Questions on Topic Eleven: The Normal Distribution 2

Questions' on Topic Eleven:The Normal Distribution

c1tiple-Choice Questions

(A) Th~ area under the standard normal curve between 0 a~d 2 is'twice thea;;::; area between 0 and 1. 'eYThe area uncler the standard normal curve between 0 and 2 is half the

' •. area between -2'and 2,e' " ',',(C) For the standard normal curve, the interquartile range is approximately 3.(D) For the standard normal curve, the range is 6.(E) For the standard notmal curve, the area to the left ofO.l is the same as :J;

" the area to the right of 0.9, Ie.. __'\ ~ - ~cort-~Populati~ns PI and P2 are normally distributed and have identical means.Howevtr, the standard deviation of Plis twice the standard deviation of P2., f.2.. 'What can be said ab,our the P,ercenr;;; of observations falling within tw~, ' \ Istandard deviations of th,e mean for each population? '" ' " " '

, ,

(A) The percentage for PI is twice the percentage for P2.(B) TRe percentage for PI is greater, but not twice asgreat, as the percent-

age for P2. '(C) The percentage for P2 is twice the percentage for PI. .(D) The percentage for P2 is grearer, but not twice as great, as the....l Jr pu...,....f1(JYt 5~ percentage for PI. '.. A ll- -.(\O'rW Vtl5 I " fI@VThe percentages are identical. 'f\QA)-t a-'vJ(f1.J q~ @

~l( OVJsevu(V~l1Y0"., . lJ) i1h VI'- ;<, o:f7{)Y\ ( ,

fa..JLvN),S-t(,LI\OlMA~~-em-lCLJ\

\: Which cifthe following is a true statement?

~,@)rhe area undet a nor~al cu;ve isalwa;sequ~l ;01, no'm:atter'wh~t the, mean and standard deviation are., " , '" ",(B) All bell-shaped curves are normal distributions for some choice of

~ and cr.(C) The smaller the standard deviation of a norrn.alcurve, the lowet and

, more spread out the graph.(D) Depending upon the value of ~hestandard deviation, n.ormal curves

" with different means may he centered arounq rhe same number:'(E) Dependin.g upon the value of the standard deviation, the mean and,

median of a particular normal distribution may be different. ,

Which of the following is a.true statement?

,

252015

b

105

a

oWhich has the larger mean and which has the larger standard deviation?

(A) Larger mean, a; larger standard deviarion, a(B) Larger mean, a; larger standard deviation, b

..((f:; Larger mean, b; larger standard deviation, a.(D) Larger mean, b; larger standard deviation, b(E) Larger mean, b; same standard deviation

4. Consider the following two normal curves:

..::'1-~.~Ji.~

~2.'•

..:~'I

.~,~-'.~

<~t&-i,';j~.~;;~~:,~~jAi~1

-{.l~""~.•.•~

.-:~

i".~In Questions 5-15, assume the given distributions are normal. '.'~~

5. A trucking firm determines that its fleet of trucks averages a mean of 12.4 :Jmiles per gallon wirh a standard devia\ion of 1.2 miles per gallon on cross-:,Icountry hauls. What is the probabiliry that o~ of rhe trucks averages fewer)~'than 10 miles per gallon? H'x L.. l 0 '),~

(A) .0082 )0 -L~4 _ d- fi~:~~~~ I. ~. ;-1 '-.-(D) .5228 -0,0 ~ ~g" ':1(E) .9772 n()fmalcdf (-00, -..L);;: :,

6. A factory dumps an average of 2.43 tons of pollutants into a river every wee~~If the standard deviarion is 0.88 tons, what is the probabiliry that in a week~t~e than 3 tons are (dumped? - . .~~

W' .2578 P ><) 3) . '\1(B) .2843 . J J"3 ..Jli

.' (C) .6500 5_- ;) ,-' .-/ Q/04 7$ ..1..•• (D).7157 6.~ .~_'hJcd.~.&47f;I~)':~

(E) .7422 flOl¥YUU -=, .:::I5~0 .17. An electronic product takes an average of 3.4 hours to move through an::~'!Jl

assembly line. If the standard deviation is 0.5 hour, what is the probabiliryilthat an item will take between"5 and 411.0urs?,:J

(A) .2119 , 1-7'2 \ . _) i~(B) .2295 <J ;J ( ':? 4 . A~"(C) .3270 Vl ()r'fY\.-~Jc J+ )\ Lj j)' ) (). s ,;~I

.~.~

~ .3811,;,;~,'~~.6730"t -

__________.•""1-

.'

278 AP Statistics

.-

i ! 1 ).! • I: : .

Questions on Topic Eteven: The Normal Distribution 279

J:1emeans~ore on~ -college,placemenr exam is 500 wirh a standard devia-",n'of 100. Ninety-fivepercent'ofthe test takers SCOreabove what? <t{ ::. ,:>((Z)

'.~'~~" 1-,9S-~.u1U{\<lfll'\{6S'~~X~ () -0-1 (Ji)

y:~~ .~i'. '..-).(,'1 .>Ll.63)-t-\(f() -. 3~)-,)~')664

he averagenoise level ina restaurant is 30 decibels with a srandard devla-9rof~ decibels,:Ninet}'-~ine percenr of the time it is below wIpe yalue?

~)20.7- : ,.. ,': . ~13)32.0 . . . ..' .' .. ' -,' .:-'" .~)33.4 . [¥1U fl(JyiV){;):~q):-, l' '.''3(v .

378 .. ... ' ". ,'*'. 39:3 (X,3 Jl.() (l-j)t'3<J:;' ~qi.~<J'

he ~~an in~ome per household in a certain state is $9500 With a standard,eviation of $1750. The middle 95% of incomes are be~een what'rwo:i,hies?. ..)).-;QS1TD [-, q~er ':- l"15':"0.. Q. 0 ;L.5'K-~~~i08~ ~~q $~~ir:~~~~;;~~~....' .~,~~~'7'q~&()'

(D) $7260 and $11,740 . GYlUr1DiIl1 (,CJ~q~X~(E.) $8049 and$IO,951.. . -1-,% ll;S())-t9rn=- /p()1oOne company produces movi~ rrailel:swith mean 150 sec6nds and standardp~viation 40 seconds, while a second company produces~aileisw,ith m~n ..120s~condsand standard deviation 30 seconds. What is the probability that .~:::- ( ~ ~c..tWo rand?mly selectedtrailers,prteproduced byeachcompany,;wilLcombine (J,J'::;. 'to <;-i.e.to less than three,minutes?' .. J-{,-r::k!,'t.::'1S7J+,IU), . ,~ u_.'/;;uJ:;Lc~:~~~'.~J:~'i~~L.~?g;o;l~~C';'. . ;;-~go '

-,(<.::).099 - s-D " ~ . 0-; -r0J- :: .Jt). + ":$0 .'. '. . ~ .

':{~r:~~r'.R,03~t .'..' '~~03;~~;; .~;; ~ l'/JS£L

i,'Jay Olshansky ffom the University of Chicago was quoted in Cha~teN~wsasjarguing tpat for the average life expectancy wreachl9J),.l8% of people. wQuldhayeto .livqo .120.WQat standa,rd dev:iationishe assllInin~ forthiS -_statement to 'make sense? ' .' '. '.' .....:. ...• . . { ,m--r'\ . ., ,.'.'.,.'.t.Vl.' , .lr'..'.'VL.~..'.•..y.W\.....,'.r. ,•.t.....<t>,',',.' .. ;(,.-.•... ):: I ;Ln, -"..... U v.~«AY21.? 'p(X;lJ-tJ ') u . ":'=-t..I-"---

(B) 24.4 '" .... ':'-:.b q Iy, Lf7:-; - ?6(C) 25:2 -:E?(D) 35.0 .'.

. .. (E) 111.\

II

I~ I

I'L ~ x: _.-M-.c-~ o

-1,;L'6:: ;;2. '-j g- -/I\..(T

o'

I AP Statistics

.~

'.~

:; -,~' H:-()-,;;J..' --L{~ 7 x~ /<0 .~

13. Cucumbers grown on a cert~n farm have weights with a standard deviatioir\~of 2 ounces. ,What is the mean weighr ifJ2-0/oof the cucumbers weigh tess .,~than 16 ounces? .' ( '6))~ /0 - --'-'{ '•.~.~.••

VWl13.92 . GJ!lU ()6YVVI ' ~(i~ 14.30 J, b %:: If,,-..M, .~

(C) 14.40 ---:; ~~(0) 14.88 ,~(E) 15.70 --»I:; 13.0;;).. ':1:'

1,r-C) 14. If 75% of all families spend more than $75 weekly for food, while 15% ':'"--~_,)(~ J ''2> ~---'-- .-u _ ~ _ .~ spend~ore than $150, what isthe mean weekly(expe diture and what is the,'

~. -:x('~ starlaard deviation.? 75-<;!a lVI Y Vl6Yf7) (;l_ "- -0, (o/<fy Ibb145-1~ (A),u=83.33,(j=12.44/s<?o (/Y'-UY1ff'''' (,<;')); I,G)&' I

t5. ~ (B) J1 = 56.26, (j = 11.85 J;{.- ~(O'1CJ ~ 7S :11,_~=--_ iQ.,u = 118.52, (j = 56.26 I

~-O,(,,l<>: -,S- -;;lu. ~ 15t-O.k>16 @P,u= 104.39, (j~ J{ +- I, oLl (J' -;;{<;(J .~.e.t.\- 1.iY-l<S":: 1<)0 (E) ,u = 139.45, (j = 83.33 ?,.!ffl

•.•'"0 15. A coffee machine can be adjusted to deliver any fixed number of ounces of'1" t CI("1(j" + 1.0'I<:r;1:> coffee. If the machine has a standard deviation in delivery equal to 0.4 oun

/./1 ~ ~ it:; what should be the mean setting so that an 8-ounce cup will overflow onlyC>'=- 4 '3S(" 05% of the time? l~vo05 :: C(qS- ..

@6.97 ounces 1 f\V ~ IV1 (. qqD" r-"u.(B) 7.22 ounces . / 0 ?-f(C) 7.34 ounces ;;(, ",--7515cO,'-l)-3:_M..(D) ~.8.0 ounces c?I _ .0, q ("q '.(E) 9.03 ounces - , ,,(.0

vJY: ~~ ,I.t 16. Assume that a baseball team hasari average pitcher, that is, one whose pro:.;{\J11 Qrf bility of winning any decision i~. If this pitcher has ~isions in a se~'

(}J son, what is the probability that he will win at least 20 games? . ')..

@2).0505 ...tl;::V1" P(.x7AOJ P(l(~;ZC}.5

i~):~~~:~-;;~~,C;) -I!1s"'JE : /,(,'13",

(0l}.333 O~&. 'J1»C'fJ->::'::ccl-! {I,04qCO)-.i(E)..,4300 . :: Vl:? :: ,a,/)1 r1 (), 0507 '.~

17. Given.that 10o?o:,fthe nails made using a certain manufacturing process ~•.a length lessthan 2.48 inches, while 5% have a length greater than 2.54.inches, what are the mean and standard deviation of the lengths of the nai~me that the len..gthshavea normal distribution.

W,u = 2.506~.O20--5~(B) ,u" 2.506, (j = 0.0410(C) ,u = 2.516, (j = 0.0825(O),u = 2.516, (j = 0.1653(E) The mean and standard deviation cannot be computed from the info

mation given.:= J.A't -1-1,2'60'