7 476 467 457 447 442 436 431 425 420 414 8 430 420 410 400 395 389 384 378 373 367 9 396 387 377 367 361 356 351 345 339 333

372 362 352 342 337 331 326 320 314 308S 10 353 343 333 323 317 312 306 300 294Q 11 288 337 328 318 307 302 296 291 285 279 272 325 315 305 295 289 284 278 272 266 260E ~

)i 14 315 305 295 284 279 273 267 261 255 249 o 15 306 296 286 276 270 264 259 252 246 240rIl ~ 16 232299 289 279 268 263 257 251 245 238 ~ ~17 292 282 272 262 256 250 244 238 232 225

287 277 267 256 250 244 238 232 226~ 18 219 Q 19 282 272 262 251 245 239 233 227 220 21~=

277 268 257 246 241 235 229 222 216 209~ 20middot 273 bull =I~~~ 253 242 237 231 ~ 225 218 211lt 21 204 270 middot260 250 239 233 227 21 214 208 200~ 22 267 257 247 236 230 224 $ 218 211 204 197o 23

z 24 264 254 244 D3 227 221 ~ Ii 215 208 201 194 261 251 241 230 224 218 212 205 198 191~ 25

26 259 249 239 228 222 216 209 203 )95 I 188 27 257 247 236 225 219 213 207 200 i~~3 185 28 2d5tf i 245middot 234 223217 211 205 198 191 183 29 253 243 232 221 215 209 203 196 189 181 30 251 241 231 220 214 207 201 194 187 179

40 164239 229 218 207 201 194 188 180 172 60 227 217 206 194 188 182 174 167 158 148

120 216 205 194 182 176 169 161 153 143 131 I

QO 100205 194 I 183 171 I 164 157 148 139 127

Page 12: STA220_Final_2012S.pdf

Full Name Student Number

10 marks 9 A public school is considering adding to the curriculum a course that it believes will improve the reading comprehension skills of its students The school has decided to conduct an experiment to evaluate the course before it is offered to the general student body Six students are randomly selected and their reading scores are recorded before and after the course These scores are composite measurements ofboth speed and comprehension (where higher scores indicate higher

h )Th I dblcompre enSlOn e resu ts are summanze e ow Student Number Score After Course Score Before Course

1 221 211 2 231 216 3 203 191 4 216 224 5 207 201 6 203 178

The school will implement the new course if there is sufficient evidence at the 5 significance level that the course improves reading comprehension

[2 m) (a) Which type of hypothesis test is appropriate for this situation and why

[6m (b) Conduct the hypothesis test of interest Include all steps requiredfor a Hypothesis Test

[2 m (c) List the assumption(s) required to use this type ofhypothesis test

Page 12 of 12 --END OF EXAM--shy

794 Appendix A Tables

TABLE IV Normal Curve Areas

o 1

00 01 bull02z 03 OS04 06 07 OS 09

0 0000 0040 0080 0120 01990160 0239 0279 0319 03591 0398 0438 0478 0517 05960557 0636 0675 0714 0753

07932 0832 0871 0910 0948 0987 1026 1064 1103 1141 3 1179 1217 1255 1293 13681331 1406 1443 1480 1517

15544 1591 1628 1664 1700 1736 1772 1808 1844 1879

1915oS 1950 1985 2019 2054 2088 2123 2157 2190 2224

22576 2291 2324 2357 24222389 2454 2486 2517 2549

25807 2611 2642 2673 2704 2734 2764 2794 2823 2852

28818 2910 2939 2967 2995 3023 3051 3078 3106 3133 3212 9 y3159 3186 3238 3264 3289 3315 3340 3365 3389

10 3413 3438 3461 3485 35313508 3554 3577 3599 3621 372911 3643 3665 3686 3708 3749 3770 3790 - 3810 3830

3849 386912 3888 3907 3925 3944 3962 3980 3997 4015

4032 4049 13 bulliQ6timiddot 4082 41154099 4131 4147 4162 4177 42071~m224192L4 4236 4251 4265 43194179 4292 4306

4332 4345 4357 4370 4382 4394middot 4406 4418 4429lS 4441 ltIJ

4452L6 4463 4474 4484 4495 4505 4515 4525 4535 4345

4554L7 4564 4573 4582 4591 4599 4608 4616 4625 4633

464118 4656 46644P49 4671 4678 4686 4693 47064699

4713 4761 L9 4719 4732 ~ 47674738 4744 4750 4726 4756 478820 4772 4793 4798 4893 4808 4812 4817~11amp 4783 shy

4821U 4826 4830 4834 48424838 48574846 4850 4854 486122 4864 4868 48874871 4875 4878 48904881 4884

23 4893 4896 4898 49164901 4913 4906 4909 49114904 24 4918 4922492Q 49364925shy 49294927 49344931 4932

4943493825 49524940 4941 4945 4946 49514948 4949 495326 4955 49644956 4957 4959 4960 4961 49634962 4965 497427 49674966 4968 4969 4970 49734971 4972 4974 498128 4975 4976 4977 4977 4978 49804979 4979

29 49864981 4982 4982 4983 4984 4984 49864985~49B5 30 4987 4987middot4987 4988 4988 4989 4990 49904989 4989 i

Source Abridged from Table I ofA Hald Statistical Table$ and Fonnulas (New YorkWiley) 1952 Reproduced by permission of A Hald

796 -~~ - -

o ~~~-TABLE VI Critical Values of t

f(t)

bull

Degrees of -Freedom tJWtlOO t050 tOO51010 tooi tooos

1 63143078 12706 31821 63657 31831 63662 2 1886 2920 4303 6965 9925 22326 31598 3 4541 1638 2353 3-182 5841 10213 12924 4 1533 2132 2776 3747 4604 7173 8610 5 1476 2015 2571 3365 4032 5893 6869 6 19431440 2447 3143 3707 5208 5959 7 1415 23651895 2998 3499 4785 5408

1397 1860 2306i 2896 3355 4501 5041 9 1383 1833 2262 2821 3250 4297 4781

10 18121372 2228 2764 3169 _ 4144 4587 U 1363 1796 2201 2718 3106 4025 4431 12 1356 1782 2179 shy 2681 3055 3930 4318

13~~ ~_ ~-i77113 i1reg 2650 3012 3852 4221 14 1345 21451761 2024 ~ 2977 41403787 15 1341 1753 2131 2602 29fl7 4073 16

373~ 1337 1746 2120 - 29212583 3686 4015

17 1333 1740 2110 2567 -2898 3646 3965 18 1330 1734 2101 t28782552 3610 3922 19 1328 t729 2093 28612539 38833579 - i

li~r~5I - 20 1725 2086 2528 2845 3850 21

3552 t 1323 2080 2518 1721 2831 3527 3819

22 20741321 1717 2508 2819 3505 3~792 23 1319 1714 2069 2500 3767 24

2807 3485 1318 1711 2064 2492 3467 3745

25 2middot797

1316 1708 2060 2485 3725 26

2787 3450 1315 1706 2056 2479 2779 3435 3707

27 1314 1703 2052 2473 2771 36903421 28 1313 170f 1 2048 2763 3674 29

~467 3408 1311 1699 2045 - 2462 2756 36593396

30 1310 1697 2D42 2457 - 2750 3646 40

3385 1303 1684 2021 2423 2704 3551

60 3307

1296 -16n 2000 2390 2660 3460 120

3232 1289 1658 1980 2358 2617 33733160

00 1645 1282 1960 2326 2576 3090 3291

Source 1his table is reproduced with the lcind permtsSioil of~e frustees of Biometrika from E S Pearson and R O Hartley (eds) The Biometrika Tables for Statisticians Vol 1 3d ed Biometrika 1966

r7~~

c

~ ApPENDIX A Tables 801

i~ 1lt TABLE IX Percentage Points of the F-distribution a = 05

~ liZ 1 2

1 1614 1995 2 1851 1900 3 1013 955 4 771 694

II

NUMERATOR DEGREES OF FREEDOM I11

3 4 9 I5 6 7 8

5

2157 2246 2302 2340 1916 1925 1930 1933

928 912 901 894 659 639 626 616

~ 661 579 541 519~ 505 495 6 599 514 476 453 439 428~7 559 474 435 412 397 387

5328 446 407 384 369 358 9 512 426 386 363 348 337~

10 -496 4100 371 348 333 322Q 11 484 7f~i 359398--~ 336 320 309

34938~middot12middot 475 326 300 ~~311~ 13 381467 341 318 303 292 0 fi

14 460 374 334 311 296 285 ~ 15 454 368 329 306 290foil 279 rr

449 363~ 16 324 301 285 274

~ 17 445 )20 296 281359 270 foil 18 441 3-16 293li))~5 277 266Q

438 352middot 313 290= 19 274 263 0 20 435 349 310 287 271 260 ~ 21 432 347 257307 284 268

i 22 430 344 305 282 266 255 303 0 23 428 342 280 264 253

Z 24 426 340 301 278 262 251foil Q 424middot25 339 299 276 260 249

26 423 337 298 274 259 247 27 421 335 296 273 257 246 28 420 334 295 271 245 256 29 418 333 293I 270 255 243 30 417 332 292 269 253 242 40 408 323 284 261 245 234 60 40q 315 276 253 237 225

120 307 392 268 245 229 217 00 384 300 260 237 221 210

2368 2389 1935 1937 889 885 609 604 488 482 421 415 379 373 350 344 329 323 314 307 301 295 291 285

283 277 271 1276 270 265 271 264 259 i266 259 254 1rII261 255 249

251 258 ~i-middot246 f 254 248 242 I

I 251 245 239 249 242 237 246 240 234 244 237 232 242 236 230 240 234 228 239 232 2]7 237 231 225 236 229 224 235 228 222 233 227 221

225 218 212 I

1217 210 204 209 202 196 201 194 188

2405 1938 881 600 477 410 368 339 318 302 290 280 ji

Source From M Merrington and eM Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 19433373-88

i (continued) I

I

I iii 11 iii

II I

I l

iINi II

802 Appendix A Tables

TABLE IX Continued

~1 NUMERATOR DEGREES OF FREEDOM Vl ~r--l-0----12----15-----r--2-0----24---r--30----40----60~--~- ~ OQ

1 2 3 4 S 6 7 8 9

~ 10 11

~ ~ ~ 14 o 15 rIJ (

Iiil 16 ~ 17 ~ 18 ~ 19 = 20

~ 21

i ii shy~ 24 ~ 2S

26 27 28 29 30

middot40 60

120 OQ

2419 1940 879 596 474 406 364 335 ~314

298 285 275 267 260 254 249 245 241 238 235 232 230 227 225 224 222 220 219 218 216

208 199 191 1~

2439 2459 2480 2491 1941 1943 1945 1945 874 870 866 864 591 586 580 577 468 462 456 453 400 394 387 384 357 351 344 341 328 322 315 312 307 301 294 290 291 285 277 274 279 272 265 261 269 262 254 251 260 253 246 242 253 246 239 235 248 240 233 229 242 235 228 224 238 231 223 219 234 227 219 215 231 223 216 211 iJi ~r~ 220 212 208 225 218 210 205 223 215 207 203 220 213 205 201 218 211 203 198 216 209 201 196

215 2Q7 199 195j 2i3 206 197 193 212 204 196 191 210 203 194 190 209 201 193 189

200 192184 179 192 184 175 170 183 i 175 166 161 U5 1m 1~ 1~

2501 1946 862 575 450 381 33S 308 286 270 257 247

238 231 225 219 215 211 207 204 201 198 196 194 192 190 188 187 185 184

174 165 155 146

2511 Imiddotmiddotmiddot

1947 859 572 446 377middot 334 304 283 266 253 243 234 227 220 215 210 206 203

~199 196 f 194 1~1 189 187 185 184 182 181 179

169 159 150 139

2522 1948 857 5~69

443 374 330 301 279 262 249 238 230 222

216 211 206 202 198 195 192 189 186 184 182 180 179 177 175 174

164 153 143 132

2533 2543 1949 1950 855 853 566 563 440 436 370 367 327 323 297 293 275 271 258 254 245 240 234 230 225 221 218 213 211 207 206 201 201 196 197 192 193 188 190 184 1~7 181 184 178

181 176 179 173 177 ~ 171

169 17~ 173 167 171 165

164170 162168

151158 147 139

125135 100122 i

ApPENDIX A Tables~c63 y

TABLE X Percentage Points of the F-distribution a = 025

f(F)

o FfJ25

- -NUMERATOR DEGREES OF FREEDOM ~ Jz 1 2 3 4 5 6 7 8 9

1 6478 7995 8642 8996 9218 9371 9482 9567 9633 2 3851 3900 3917 3925 3930 3933 3936 3937 3939 3 1744 1604 1544 1510 1488 1473 1462 1454 1447 4 1222 1065 998 960 936 920 907 898 890 5 1001 843 776 739 715 698 685 676 668 ( 881 726 660 623 599 582 570 560 552

589 7 807 654 552 529 512 499 490 482 8 757 606 542 505 482 465 453 443 436 9 721 571 508 472 448 432 420 410 403

gt1 10 - 694 483 447 424 407 395 385 378-4~~i 0 003f2~~ 11 672 463 428 404 88 376 366 359fIl

12- 655 510 447 412 389 373 351 344~61f l~ 13 641 497 435 400 377 360 348 339 331

0 ~ 14 630 486 424 389 366 329338-~$P 321 IIl 15 620 477 A15 380 358 34-1 329 320 312fIl

16 61_2 469 408 373 350 334 322 305 31~r-~ ~ 17 462- 4016Cl1) 366 344 328 316 298I j r~ 30(hmiddotfIl 45618 598 395 361 338 322 310 301 293~

592 451 390 356 333= 19 317 305 296 288

~ 20 587 446 386 351 329 313 301 291 284 21 583 442 382 348 325 309 297 287 280 22 - 579 438 378 344 322 305 293 284 276~

0 23 575 435 375 341 318 302 290 281 273 ~ 24 572 432 372 338 315 299 287 278 270 ~ 25 569 429 369 335 313 297 285 275 268

26 566 427 367 333 310 294 282 273 265 27 563 424 365 331 - 292 308 280 271 263 28 561 422 363 329 306 290 278 269 261 29 559 420 361 327 304 288 276 267 259 30 557 418 359 325 303 287 275 265 257

40 542 405 346 290 313 274 262 253 245 60 529 393 334 301 279 263 251 241 233

120 515 380 323 289 267 252 239 230 222 00 502 369 312 279 257 241 229 219 211

SourceFrom M Merrington and C M Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 1943337388

(continued)

4 804 Appendix A Tables

TABLE X Continued

~Vl NUMERATOR DEGREES OF FREEDOM V2~r---l-0--r--U--~-~~--~~20-----~~---~------~~middotmiddot----6-0-----U~O-----~---

9686 9767 9849 9931 9972 1001 1006 _1010 1~014-1 1018 2 3940 3941 3943 3945 3946 3946 3947 3948 394~ 3950 3 1442 1434 1425 1417 1412 1408 1404 1399 1395 1390

884 875 866 856 851 846 841 836 8314 826 662 652 643 633 628 623 618 612 6075 602 546 537 527 517 512 507 501 496 4906 485

7 476 467 457 447 442 436 431 425 420 414 8 430 420 410 400 395 389 384 378 373 367 9 396 387 377 367 361 356 351 345 339 333

372 362 352 342 337 331 326 320 314 308S 10 353 343 333 323 317 312 306 300 294Q 11 288 337 328 318 307 302 296 291 285 279 272 325 315 305 295 289 284 278 272 266 260E ~

)i 14 315 305 295 284 279 273 267 261 255 249 o 15 306 296 286 276 270 264 259 252 246 240rIl ~ 16 232299 289 279 268 263 257 251 245 238 ~ ~17 292 282 272 262 256 250 244 238 232 225

287 277 267 256 250 244 238 232 226~ 18 219 Q 19 282 272 262 251 245 239 233 227 220 21~=

277 268 257 246 241 235 229 222 216 209~ 20middot 273 bull =I~~~ 253 242 237 231 ~ 225 218 211lt 21 204 270 middot260 250 239 233 227 21 214 208 200~ 22 267 257 247 236 230 224 $ 218 211 204 197o 23

z 24 264 254 244 D3 227 221 ~ Ii 215 208 201 194 261 251 241 230 224 218 212 205 198 191~ 25

26 259 249 239 228 222 216 209 203 )95 I 188 27 257 247 236 225 219 213 207 200 i~~3 185 28 2d5tf i 245middot 234 223217 211 205 198 191 183 29 253 243 232 221 215 209 203 196 189 181 30 251 241 231 220 214 207 201 194 187 179

40 164239 229 218 207 201 194 188 180 172 60 227 217 206 194 188 182 174 167 158 148

120 216 205 194 182 176 169 161 153 143 131 I

QO 100205 194 I 183 171 I 164 157 148 139 127

Page 13: STA220_Final_2012S.pdf

794 Appendix A Tables

TABLE IV Normal Curve Areas

o 1

00 01 bull02z 03 OS04 06 07 OS 09

0 0000 0040 0080 0120 01990160 0239 0279 0319 03591 0398 0438 0478 0517 05960557 0636 0675 0714 0753

07932 0832 0871 0910 0948 0987 1026 1064 1103 1141 3 1179 1217 1255 1293 13681331 1406 1443 1480 1517

15544 1591 1628 1664 1700 1736 1772 1808 1844 1879

1915oS 1950 1985 2019 2054 2088 2123 2157 2190 2224

22576 2291 2324 2357 24222389 2454 2486 2517 2549

25807 2611 2642 2673 2704 2734 2764 2794 2823 2852

28818 2910 2939 2967 2995 3023 3051 3078 3106 3133 3212 9 y3159 3186 3238 3264 3289 3315 3340 3365 3389

10 3413 3438 3461 3485 35313508 3554 3577 3599 3621 372911 3643 3665 3686 3708 3749 3770 3790 - 3810 3830

3849 386912 3888 3907 3925 3944 3962 3980 3997 4015

4032 4049 13 bulliQ6timiddot 4082 41154099 4131 4147 4162 4177 42071~m224192L4 4236 4251 4265 43194179 4292 4306

4332 4345 4357 4370 4382 4394middot 4406 4418 4429lS 4441 ltIJ

4452L6 4463 4474 4484 4495 4505 4515 4525 4535 4345

4554L7 4564 4573 4582 4591 4599 4608 4616 4625 4633

464118 4656 46644P49 4671 4678 4686 4693 47064699

4713 4761 L9 4719 4732 ~ 47674738 4744 4750 4726 4756 478820 4772 4793 4798 4893 4808 4812 4817~11amp 4783 shy

4821U 4826 4830 4834 48424838 48574846 4850 4854 486122 4864 4868 48874871 4875 4878 48904881 4884

23 4893 4896 4898 49164901 4913 4906 4909 49114904 24 4918 4922492Q 49364925shy 49294927 49344931 4932

4943493825 49524940 4941 4945 4946 49514948 4949 495326 4955 49644956 4957 4959 4960 4961 49634962 4965 497427 49674966 4968 4969 4970 49734971 4972 4974 498128 4975 4976 4977 4977 4978 49804979 4979

29 49864981 4982 4982 4983 4984 4984 49864985~49B5 30 4987 4987middot4987 4988 4988 4989 4990 49904989 4989 i

Source Abridged from Table I ofA Hald Statistical Table$ and Fonnulas (New YorkWiley) 1952 Reproduced by permission of A Hald

796 -~~ - -

o ~~~-TABLE VI Critical Values of t

f(t)

bull

Degrees of -Freedom tJWtlOO t050 tOO51010 tooi tooos

1 63143078 12706 31821 63657 31831 63662 2 1886 2920 4303 6965 9925 22326 31598 3 4541 1638 2353 3-182 5841 10213 12924 4 1533 2132 2776 3747 4604 7173 8610 5 1476 2015 2571 3365 4032 5893 6869 6 19431440 2447 3143 3707 5208 5959 7 1415 23651895 2998 3499 4785 5408

1397 1860 2306i 2896 3355 4501 5041 9 1383 1833 2262 2821 3250 4297 4781

10 18121372 2228 2764 3169 _ 4144 4587 U 1363 1796 2201 2718 3106 4025 4431 12 1356 1782 2179 shy 2681 3055 3930 4318

13~~ ~_ ~-i77113 i1reg 2650 3012 3852 4221 14 1345 21451761 2024 ~ 2977 41403787 15 1341 1753 2131 2602 29fl7 4073 16

373~ 1337 1746 2120 - 29212583 3686 4015

17 1333 1740 2110 2567 -2898 3646 3965 18 1330 1734 2101 t28782552 3610 3922 19 1328 t729 2093 28612539 38833579 - i

li~r~5I - 20 1725 2086 2528 2845 3850 21

3552 t 1323 2080 2518 1721 2831 3527 3819

22 20741321 1717 2508 2819 3505 3~792 23 1319 1714 2069 2500 3767 24

2807 3485 1318 1711 2064 2492 3467 3745

25 2middot797

1316 1708 2060 2485 3725 26

2787 3450 1315 1706 2056 2479 2779 3435 3707

27 1314 1703 2052 2473 2771 36903421 28 1313 170f 1 2048 2763 3674 29

~467 3408 1311 1699 2045 - 2462 2756 36593396

30 1310 1697 2D42 2457 - 2750 3646 40

3385 1303 1684 2021 2423 2704 3551

60 3307

1296 -16n 2000 2390 2660 3460 120

3232 1289 1658 1980 2358 2617 33733160

00 1645 1282 1960 2326 2576 3090 3291

Source 1his table is reproduced with the lcind permtsSioil of~e frustees of Biometrika from E S Pearson and R O Hartley (eds) The Biometrika Tables for Statisticians Vol 1 3d ed Biometrika 1966

r7~~

c

~ ApPENDIX A Tables 801

i~ 1lt TABLE IX Percentage Points of the F-distribution a = 05

~ liZ 1 2

1 1614 1995 2 1851 1900 3 1013 955 4 771 694

II

NUMERATOR DEGREES OF FREEDOM I11

3 4 9 I5 6 7 8

5

2157 2246 2302 2340 1916 1925 1930 1933

928 912 901 894 659 639 626 616

~ 661 579 541 519~ 505 495 6 599 514 476 453 439 428~7 559 474 435 412 397 387

5328 446 407 384 369 358 9 512 426 386 363 348 337~

10 -496 4100 371 348 333 322Q 11 484 7f~i 359398--~ 336 320 309

34938~middot12middot 475 326 300 ~~311~ 13 381467 341 318 303 292 0 fi

14 460 374 334 311 296 285 ~ 15 454 368 329 306 290foil 279 rr

449 363~ 16 324 301 285 274

~ 17 445 )20 296 281359 270 foil 18 441 3-16 293li))~5 277 266Q

438 352middot 313 290= 19 274 263 0 20 435 349 310 287 271 260 ~ 21 432 347 257307 284 268

i 22 430 344 305 282 266 255 303 0 23 428 342 280 264 253

Z 24 426 340 301 278 262 251foil Q 424middot25 339 299 276 260 249

26 423 337 298 274 259 247 27 421 335 296 273 257 246 28 420 334 295 271 245 256 29 418 333 293I 270 255 243 30 417 332 292 269 253 242 40 408 323 284 261 245 234 60 40q 315 276 253 237 225

120 307 392 268 245 229 217 00 384 300 260 237 221 210

2368 2389 1935 1937 889 885 609 604 488 482 421 415 379 373 350 344 329 323 314 307 301 295 291 285

283 277 271 1276 270 265 271 264 259 i266 259 254 1rII261 255 249

251 258 ~i-middot246 f 254 248 242 I

I 251 245 239 249 242 237 246 240 234 244 237 232 242 236 230 240 234 228 239 232 2]7 237 231 225 236 229 224 235 228 222 233 227 221

225 218 212 I

1217 210 204 209 202 196 201 194 188

2405 1938 881 600 477 410 368 339 318 302 290 280 ji

Source From M Merrington and eM Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 19433373-88

i (continued) I

I

I iii 11 iii

II I

I l

iINi II

802 Appendix A Tables

TABLE IX Continued

~1 NUMERATOR DEGREES OF FREEDOM Vl ~r--l-0----12----15-----r--2-0----24---r--30----40----60~--~- ~ OQ

1 2 3 4 S 6 7 8 9

~ 10 11

~ ~ ~ 14 o 15 rIJ (

Iiil 16 ~ 17 ~ 18 ~ 19 = 20

~ 21

i ii shy~ 24 ~ 2S

26 27 28 29 30

middot40 60

120 OQ

2419 1940 879 596 474 406 364 335 ~314

298 285 275 267 260 254 249 245 241 238 235 232 230 227 225 224 222 220 219 218 216

208 199 191 1~

2439 2459 2480 2491 1941 1943 1945 1945 874 870 866 864 591 586 580 577 468 462 456 453 400 394 387 384 357 351 344 341 328 322 315 312 307 301 294 290 291 285 277 274 279 272 265 261 269 262 254 251 260 253 246 242 253 246 239 235 248 240 233 229 242 235 228 224 238 231 223 219 234 227 219 215 231 223 216 211 iJi ~r~ 220 212 208 225 218 210 205 223 215 207 203 220 213 205 201 218 211 203 198 216 209 201 196

215 2Q7 199 195j 2i3 206 197 193 212 204 196 191 210 203 194 190 209 201 193 189

200 192184 179 192 184 175 170 183 i 175 166 161 U5 1m 1~ 1~

2501 1946 862 575 450 381 33S 308 286 270 257 247

238 231 225 219 215 211 207 204 201 198 196 194 192 190 188 187 185 184

174 165 155 146

2511 Imiddotmiddotmiddot

1947 859 572 446 377middot 334 304 283 266 253 243 234 227 220 215 210 206 203

~199 196 f 194 1~1 189 187 185 184 182 181 179

169 159 150 139

2522 1948 857 5~69

443 374 330 301 279 262 249 238 230 222

216 211 206 202 198 195 192 189 186 184 182 180 179 177 175 174

164 153 143 132

2533 2543 1949 1950 855 853 566 563 440 436 370 367 327 323 297 293 275 271 258 254 245 240 234 230 225 221 218 213 211 207 206 201 201 196 197 192 193 188 190 184 1~7 181 184 178

181 176 179 173 177 ~ 171

169 17~ 173 167 171 165

164170 162168

151158 147 139

125135 100122 i

ApPENDIX A Tables~c63 y

TABLE X Percentage Points of the F-distribution a = 025

f(F)

o FfJ25

- -NUMERATOR DEGREES OF FREEDOM ~ Jz 1 2 3 4 5 6 7 8 9

1 6478 7995 8642 8996 9218 9371 9482 9567 9633 2 3851 3900 3917 3925 3930 3933 3936 3937 3939 3 1744 1604 1544 1510 1488 1473 1462 1454 1447 4 1222 1065 998 960 936 920 907 898 890 5 1001 843 776 739 715 698 685 676 668 ( 881 726 660 623 599 582 570 560 552

589 7 807 654 552 529 512 499 490 482 8 757 606 542 505 482 465 453 443 436 9 721 571 508 472 448 432 420 410 403

gt1 10 - 694 483 447 424 407 395 385 378-4~~i 0 003f2~~ 11 672 463 428 404 88 376 366 359fIl

12- 655 510 447 412 389 373 351 344~61f l~ 13 641 497 435 400 377 360 348 339 331

0 ~ 14 630 486 424 389 366 329338-~$P 321 IIl 15 620 477 A15 380 358 34-1 329 320 312fIl

16 61_2 469 408 373 350 334 322 305 31~r-~ ~ 17 462- 4016Cl1) 366 344 328 316 298I j r~ 30(hmiddotfIl 45618 598 395 361 338 322 310 301 293~

592 451 390 356 333= 19 317 305 296 288

~ 20 587 446 386 351 329 313 301 291 284 21 583 442 382 348 325 309 297 287 280 22 - 579 438 378 344 322 305 293 284 276~

0 23 575 435 375 341 318 302 290 281 273 ~ 24 572 432 372 338 315 299 287 278 270 ~ 25 569 429 369 335 313 297 285 275 268

26 566 427 367 333 310 294 282 273 265 27 563 424 365 331 - 292 308 280 271 263 28 561 422 363 329 306 290 278 269 261 29 559 420 361 327 304 288 276 267 259 30 557 418 359 325 303 287 275 265 257

40 542 405 346 290 313 274 262 253 245 60 529 393 334 301 279 263 251 241 233

120 515 380 323 289 267 252 239 230 222 00 502 369 312 279 257 241 229 219 211

SourceFrom M Merrington and C M Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 1943337388

(continued)

4 804 Appendix A Tables

TABLE X Continued

~Vl NUMERATOR DEGREES OF FREEDOM V2~r---l-0--r--U--~-~~--~~20-----~~---~------~~middotmiddot----6-0-----U~O-----~---

9686 9767 9849 9931 9972 1001 1006 _1010 1~014-1 1018 2 3940 3941 3943 3945 3946 3946 3947 3948 394~ 3950 3 1442 1434 1425 1417 1412 1408 1404 1399 1395 1390

884 875 866 856 851 846 841 836 8314 826 662 652 643 633 628 623 618 612 6075 602 546 537 527 517 512 507 501 496 4906 485

7 476 467 457 447 442 436 431 425 420 414 8 430 420 410 400 395 389 384 378 373 367 9 396 387 377 367 361 356 351 345 339 333

372 362 352 342 337 331 326 320 314 308S 10 353 343 333 323 317 312 306 300 294Q 11 288 337 328 318 307 302 296 291 285 279 272 325 315 305 295 289 284 278 272 266 260E ~

)i 14 315 305 295 284 279 273 267 261 255 249 o 15 306 296 286 276 270 264 259 252 246 240rIl ~ 16 232299 289 279 268 263 257 251 245 238 ~ ~17 292 282 272 262 256 250 244 238 232 225

287 277 267 256 250 244 238 232 226~ 18 219 Q 19 282 272 262 251 245 239 233 227 220 21~=

277 268 257 246 241 235 229 222 216 209~ 20middot 273 bull =I~~~ 253 242 237 231 ~ 225 218 211lt 21 204 270 middot260 250 239 233 227 21 214 208 200~ 22 267 257 247 236 230 224 $ 218 211 204 197o 23

z 24 264 254 244 D3 227 221 ~ Ii 215 208 201 194 261 251 241 230 224 218 212 205 198 191~ 25

26 259 249 239 228 222 216 209 203 )95 I 188 27 257 247 236 225 219 213 207 200 i~~3 185 28 2d5tf i 245middot 234 223217 211 205 198 191 183 29 253 243 232 221 215 209 203 196 189 181 30 251 241 231 220 214 207 201 194 187 179

40 164239 229 218 207 201 194 188 180 172 60 227 217 206 194 188 182 174 167 158 148

120 216 205 194 182 176 169 161 153 143 131 I

QO 100205 194 I 183 171 I 164 157 148 139 127

Page 14: STA220_Final_2012S.pdf

796 -~~ - -

o ~~~-TABLE VI Critical Values of t

f(t)

bull

Degrees of -Freedom tJWtlOO t050 tOO51010 tooi tooos

1 63143078 12706 31821 63657 31831 63662 2 1886 2920 4303 6965 9925 22326 31598 3 4541 1638 2353 3-182 5841 10213 12924 4 1533 2132 2776 3747 4604 7173 8610 5 1476 2015 2571 3365 4032 5893 6869 6 19431440 2447 3143 3707 5208 5959 7 1415 23651895 2998 3499 4785 5408

1397 1860 2306i 2896 3355 4501 5041 9 1383 1833 2262 2821 3250 4297 4781

10 18121372 2228 2764 3169 _ 4144 4587 U 1363 1796 2201 2718 3106 4025 4431 12 1356 1782 2179 shy 2681 3055 3930 4318

13~~ ~_ ~-i77113 i1reg 2650 3012 3852 4221 14 1345 21451761 2024 ~ 2977 41403787 15 1341 1753 2131 2602 29fl7 4073 16

373~ 1337 1746 2120 - 29212583 3686 4015

17 1333 1740 2110 2567 -2898 3646 3965 18 1330 1734 2101 t28782552 3610 3922 19 1328 t729 2093 28612539 38833579 - i

li~r~5I - 20 1725 2086 2528 2845 3850 21

3552 t 1323 2080 2518 1721 2831 3527 3819

22 20741321 1717 2508 2819 3505 3~792 23 1319 1714 2069 2500 3767 24

2807 3485 1318 1711 2064 2492 3467 3745

25 2middot797

1316 1708 2060 2485 3725 26

2787 3450 1315 1706 2056 2479 2779 3435 3707

27 1314 1703 2052 2473 2771 36903421 28 1313 170f 1 2048 2763 3674 29

~467 3408 1311 1699 2045 - 2462 2756 36593396

30 1310 1697 2D42 2457 - 2750 3646 40

3385 1303 1684 2021 2423 2704 3551

60 3307

1296 -16n 2000 2390 2660 3460 120

3232 1289 1658 1980 2358 2617 33733160

00 1645 1282 1960 2326 2576 3090 3291

Source 1his table is reproduced with the lcind permtsSioil of~e frustees of Biometrika from E S Pearson and R O Hartley (eds) The Biometrika Tables for Statisticians Vol 1 3d ed Biometrika 1966

r7~~

c

~ ApPENDIX A Tables 801

i~ 1lt TABLE IX Percentage Points of the F-distribution a = 05

~ liZ 1 2

1 1614 1995 2 1851 1900 3 1013 955 4 771 694

II

NUMERATOR DEGREES OF FREEDOM I11

3 4 9 I5 6 7 8

5

2157 2246 2302 2340 1916 1925 1930 1933

928 912 901 894 659 639 626 616

~ 661 579 541 519~ 505 495 6 599 514 476 453 439 428~7 559 474 435 412 397 387

5328 446 407 384 369 358 9 512 426 386 363 348 337~

10 -496 4100 371 348 333 322Q 11 484 7f~i 359398--~ 336 320 309

34938~middot12middot 475 326 300 ~~311~ 13 381467 341 318 303 292 0 fi

14 460 374 334 311 296 285 ~ 15 454 368 329 306 290foil 279 rr

449 363~ 16 324 301 285 274

~ 17 445 )20 296 281359 270 foil 18 441 3-16 293li))~5 277 266Q

438 352middot 313 290= 19 274 263 0 20 435 349 310 287 271 260 ~ 21 432 347 257307 284 268

i 22 430 344 305 282 266 255 303 0 23 428 342 280 264 253

Z 24 426 340 301 278 262 251foil Q 424middot25 339 299 276 260 249

26 423 337 298 274 259 247 27 421 335 296 273 257 246 28 420 334 295 271 245 256 29 418 333 293I 270 255 243 30 417 332 292 269 253 242 40 408 323 284 261 245 234 60 40q 315 276 253 237 225

120 307 392 268 245 229 217 00 384 300 260 237 221 210

2368 2389 1935 1937 889 885 609 604 488 482 421 415 379 373 350 344 329 323 314 307 301 295 291 285

283 277 271 1276 270 265 271 264 259 i266 259 254 1rII261 255 249

251 258 ~i-middot246 f 254 248 242 I

I 251 245 239 249 242 237 246 240 234 244 237 232 242 236 230 240 234 228 239 232 2]7 237 231 225 236 229 224 235 228 222 233 227 221

225 218 212 I

1217 210 204 209 202 196 201 194 188

2405 1938 881 600 477 410 368 339 318 302 290 280 ji

Source From M Merrington and eM Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 19433373-88

i (continued) I

I

I iii 11 iii

II I

I l

iINi II

802 Appendix A Tables

TABLE IX Continued

~1 NUMERATOR DEGREES OF FREEDOM Vl ~r--l-0----12----15-----r--2-0----24---r--30----40----60~--~- ~ OQ

1 2 3 4 S 6 7 8 9

~ 10 11

~ ~ ~ 14 o 15 rIJ (

Iiil 16 ~ 17 ~ 18 ~ 19 = 20

~ 21

i ii shy~ 24 ~ 2S

26 27 28 29 30

middot40 60

120 OQ

2419 1940 879 596 474 406 364 335 ~314

298 285 275 267 260 254 249 245 241 238 235 232 230 227 225 224 222 220 219 218 216

208 199 191 1~

2439 2459 2480 2491 1941 1943 1945 1945 874 870 866 864 591 586 580 577 468 462 456 453 400 394 387 384 357 351 344 341 328 322 315 312 307 301 294 290 291 285 277 274 279 272 265 261 269 262 254 251 260 253 246 242 253 246 239 235 248 240 233 229 242 235 228 224 238 231 223 219 234 227 219 215 231 223 216 211 iJi ~r~ 220 212 208 225 218 210 205 223 215 207 203 220 213 205 201 218 211 203 198 216 209 201 196

215 2Q7 199 195j 2i3 206 197 193 212 204 196 191 210 203 194 190 209 201 193 189

200 192184 179 192 184 175 170 183 i 175 166 161 U5 1m 1~ 1~

2501 1946 862 575 450 381 33S 308 286 270 257 247

238 231 225 219 215 211 207 204 201 198 196 194 192 190 188 187 185 184

174 165 155 146

2511 Imiddotmiddotmiddot

1947 859 572 446 377middot 334 304 283 266 253 243 234 227 220 215 210 206 203

~199 196 f 194 1~1 189 187 185 184 182 181 179

169 159 150 139

2522 1948 857 5~69

443 374 330 301 279 262 249 238 230 222

216 211 206 202 198 195 192 189 186 184 182 180 179 177 175 174

164 153 143 132

2533 2543 1949 1950 855 853 566 563 440 436 370 367 327 323 297 293 275 271 258 254 245 240 234 230 225 221 218 213 211 207 206 201 201 196 197 192 193 188 190 184 1~7 181 184 178

181 176 179 173 177 ~ 171

169 17~ 173 167 171 165

164170 162168

151158 147 139

125135 100122 i

ApPENDIX A Tables~c63 y

TABLE X Percentage Points of the F-distribution a = 025

f(F)

o FfJ25

- -NUMERATOR DEGREES OF FREEDOM ~ Jz 1 2 3 4 5 6 7 8 9

1 6478 7995 8642 8996 9218 9371 9482 9567 9633 2 3851 3900 3917 3925 3930 3933 3936 3937 3939 3 1744 1604 1544 1510 1488 1473 1462 1454 1447 4 1222 1065 998 960 936 920 907 898 890 5 1001 843 776 739 715 698 685 676 668 ( 881 726 660 623 599 582 570 560 552

589 7 807 654 552 529 512 499 490 482 8 757 606 542 505 482 465 453 443 436 9 721 571 508 472 448 432 420 410 403

gt1 10 - 694 483 447 424 407 395 385 378-4~~i 0 003f2~~ 11 672 463 428 404 88 376 366 359fIl

12- 655 510 447 412 389 373 351 344~61f l~ 13 641 497 435 400 377 360 348 339 331

0 ~ 14 630 486 424 389 366 329338-~$P 321 IIl 15 620 477 A15 380 358 34-1 329 320 312fIl

16 61_2 469 408 373 350 334 322 305 31~r-~ ~ 17 462- 4016Cl1) 366 344 328 316 298I j r~ 30(hmiddotfIl 45618 598 395 361 338 322 310 301 293~

592 451 390 356 333= 19 317 305 296 288

~ 20 587 446 386 351 329 313 301 291 284 21 583 442 382 348 325 309 297 287 280 22 - 579 438 378 344 322 305 293 284 276~

0 23 575 435 375 341 318 302 290 281 273 ~ 24 572 432 372 338 315 299 287 278 270 ~ 25 569 429 369 335 313 297 285 275 268

26 566 427 367 333 310 294 282 273 265 27 563 424 365 331 - 292 308 280 271 263 28 561 422 363 329 306 290 278 269 261 29 559 420 361 327 304 288 276 267 259 30 557 418 359 325 303 287 275 265 257

40 542 405 346 290 313 274 262 253 245 60 529 393 334 301 279 263 251 241 233

120 515 380 323 289 267 252 239 230 222 00 502 369 312 279 257 241 229 219 211

SourceFrom M Merrington and C M Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 1943337388

(continued)

4 804 Appendix A Tables

TABLE X Continued

~Vl NUMERATOR DEGREES OF FREEDOM V2~r---l-0--r--U--~-~~--~~20-----~~---~------~~middotmiddot----6-0-----U~O-----~---

9686 9767 9849 9931 9972 1001 1006 _1010 1~014-1 1018 2 3940 3941 3943 3945 3946 3946 3947 3948 394~ 3950 3 1442 1434 1425 1417 1412 1408 1404 1399 1395 1390

884 875 866 856 851 846 841 836 8314 826 662 652 643 633 628 623 618 612 6075 602 546 537 527 517 512 507 501 496 4906 485

7 476 467 457 447 442 436 431 425 420 414 8 430 420 410 400 395 389 384 378 373 367 9 396 387 377 367 361 356 351 345 339 333

372 362 352 342 337 331 326 320 314 308S 10 353 343 333 323 317 312 306 300 294Q 11 288 337 328 318 307 302 296 291 285 279 272 325 315 305 295 289 284 278 272 266 260E ~

)i 14 315 305 295 284 279 273 267 261 255 249 o 15 306 296 286 276 270 264 259 252 246 240rIl ~ 16 232299 289 279 268 263 257 251 245 238 ~ ~17 292 282 272 262 256 250 244 238 232 225

287 277 267 256 250 244 238 232 226~ 18 219 Q 19 282 272 262 251 245 239 233 227 220 21~=

277 268 257 246 241 235 229 222 216 209~ 20middot 273 bull =I~~~ 253 242 237 231 ~ 225 218 211lt 21 204 270 middot260 250 239 233 227 21 214 208 200~ 22 267 257 247 236 230 224 $ 218 211 204 197o 23

z 24 264 254 244 D3 227 221 ~ Ii 215 208 201 194 261 251 241 230 224 218 212 205 198 191~ 25

26 259 249 239 228 222 216 209 203 )95 I 188 27 257 247 236 225 219 213 207 200 i~~3 185 28 2d5tf i 245middot 234 223217 211 205 198 191 183 29 253 243 232 221 215 209 203 196 189 181 30 251 241 231 220 214 207 201 194 187 179

40 164239 229 218 207 201 194 188 180 172 60 227 217 206 194 188 182 174 167 158 148

120 216 205 194 182 176 169 161 153 143 131 I

QO 100205 194 I 183 171 I 164 157 148 139 127

Page 15: STA220_Final_2012S.pdf

r7~~

c

~ ApPENDIX A Tables 801

i~ 1lt TABLE IX Percentage Points of the F-distribution a = 05

~ liZ 1 2

1 1614 1995 2 1851 1900 3 1013 955 4 771 694

II

NUMERATOR DEGREES OF FREEDOM I11

3 4 9 I5 6 7 8

5

2157 2246 2302 2340 1916 1925 1930 1933

928 912 901 894 659 639 626 616

~ 661 579 541 519~ 505 495 6 599 514 476 453 439 428~7 559 474 435 412 397 387

5328 446 407 384 369 358 9 512 426 386 363 348 337~

10 -496 4100 371 348 333 322Q 11 484 7f~i 359398--~ 336 320 309

34938~middot12middot 475 326 300 ~~311~ 13 381467 341 318 303 292 0 fi

14 460 374 334 311 296 285 ~ 15 454 368 329 306 290foil 279 rr

449 363~ 16 324 301 285 274

~ 17 445 )20 296 281359 270 foil 18 441 3-16 293li))~5 277 266Q

438 352middot 313 290= 19 274 263 0 20 435 349 310 287 271 260 ~ 21 432 347 257307 284 268

i 22 430 344 305 282 266 255 303 0 23 428 342 280 264 253

Z 24 426 340 301 278 262 251foil Q 424middot25 339 299 276 260 249

26 423 337 298 274 259 247 27 421 335 296 273 257 246 28 420 334 295 271 245 256 29 418 333 293I 270 255 243 30 417 332 292 269 253 242 40 408 323 284 261 245 234 60 40q 315 276 253 237 225

120 307 392 268 245 229 217 00 384 300 260 237 221 210

2368 2389 1935 1937 889 885 609 604 488 482 421 415 379 373 350 344 329 323 314 307 301 295 291 285

283 277 271 1276 270 265 271 264 259 i266 259 254 1rII261 255 249

251 258 ~i-middot246 f 254 248 242 I

I 251 245 239 249 242 237 246 240 234 244 237 232 242 236 230 240 234 228 239 232 2]7 237 231 225 236 229 224 235 228 222 233 227 221

225 218 212 I

1217 210 204 209 202 196 201 194 188

2405 1938 881 600 477 410 368 339 318 302 290 280 ji

Source From M Merrington and eM Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 19433373-88

i (continued) I

I

I iii 11 iii

II I

I l

iINi II

802 Appendix A Tables

TABLE IX Continued

~1 NUMERATOR DEGREES OF FREEDOM Vl ~r--l-0----12----15-----r--2-0----24---r--30----40----60~--~- ~ OQ

1 2 3 4 S 6 7 8 9

~ 10 11

~ ~ ~ 14 o 15 rIJ (

Iiil 16 ~ 17 ~ 18 ~ 19 = 20

~ 21

i ii shy~ 24 ~ 2S

26 27 28 29 30

middot40 60

120 OQ

2419 1940 879 596 474 406 364 335 ~314

298 285 275 267 260 254 249 245 241 238 235 232 230 227 225 224 222 220 219 218 216

208 199 191 1~

2439 2459 2480 2491 1941 1943 1945 1945 874 870 866 864 591 586 580 577 468 462 456 453 400 394 387 384 357 351 344 341 328 322 315 312 307 301 294 290 291 285 277 274 279 272 265 261 269 262 254 251 260 253 246 242 253 246 239 235 248 240 233 229 242 235 228 224 238 231 223 219 234 227 219 215 231 223 216 211 iJi ~r~ 220 212 208 225 218 210 205 223 215 207 203 220 213 205 201 218 211 203 198 216 209 201 196

215 2Q7 199 195j 2i3 206 197 193 212 204 196 191 210 203 194 190 209 201 193 189

200 192184 179 192 184 175 170 183 i 175 166 161 U5 1m 1~ 1~

2501 1946 862 575 450 381 33S 308 286 270 257 247

238 231 225 219 215 211 207 204 201 198 196 194 192 190 188 187 185 184

174 165 155 146

2511 Imiddotmiddotmiddot

1947 859 572 446 377middot 334 304 283 266 253 243 234 227 220 215 210 206 203

~199 196 f 194 1~1 189 187 185 184 182 181 179

169 159 150 139

2522 1948 857 5~69

443 374 330 301 279 262 249 238 230 222

216 211 206 202 198 195 192 189 186 184 182 180 179 177 175 174

164 153 143 132

2533 2543 1949 1950 855 853 566 563 440 436 370 367 327 323 297 293 275 271 258 254 245 240 234 230 225 221 218 213 211 207 206 201 201 196 197 192 193 188 190 184 1~7 181 184 178

181 176 179 173 177 ~ 171

169 17~ 173 167 171 165

164170 162168

151158 147 139

125135 100122 i

ApPENDIX A Tables~c63 y

TABLE X Percentage Points of the F-distribution a = 025

f(F)

o FfJ25

- -NUMERATOR DEGREES OF FREEDOM ~ Jz 1 2 3 4 5 6 7 8 9

1 6478 7995 8642 8996 9218 9371 9482 9567 9633 2 3851 3900 3917 3925 3930 3933 3936 3937 3939 3 1744 1604 1544 1510 1488 1473 1462 1454 1447 4 1222 1065 998 960 936 920 907 898 890 5 1001 843 776 739 715 698 685 676 668 ( 881 726 660 623 599 582 570 560 552

589 7 807 654 552 529 512 499 490 482 8 757 606 542 505 482 465 453 443 436 9 721 571 508 472 448 432 420 410 403

gt1 10 - 694 483 447 424 407 395 385 378-4~~i 0 003f2~~ 11 672 463 428 404 88 376 366 359fIl

12- 655 510 447 412 389 373 351 344~61f l~ 13 641 497 435 400 377 360 348 339 331

0 ~ 14 630 486 424 389 366 329338-~$P 321 IIl 15 620 477 A15 380 358 34-1 329 320 312fIl

16 61_2 469 408 373 350 334 322 305 31~r-~ ~ 17 462- 4016Cl1) 366 344 328 316 298I j r~ 30(hmiddotfIl 45618 598 395 361 338 322 310 301 293~

592 451 390 356 333= 19 317 305 296 288

~ 20 587 446 386 351 329 313 301 291 284 21 583 442 382 348 325 309 297 287 280 22 - 579 438 378 344 322 305 293 284 276~

0 23 575 435 375 341 318 302 290 281 273 ~ 24 572 432 372 338 315 299 287 278 270 ~ 25 569 429 369 335 313 297 285 275 268

26 566 427 367 333 310 294 282 273 265 27 563 424 365 331 - 292 308 280 271 263 28 561 422 363 329 306 290 278 269 261 29 559 420 361 327 304 288 276 267 259 30 557 418 359 325 303 287 275 265 257

40 542 405 346 290 313 274 262 253 245 60 529 393 334 301 279 263 251 241 233

120 515 380 323 289 267 252 239 230 222 00 502 369 312 279 257 241 229 219 211

SourceFrom M Merrington and C M Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 1943337388

(continued)

4 804 Appendix A Tables

TABLE X Continued

~Vl NUMERATOR DEGREES OF FREEDOM V2~r---l-0--r--U--~-~~--~~20-----~~---~------~~middotmiddot----6-0-----U~O-----~---

9686 9767 9849 9931 9972 1001 1006 _1010 1~014-1 1018 2 3940 3941 3943 3945 3946 3946 3947 3948 394~ 3950 3 1442 1434 1425 1417 1412 1408 1404 1399 1395 1390

884 875 866 856 851 846 841 836 8314 826 662 652 643 633 628 623 618 612 6075 602 546 537 527 517 512 507 501 496 4906 485

7 476 467 457 447 442 436 431 425 420 414 8 430 420 410 400 395 389 384 378 373 367 9 396 387 377 367 361 356 351 345 339 333

372 362 352 342 337 331 326 320 314 308S 10 353 343 333 323 317 312 306 300 294Q 11 288 337 328 318 307 302 296 291 285 279 272 325 315 305 295 289 284 278 272 266 260E ~

)i 14 315 305 295 284 279 273 267 261 255 249 o 15 306 296 286 276 270 264 259 252 246 240rIl ~ 16 232299 289 279 268 263 257 251 245 238 ~ ~17 292 282 272 262 256 250 244 238 232 225

287 277 267 256 250 244 238 232 226~ 18 219 Q 19 282 272 262 251 245 239 233 227 220 21~=

277 268 257 246 241 235 229 222 216 209~ 20middot 273 bull =I~~~ 253 242 237 231 ~ 225 218 211lt 21 204 270 middot260 250 239 233 227 21 214 208 200~ 22 267 257 247 236 230 224 $ 218 211 204 197o 23

z 24 264 254 244 D3 227 221 ~ Ii 215 208 201 194 261 251 241 230 224 218 212 205 198 191~ 25

26 259 249 239 228 222 216 209 203 )95 I 188 27 257 247 236 225 219 213 207 200 i~~3 185 28 2d5tf i 245middot 234 223217 211 205 198 191 183 29 253 243 232 221 215 209 203 196 189 181 30 251 241 231 220 214 207 201 194 187 179

40 164239 229 218 207 201 194 188 180 172 60 227 217 206 194 188 182 174 167 158 148

120 216 205 194 182 176 169 161 153 143 131 I

QO 100205 194 I 183 171 I 164 157 148 139 127

Page 16: STA220_Final_2012S.pdf

802 Appendix A Tables

TABLE IX Continued

~1 NUMERATOR DEGREES OF FREEDOM Vl ~r--l-0----12----15-----r--2-0----24---r--30----40----60~--~- ~ OQ

1 2 3 4 S 6 7 8 9

~ 10 11

~ ~ ~ 14 o 15 rIJ (

Iiil 16 ~ 17 ~ 18 ~ 19 = 20

~ 21

i ii shy~ 24 ~ 2S

26 27 28 29 30

middot40 60

120 OQ

2419 1940 879 596 474 406 364 335 ~314

298 285 275 267 260 254 249 245 241 238 235 232 230 227 225 224 222 220 219 218 216

208 199 191 1~

2439 2459 2480 2491 1941 1943 1945 1945 874 870 866 864 591 586 580 577 468 462 456 453 400 394 387 384 357 351 344 341 328 322 315 312 307 301 294 290 291 285 277 274 279 272 265 261 269 262 254 251 260 253 246 242 253 246 239 235 248 240 233 229 242 235 228 224 238 231 223 219 234 227 219 215 231 223 216 211 iJi ~r~ 220 212 208 225 218 210 205 223 215 207 203 220 213 205 201 218 211 203 198 216 209 201 196

215 2Q7 199 195j 2i3 206 197 193 212 204 196 191 210 203 194 190 209 201 193 189

200 192184 179 192 184 175 170 183 i 175 166 161 U5 1m 1~ 1~

2501 1946 862 575 450 381 33S 308 286 270 257 247

238 231 225 219 215 211 207 204 201 198 196 194 192 190 188 187 185 184

174 165 155 146

2511 Imiddotmiddotmiddot

1947 859 572 446 377middot 334 304 283 266 253 243 234 227 220 215 210 206 203

~199 196 f 194 1~1 189 187 185 184 182 181 179

169 159 150 139

2522 1948 857 5~69

443 374 330 301 279 262 249 238 230 222

216 211 206 202 198 195 192 189 186 184 182 180 179 177 175 174

164 153 143 132

2533 2543 1949 1950 855 853 566 563 440 436 370 367 327 323 297 293 275 271 258 254 245 240 234 230 225 221 218 213 211 207 206 201 201 196 197 192 193 188 190 184 1~7 181 184 178

181 176 179 173 177 ~ 171

169 17~ 173 167 171 165

164170 162168

151158 147 139

125135 100122 i

ApPENDIX A Tables~c63 y

TABLE X Percentage Points of the F-distribution a = 025

f(F)

o FfJ25

- -NUMERATOR DEGREES OF FREEDOM ~ Jz 1 2 3 4 5 6 7 8 9

1 6478 7995 8642 8996 9218 9371 9482 9567 9633 2 3851 3900 3917 3925 3930 3933 3936 3937 3939 3 1744 1604 1544 1510 1488 1473 1462 1454 1447 4 1222 1065 998 960 936 920 907 898 890 5 1001 843 776 739 715 698 685 676 668 ( 881 726 660 623 599 582 570 560 552

589 7 807 654 552 529 512 499 490 482 8 757 606 542 505 482 465 453 443 436 9 721 571 508 472 448 432 420 410 403

gt1 10 - 694 483 447 424 407 395 385 378-4~~i 0 003f2~~ 11 672 463 428 404 88 376 366 359fIl

12- 655 510 447 412 389 373 351 344~61f l~ 13 641 497 435 400 377 360 348 339 331

0 ~ 14 630 486 424 389 366 329338-~$P 321 IIl 15 620 477 A15 380 358 34-1 329 320 312fIl

16 61_2 469 408 373 350 334 322 305 31~r-~ ~ 17 462- 4016Cl1) 366 344 328 316 298I j r~ 30(hmiddotfIl 45618 598 395 361 338 322 310 301 293~

592 451 390 356 333= 19 317 305 296 288

~ 20 587 446 386 351 329 313 301 291 284 21 583 442 382 348 325 309 297 287 280 22 - 579 438 378 344 322 305 293 284 276~

0 23 575 435 375 341 318 302 290 281 273 ~ 24 572 432 372 338 315 299 287 278 270 ~ 25 569 429 369 335 313 297 285 275 268

26 566 427 367 333 310 294 282 273 265 27 563 424 365 331 - 292 308 280 271 263 28 561 422 363 329 306 290 278 269 261 29 559 420 361 327 304 288 276 267 259 30 557 418 359 325 303 287 275 265 257

40 542 405 346 290 313 274 262 253 245 60 529 393 334 301 279 263 251 241 233

120 515 380 323 289 267 252 239 230 222 00 502 369 312 279 257 241 229 219 211

SourceFrom M Merrington and C M Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 1943337388

(continued)

4 804 Appendix A Tables

TABLE X Continued

~Vl NUMERATOR DEGREES OF FREEDOM V2~r---l-0--r--U--~-~~--~~20-----~~---~------~~middotmiddot----6-0-----U~O-----~---

9686 9767 9849 9931 9972 1001 1006 _1010 1~014-1 1018 2 3940 3941 3943 3945 3946 3946 3947 3948 394~ 3950 3 1442 1434 1425 1417 1412 1408 1404 1399 1395 1390

884 875 866 856 851 846 841 836 8314 826 662 652 643 633 628 623 618 612 6075 602 546 537 527 517 512 507 501 496 4906 485

7 476 467 457 447 442 436 431 425 420 414 8 430 420 410 400 395 389 384 378 373 367 9 396 387 377 367 361 356 351 345 339 333

372 362 352 342 337 331 326 320 314 308S 10 353 343 333 323 317 312 306 300 294Q 11 288 337 328 318 307 302 296 291 285 279 272 325 315 305 295 289 284 278 272 266 260E ~

)i 14 315 305 295 284 279 273 267 261 255 249 o 15 306 296 286 276 270 264 259 252 246 240rIl ~ 16 232299 289 279 268 263 257 251 245 238 ~ ~17 292 282 272 262 256 250 244 238 232 225

287 277 267 256 250 244 238 232 226~ 18 219 Q 19 282 272 262 251 245 239 233 227 220 21~=

277 268 257 246 241 235 229 222 216 209~ 20middot 273 bull =I~~~ 253 242 237 231 ~ 225 218 211lt 21 204 270 middot260 250 239 233 227 21 214 208 200~ 22 267 257 247 236 230 224 $ 218 211 204 197o 23

z 24 264 254 244 D3 227 221 ~ Ii 215 208 201 194 261 251 241 230 224 218 212 205 198 191~ 25

26 259 249 239 228 222 216 209 203 )95 I 188 27 257 247 236 225 219 213 207 200 i~~3 185 28 2d5tf i 245middot 234 223217 211 205 198 191 183 29 253 243 232 221 215 209 203 196 189 181 30 251 241 231 220 214 207 201 194 187 179

40 164239 229 218 207 201 194 188 180 172 60 227 217 206 194 188 182 174 167 158 148

120 216 205 194 182 176 169 161 153 143 131 I

QO 100205 194 I 183 171 I 164 157 148 139 127

Page 17: STA220_Final_2012S.pdf

ApPENDIX A Tables~c63 y

TABLE X Percentage Points of the F-distribution a = 025

f(F)

o FfJ25

- -NUMERATOR DEGREES OF FREEDOM ~ Jz 1 2 3 4 5 6 7 8 9

1 6478 7995 8642 8996 9218 9371 9482 9567 9633 2 3851 3900 3917 3925 3930 3933 3936 3937 3939 3 1744 1604 1544 1510 1488 1473 1462 1454 1447 4 1222 1065 998 960 936 920 907 898 890 5 1001 843 776 739 715 698 685 676 668 ( 881 726 660 623 599 582 570 560 552

589 7 807 654 552 529 512 499 490 482 8 757 606 542 505 482 465 453 443 436 9 721 571 508 472 448 432 420 410 403

gt1 10 - 694 483 447 424 407 395 385 378-4~~i 0 003f2~~ 11 672 463 428 404 88 376 366 359fIl

12- 655 510 447 412 389 373 351 344~61f l~ 13 641 497 435 400 377 360 348 339 331

0 ~ 14 630 486 424 389 366 329338-~$P 321 IIl 15 620 477 A15 380 358 34-1 329 320 312fIl

16 61_2 469 408 373 350 334 322 305 31~r-~ ~ 17 462- 4016Cl1) 366 344 328 316 298I j r~ 30(hmiddotfIl 45618 598 395 361 338 322 310 301 293~

592 451 390 356 333= 19 317 305 296 288

~ 20 587 446 386 351 329 313 301 291 284 21 583 442 382 348 325 309 297 287 280 22 - 579 438 378 344 322 305 293 284 276~

0 23 575 435 375 341 318 302 290 281 273 ~ 24 572 432 372 338 315 299 287 278 270 ~ 25 569 429 369 335 313 297 285 275 268

26 566 427 367 333 310 294 282 273 265 27 563 424 365 331 - 292 308 280 271 263 28 561 422 363 329 306 290 278 269 261 29 559 420 361 327 304 288 276 267 259 30 557 418 359 325 303 287 275 265 257

40 542 405 346 290 313 274 262 253 245 60 529 393 334 301 279 263 251 241 233

120 515 380 323 289 267 252 239 230 222 00 502 369 312 279 257 241 229 219 211

SourceFrom M Merrington and C M Thompson Tables of Percentage Points of the Inverted Beta (F)-Distribution Biometrika 1943337388

(continued)

4 804 Appendix A Tables

TABLE X Continued

~Vl NUMERATOR DEGREES OF FREEDOM V2~r---l-0--r--U--~-~~--~~20-----~~---~------~~middotmiddot----6-0-----U~O-----~---

9686 9767 9849 9931 9972 1001 1006 _1010 1~014-1 1018 2 3940 3941 3943 3945 3946 3946 3947 3948 394~ 3950 3 1442 1434 1425 1417 1412 1408 1404 1399 1395 1390

884 875 866 856 851 846 841 836 8314 826 662 652 643 633 628 623 618 612 6075 602 546 537 527 517 512 507 501 496 4906 485

7 476 467 457 447 442 436 431 425 420 414 8 430 420 410 400 395 389 384 378 373 367 9 396 387 377 367 361 356 351 345 339 333

372 362 352 342 337 331 326 320 314 308S 10 353 343 333 323 317 312 306 300 294Q 11 288 337 328 318 307 302 296 291 285 279 272 325 315 305 295 289 284 278 272 266 260E ~

)i 14 315 305 295 284 279 273 267 261 255 249 o 15 306 296 286 276 270 264 259 252 246 240rIl ~ 16 232299 289 279 268 263 257 251 245 238 ~ ~17 292 282 272 262 256 250 244 238 232 225

287 277 267 256 250 244 238 232 226~ 18 219 Q 19 282 272 262 251 245 239 233 227 220 21~=

277 268 257 246 241 235 229 222 216 209~ 20middot 273 bull =I~~~ 253 242 237 231 ~ 225 218 211lt 21 204 270 middot260 250 239 233 227 21 214 208 200~ 22 267 257 247 236 230 224 $ 218 211 204 197o 23

z 24 264 254 244 D3 227 221 ~ Ii 215 208 201 194 261 251 241 230 224 218 212 205 198 191~ 25

26 259 249 239 228 222 216 209 203 )95 I 188 27 257 247 236 225 219 213 207 200 i~~3 185 28 2d5tf i 245middot 234 223217 211 205 198 191 183 29 253 243 232 221 215 209 203 196 189 181 30 251 241 231 220 214 207 201 194 187 179

40 164239 229 218 207 201 194 188 180 172 60 227 217 206 194 188 182 174 167 158 148

120 216 205 194 182 176 169 161 153 143 131 I

QO 100205 194 I 183 171 I 164 157 148 139 127

Page 18: STA220_Final_2012S.pdf

4 804 Appendix A Tables

TABLE X Continued

~Vl NUMERATOR DEGREES OF FREEDOM V2~r---l-0--r--U--~-~~--~~20-----~~---~------~~middotmiddot----6-0-----U~O-----~---

9686 9767 9849 9931 9972 1001 1006 _1010 1~014-1 1018 2 3940 3941 3943 3945 3946 3946 3947 3948 394~ 3950 3 1442 1434 1425 1417 1412 1408 1404 1399 1395 1390

884 875 866 856 851 846 841 836 8314 826 662 652 643 633 628 623 618 612 6075 602 546 537 527 517 512 507 501 496 4906 485

7 476 467 457 447 442 436 431 425 420 414 8 430 420 410 400 395 389 384 378 373 367 9 396 387 377 367 361 356 351 345 339 333

372 362 352 342 337 331 326 320 314 308S 10 353 343 333 323 317 312 306 300 294Q 11 288 337 328 318 307 302 296 291 285 279 272 325 315 305 295 289 284 278 272 266 260E ~

)i 14 315 305 295 284 279 273 267 261 255 249 o 15 306 296 286 276 270 264 259 252 246 240rIl ~ 16 232299 289 279 268 263 257 251 245 238 ~ ~17 292 282 272 262 256 250 244 238 232 225

287 277 267 256 250 244 238 232 226~ 18 219 Q 19 282 272 262 251 245 239 233 227 220 21~=

277 268 257 246 241 235 229 222 216 209~ 20middot 273 bull =I~~~ 253 242 237 231 ~ 225 218 211lt 21 204 270 middot260 250 239 233 227 21 214 208 200~ 22 267 257 247 236 230 224 $ 218 211 204 197o 23

z 24 264 254 244 D3 227 221 ~ Ii 215 208 201 194 261 251 241 230 224 218 212 205 198 191~ 25

26 259 249 239 228 222 216 209 203 )95 I 188 27 257 247 236 225 219 213 207 200 i~~3 185 28 2d5tf i 245middot 234 223217 211 205 198 191 183 29 253 243 232 221 215 209 203 196 189 181 30 251 241 231 220 214 207 201 194 187 179

40 164239 229 218 207 201 194 188 180 172 60 227 217 206 194 188 182 174 167 158 148

120 216 205 194 182 176 169 161 153 143 131 I

QO 100205 194 I 183 171 I 164 157 148 139 127