Statistics Final Exam Review Chapters 3, 4, 5 Practice Problems with Solutions

Statistics Final Exam Review

Chapters 3, 4, 5

Practice Problems with Solutions

Chapter 5 Practice Problems

1) Find the area under the standard normal curve between z = 0 and z = 3. 1

A) 0.9987 B) 0.4987 C) 0.0010 D) 0.4641

Normalcdf(0,3.1) = 0.4990 B

2) Find the area under the standard normal curve to the left of z = 1.25.

A) 0.1056 B) 0.8944 C) 0.2318 D) 0.7682

Normalcdf(-10000,1.25) = 0.8944 B

Chapter 5 Practice Problems3) Find the area of the indicated region under the standard normal curve.

A) 0.0968 B) 0.0823 C) 0.9032 D) 0.9177

Normalcdf(-1.30,1.0000) = 0.9032 C

4) For the standard normal curve, find the z-score that corresponds to the first quartile. A) 0.67 B) -0.23 C) 0.77 D) -0.67

The first quartile means that P = 0.2500. The z-score that corresponds to this percentage from the Normal Distribution chart is – 0.67. The answer is D.


5) IQ test scores are normally distributed with a mean of 99 and a standard deviation of 11. An individual's IQ score is found to be 109. Find the z-score corresponding to this value.

A) 0.91 B) 1.10 C) -1.10 D) -0.91

Given that the value of = 99, = 11, and x = 109, we can solve for the value of z using the equation z = (x - )/ . Z = 0.91 A

6) An airline knows from experience that the distribution of the number of suitcases that get lost each week on a certain route is approximately normal with μ = 15.5 and = 3.6. What is the probability that during a given week the airline will lose between 10 and 20 suitcases?

A) 0.3944 B) 0.8314 C) 0.1056 D) 0.4040

Normalcdf(10,20,15.5,3.6) = 0.8311 B


7.

With 5% of the data not included on either side, we look up the percent value of .0500 on the chart and we get that z = -1.645. The z value for the other side of the chart will be +1.645. The solution is B.

8.

Using the equation x = z + , we see that x = (15)(2.33) + 100 = 134.95. The correct choice will be D.

Chapter 5 Practice Problems9.

Solve for the value of z for each of the situations and then compare the results; the larger of the two z scores will have performed better. The first z-value will be z = (75 – 65)/8 which equals 1.25. The second z-value will be z = (75 – 70)/4 which equals 1.25. Since both z scores are the same the correct choice will be C.

Chapter 4 Practice Problems10. State whether the variable is discrete or continuous. The number of cups of coffee sold in a cafeteria during lunch

A) discrete B) continuous A) discrete

11. State whether the variable is discrete or continuous. The blood pressures of a group of students the day before their final exam

A) continuous B) discrete A) continuous


12) The random variable x represents the number of cars per household in a town of 1000 households. Find the probability of randomly selecting a household that has less than two cars.

A) 0.809 B) 0.125 C) 0.553 D) 0.428

Cars Households0 1251 4282 2563 1084 83

P(x) = (428 + 125)/1000 = 0.553 C) 0.553


x P(x) xP(x) (x - ) (x - )2 (x - )2P(x)

0 0.07 0.00 -1.23 1.513 0.106

1 0.68 0.68 -0.23 .053 0.036

2 0.21 0.42 0.77 .593 0.125

3 0.03 0.09 1.77 3.133 0.094

4 0.01 0.04 2.77 7.673 0.077

Total 1.00 1.23 0.4.38

13) The random variable x represents the number of credit cards that adults have along with the corresponding probabilities. Find the mean and standard deviation.

A) mean: 1.30; standard deviation: 0.44 B) mean: 1.23; standard deviation: 0.66

C) mean: 1.23; standard deviation: 0.44 D) mean: 1.30; standard deviation: 0.32

x P(x)0 0.071 0.682 0.213 0.034 0.01

2( ) ( ) 0.438 0.66x P x B) mean: 1.23; standard deviation: 0.66

Chapter 4 Practice Problems14) At a raffle, 10,000 tickets are sold at $5 each for three prizes valued at $4,800, $1,200, and $400. What is the expected value of one ticket?

A) $4.36 B) -$4.36 C) $0.64 D) -$0.64

1 1 1( ) ( $4,800) ( $1,200) ( $400)

10,000 9,999 9,998

999( $5) $0.48 $0.12 $0.04 ( $5.00) $4.3610000

xP x

B) -$4.36

15) A sports analyst records the winners of NASCAR Winston Cup races for a recent season. The random variable x represents the races won by a driver in one season. Use the frequency distribution to construct a probability distribution.

Wins 1 2 3 4 5 6 7Drivers 12 2 0 2 0 0 1 x 1 2 3 4 5 6 7

P(x) 0.71 0.12 0 0.12 0 0 0.06

Chapter 4 Practice Problems16) Determine whether the distribution represents a probability distribution. If not, identify any requirements that are not satisfied.

x P(x)1 0.22 0.23 0.24 0.25 0.2

It is a probability distribution, because each value is between 0 and 1 and the sum of P(x) = 1.

17. According to government data, the probability that a woman between the ages of 25 and 29 was never married is 40%. In a random survey of 10 women in this age group, what is the probability that at least eight were married?

B) 0.167P(8) + P(9) +P(10) = 0.1209 + .0403 + .0060 = .0.1672

Binompdf(10,.60,8) = 0.1209; Binompdf(10,.60,9) = 0.0403; Binompdf(10,.60,10) = 0.0060

Chapter 4 Practice Problems17) The probability that a tennis set will go to a tiebreaker is 18%. In 60 randomly

selected tennis sets, what is the mean and the standard deviation of the number of tiebreakers?

A) mean: 10.8; standard deviation: 2.98 B) mean: 10.2; standard deviation: 2.98B) C) mean: 10.8; standard deviation: 3.29 D) mean: 10.2; standard deviation: 3.29

Mean: μ = np = 60 x 0.18 = 10.8

Standard Deviation: =npq = 60x.18x.82 = 2.98

A) mean: 10.8; standard deviation: 2.98


18) Fifty-seven percent of families say that their children have an influence on their vacation plans. Consider a sample of eight families who are asked if their children influence their vacation plans. Identify the values of n, p, and q, and list the possible values of the random variable x.

n = 8; p = 0.57; q = 0.43; x = 0, 1, 2, 3, 4, 5, 6, 7, 8

19) You observe the gender of the next 100 babies born at a local hospital. You count the number of girls born. Identify the values of n, p, and q, and list the possible values of the random variable x

n = 100; p = 0.5; q = 0.5; x = 0, 1, 2, . . ., 99, 100

Chapter 4 Practice Problems20. A statistics professor finds that when he schedules an office hour at the 10:30 a.m. time slot, an average of three students arrive. Use the Poisson distribution to find the probability that in a randomly selected office hour in the 10:30 a.m. time slot exactly five students will arrive.

A) 0.0137 B) 0.0519 C) 0.0070 D) 0.10083) D

= 3; x = 5;

5 33( ) 0.1008

! 5!

xe eP x

x

Poisson(3,5) = 0.1008

Chapter 4 Practice Problems21. A mail-order company receives an average of five orders per 500 solicitations. If it sends out 100 advertisements, find the probability of receiving at least two orders. Use the Poisson distribution.

A) 0.1839 B) 0.9596 C) 0.9048 D) 0.2642

5) D0 1

1 1

1(0) 0.3679

! 0!

1(1) 0.3679

! 1!( 2) 1 (0.3679 .03679) 0.2642

x

x

e eP

x

e eP

xP x

Poisson(1,0) = 0.3679; Poisson(1,1) = 0.3679; P(0) +P(1)= 2(0.3679)= 0.7358

P( x 2) = 1 – 0.7358 = 0.2642

Chapter 4 Practice Problems22. Given: The probability that a federal income tax return is filled out incorrectly with an error in favor of the taxpayer is 20%. Question: What is the probability type that of the ten tax returns randomly selected for an audit, three returns will contain only errors favoring the taxpayer?

A) Poisson B) geometric C) binomial

7) C1. The experiment is repeated for a fixed number of trials, where

each trial is independent of other trials.

2. There are only two possible outcomes of interest for each trial. The outcomes can be classified as a success (S) or as a failure (F).

3. The probability of a success P(S) is the same for each trial.

4. The random variable x counts the number of successful trials.

Chapter 4 Practice Problems23 Given: The probability that a federal income tax return is filled out incorrectly with an error in favor of the taxpayer is 20%. Question: What is the probability type that when the ten tax returns are randomly selected for an audit, the sixth return will contain only errors favoring the taxpayer?

A) Poisson B) binomial C) geometric

8) C

Geometric distribution A discrete probability distribution. Satisfies the following conditions

a) A trial is repeated until a success occurs.b) The repeated trials are independent of each other.c) The probability of success p is constant for each trial.

Chapter 4 Practice Problems Binomial

repeated for a fix # of trials Random variable x, counts the # of successes out of n trials

Geometric repeated until a successful outcome occurs Random variable x, is the # of where the first success occurs

Poisson Counting the # times an event occurs for a specified interval The number of occurrences of an event in a specified interval

is independent of the # of occurrences of the event in other specified intervals


Binomial

Geometric

Poisson

2

np

npq

npq

Mean

Variance

Standard Deviation

Mean

Variance

Standard Deviation

Mean

Variance

Standard Deviation

22

2

1

p

q

p

q

p

2

#mean occurrences

Chapter 4 Practice Problems24. Assume the probability that you will make a sale on any given telephone call is 0.19. Find the probability that you (a) make your first sale on the fifth call, (b) make your first sale on the first, second, or third call, and (c) do not make a sale on the first three calls.

x = # of trials for 1st success

p = probability of success

q = probability of failure

P(x) = p(q) x-1

(a) geometpdf(.19,5) = 0.082

(b) geometpdf(.19,1) + geometpdf(.19,2) + geometpdf(.19,3) = 0.19 + 0.154 + 0.125 = 0.469

(c) No sales on the first 3 calls = Q(1) + Q(2) + Q(3) = 1 – 0.469 = 0.531

Chapter 4 Practice Problems24. During a 36-year period, lightning killed 2457 people in the United States. Assume that this rate holds true today and is constant throughout the year. Find the probability that tomorrow (a) no one in the United States will be struck and killed by lightning, (b) one person will be struck and killed, (c) more than one person will be struck and killed.

x = # occurrences in a given time or space

= average # of occurrences in a given time or space

( )!

xeP x

x

Poissoinpdf(,x)

= 2457/13140 = 0.1870

(a) x = 0; poissonpdf(0.1870,0) = 0.829

(b) x = 1; poissonpdf(0.1870,1) = 0.155

(c) x > 1 = 1 – (P(0) + P(1)) = 0.016

36 yrs = 13140 days


x = # occurrences in a given time or space

= average # of occurrences in a given time or space

( )!

xeP x

x

Poissoinpdf(,x)

(a) = 4, x = 3; P(3) poissonpdf(4,3) = 0.195

(b) P(x 3) = P(0) + P(1) + P(2) + P(3) = .0183 + .0733 + 0.1465 + 0.1954 = 0.433

(c) P(x > 3) = 1 – P( x 3) = 1 – 0.433 = 0.567

25. A newspaper finds that the mean number of typographical errors per page is four. Find the probability that (a) exactly three typographical errors will be found on a page, (b) at most three typographical errors will be found on a page, and (c) more than three typographical errors will be found on a page.


26. A single six-sided die is rolled. Find the probability of rolling a number less than 3.

A) 0.25 B) 0.333 C) 0.1 D) 0.5

1) B 2 correct answers/6 possible answers =0.33

27. In a survey of college students, 880 said that they have cheated on an exam and 1721 said that they have not. If one college student is selected at random, find the probability that the student has cheated on an exam.

A) 1721/2601 B) 2601/1721 C) 2601/880 D) 880/2601

4) D 880 correct answers/(1721+880) possible answers = 880/2601

Chapter 3 Practice Problems28. The distribution of blood types for 100 Americans is listed in the table. If one donor is selected at random, find the probability of selecting a person with blood type A+ or A-. Blood Type O+ O- A+ A- B+ B- AB+ AB-

Number 37 6 34 6 10 2 4 1

A) 0.34 B) 0.06 C) 0.4 D) 0.02

5) C (34 +6) correct answers/100 possible answers = 40/100 = 0.4

29. A card is picked at random from a standard deck of 52 playing cards. Find the odds that it is not a heart.

A) 1:3 B) 1:4 C) 4:1 D) 3:1

10) D 1 suit of correct answers/4 suits of possible answers = ¼. To not be a heart means that we have 4/4 – ¼ = ¾ . There are a 3:1 odds that a heart will not be chosen randomly.

Chapter 3 Practice Problems29. Classify the events as dependent or independent. Events A and B whereP(A) = 0.6, P(B) = 0.3, and P(A and B) = 0.17

A) independent B) dependent

12) B The results of P(A and B) = 0.17 indicates that the event A and the event B are interrelated and so they are dependent.

30. Find the probability of answering two true or false questions correctly if random guesses are made. Only one of the choices is correct.

A) 0.25 B) 0.75 C) 0.1 D) 0.5

15) A 1 correct answer/4 possible answers =0.25

Chapter 3 Practice Problems30. Find the probability of getting four consecutive aces when four cards are drawn without replacement from a standard deck of 52 playing cards.

31. A multiple-choice test has five questions, each with five choices for the answer. Only one of the choices is correct. You randomly guess the answer to each question. What is the probability that you do not answer any of the questions correctly?

32. The probability it will rain is 40% each day over a three-day period. What is theprobability it will rain at least one of the three days?

30. P(4-Aces) = (4/52)(3/51)(2/50)(1/49 )= 0.00000369

31. P(all five questions answers incorrect) = (4/5)(4/5)(4/5)(4/5)(4/5) = 0.32768

32. P(rain at least one day) = 1 - P(no rain all three days)= 1 - (0.60)(0.60)(0.60) = 0.7845

Chapter 3 Practice Problems32. A card is selected at random from a standard deck. Find each

probability.

(a) Randomly selecting a diamond or a 7._________________________

(b) Randomly selecting a red suit or a queen._________________________

(c) Randomly selecting a 3 or a face card_________________________33. You roll one die. Find each probability.

(a) Rolling a 6 or a number greater than 4. _________________________

(b) Rolling a number less than 5 or an odd number. _________________________

(c) Rolling a 3 or an even number. _________________________

13 4 1 160.308

52 52 52 52

26 4 2 280.538

52 52 52 52

4 12 160.308

52 52 52

1 2 1 20.333

6 6 6 6

4 3 2 50.833

6 6 6 6

1 3 40.667

6 6 6


34. Determine the probability that an 8-sided die, numbered 1-8 is rolled and the results of the roll is an even number or a number that is greater than 6.

___________________________________

4 2 1 50.625

8 8 8 8


Gender

Male Female Total

Level Associates 260 405 665

Of Bachelor’s 595 804 1399

Degree Master’s 230 329 559

Doctorate 25 23 48

Total 1110 1561 2671

(a) earned a master’s degree or is a female _________________

(b) earned an associate degree and is a male _________________

(c) is a female given that the person earned a bachelor’s degree_________________

8040.575

1399

2600.097

2671

559 1561 3290.671

2671 2671 2671

35.


1 13 140.269

52 52 52

1 12 120.005

52 51 2652

13 13 1690.064

52 51 2652

36. Using a standard 52-card deck for each situation, determine the probability.

(a) Drawing a five and a spade.

(b) Drawing a five of spades.

(c) Drawing a spade and then without replacement, drawing club.

(d) Drawing a 3 of clubs and then without replacement, drawing another club.

(e) Drawing a 4 of diamonds or a heart on the first draw.

4 1 40.019

52 4 208

10.019

52

Chapter 3 Practice Problems37. How many ways can a jury of five men and three women be selected from twelve men and ten women?

(12C5)(10C3) = 95,040

38. How many different permutations of the letters in the word PROBABILITY are there?

11!/(2!2!) = 9,979,200

39. A student must answer six questions on an exam that contains twelve questions.a) How many ways can the student do this?b) How many ways are there if the student must answer the first and last question?

(a) 12C6 = 924; (b) 10C4 = 210


40. In the California State lottery, you must select six numbers from fifty-two numbers to win the big prize. The numbers do not have to be in a particular order. What is the probability that you will win the big prize if you buy one ticket?

52C6 = 1/20,358,520 = 0.00000004911

41. From a group of 40 people, a jury of 12 people is selected. In how many different ways can a jury of 12 people be selected?

5,586,853,48040 12

40!5,586,853,480

(40 12)!(12!)C

Chapter 3 Practice Problems42. An area code consists of three digits. How many area codes are possible if (a) there are no restrictions and (b) if the first digit can not be 1 or 0. (c) What isthe probability of selecting an area code at random that ends in an odd number if the first digit cannot be a 1 or a 0?

42. (a) 10x10x10= 1000 (b) 10x10x8= 800 (c) 0.5

43. A shipment of 10 microwave ovens contains two defective units. In how many ways can a restaurant buy three of these units and receive (a) no defective units, (b) one defective unit, and (c) at least two non-defective units? (d) What is the probability of the restaurant buying at least two defective units?

43. (a) 56 (b) 56 (c) 112 (d) 0.067

(a) (8C3 )(2C0 ) = (56 )(1 )= 56

(b) (8C2 )(2C1 ) = (28 )(2 )= 56

(c) At least two good units one or fewer defective units. 56 + 56 = 112

(d) Pat least 2 defective units =8 1 2 2

10 3

8 10.067

120

C C

C

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Statistics Final Exam Review Chapters 3, 4, 5 Practice Problems with Solutions