CHAPTER 9 SAMPLING DISTRIBUTIONS MULTIPLE CHOICE...

CHAPTER 9 SAMPLING DISTRIBUTIONS

MULTIPLE CHOICE QUESTIONS

In the following multiple choice ques tions , please circle the correct answer .

1. As a general rule, the normal distribution is used to approxima t e the sampling distribution of the sample proportion only ifa. the sample size n is grea t e r than 30b. the population proportion p is close to 0.50c. the underlying population is normald. np and n (1 p ) are both grea t e r than 5ANSWER: d

2. Random samples of size 49 are taken from an infinite population whose mean is 300 and stand ard deviation is 21. The mean and standard error of the sample mean, respec tively, are:a. 300 and 21b. 300 and 3c. 70 and 230d. 49 and 21ANSWER: b

3. A normally distributed population with 200 element s has a mean of 60 and a stand ard deviation of 10. The probability that the mean of a sample of 25 eleme nt s taken from this population will be smaller than 56 isa. 0.0166b. 0.0228c. 0.3708d. 0.0394ANSWER: a

4. Given an infinite population with a mean of 75 and a standard deviation of 12, the probability that the mean of a sample of 36 observa tions , taken at random from this population, exceeds 78 isa. 0.4332b. 0.0668c. 0.0987d. 0.9013ANSWER: b

5. A population that consists of 500 observa tions has a mean of 40 and a stand ard deviation of 15. A sample of size 100 is taken at random from this population. The stand ard error of the sample mean equals:a. 2.50

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b. 12.50c. 1.343d. 1.50ANSWER: c

6. An infinite population has a mean of 60 and a standard deviation of 8. A sample of 50 observa tions will be taken at rando m from this population. The probability that the sample mean will be betwe en 57 and 62 isa. 0.9576b. 0.9960c. 0.2467d. 0.3520ANSWER: a

7. If all possible samples of size n are drawn from an infinite population with a mean of 15 and a standard deviation of 5, then the stand ard error of the sample mean equals 1.0 only for samples of sizea. 5b. 15c. 25d. 75ANSWER: c

8. If the stand ard error of the sampling distribution of the sample proportion is 0.0229 for samples of size 400, then the population proportion must be eithera. 0.4 or 0.6b. 0.5 or 0.5c. 0.2 or 0.8d. 0.3 or 0.7ANSWER: d

9. As a general rule in computing the stand ard error of the sample mean, the finite population correction factor is used only if the :a. sample size is smaller than 10% of the population sizeb. population size is smaller than 10% of the sample sizec. sample size is great e r than 1% of the population sized. population size is great e r than 1% of the sample sizeANSWER: c

10. Given that X is a binomial rando m variable, the binomial probability P(X ≥x ) is approxima t e d by the area under a normal curve to the right ofa. x 0.5b. x +0.5c. x 1d. x +1

Sampling Distributions 141

ANSWER: a

11. Consider an infinite population with a mean of 160 and a standard deviation of 25. A random sample of size 64 is taken from this population. The stand ard deviation of the sample mean equals:a. 12.649b. 25.0c. 2.56d. 3.125ANSWER: d

12. A sample of size 200 will be taken at random from an infinite population. Given that the population proportion is 0.60, the probability that the sample proportion will be grea t e r than 0.58 isa. 0.281b. 0.719c. 0.580d. 0.762ANSWER: b

13. A sample of size 40 will be taken from an infinite population whose mean and stand ard deviation are 68 and 12, respec tively. The probability that the sample mean will be larger than 70 isa. 0.3970b. 0.4332c. 0.1469d. 0.0668ANSWER: c

14. A sample of size n is select ed at random from an infinite population. As n increas e s , which of the following state m e n t s is true?a. The population stand ard deviation decrea s e sb. The stand ard error of the sample mean decreas e sc. The population stand ard deviation increase sd. The stand ard error of the sample mean increas e sANSWER: b

15. The finite population correction factor should not be used when:a. we are sampling from an infinite populationb. we are sampling from a finite populationc. sample size is great e r than 1% of the population sized. None of the above state m e n t s is correctANSWER: a

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16. If the stand ard error of the sampling distribution of the sample proportion is 0.0337 for samples of size 200, then the population proportion must be either:a. 0.25b. 0.75c. 0.20 or 0.80d. 0.35 or 0.65e. 0.30 or 0.70ANSWER: c

17. Random samples of size 81 are taken from an infinite population whose mean and standard deviation are 45 and 9, respec tively. The mean and stand ard error of the sampling distribution of the sample mean are:a. 9 and 45b. 45 and 9c. 81 and 45d. 45 and 1ANSWER: d

18. A sample of 250 observa tions will be select ed at random from an infinite population. Given that the population proportion is .25, the stand ard error of the sampling distribution of the sample proportion is :a. 0.0274b. 0.50c. 0.0316d. 0.0548ANSWER: a

19. A sample of size 25 is selected at random from a finite population. If the finite population correction factor is 0.6325, then the population size is:a. 10b. 41c. 15d. 35ANSWER: b

20. If two populations are normally distributed, the sampling distribution of the sample mean difference 21 XX − will be:a. approxima t ely normally distributedb. normally distributed only if both sample sizes are grea t e r than 30c. normally distributedd. normally distributed only if both population sizes are great e r than 30ANSWER: c


21. Two samples are selected at random from two independ e n t normally distributed populations . Sample 1 has 49 observa tions and has a mean of 10 and a stand ard deviation of 5. Sample 2 has 36 observa tions and has a mean of 12 and a standard deviation of 3. The standard error of the sampling distribution of the sample mean difference 21 XX − isa. 0.1853b. 0.7602c. 0.7331d. 0.8719ANSWER: d

22. Given a binomial distribution with n trials and probability p of a success on any trial, a conventional rule of thumb is that the normal distribution will provide an adequ a t e approxima tion of the binomial distribution ifa. np ≥5 and n (1 p ) ≥5b. np ≤5 and n (1 p ) ≤5c. np ≥5 and n (1 p ) ≤5d. np ≤5 and n (1 p ) ≥5ANSWER: a

23. If two rando m samples of sizes 1n and 2n are selected indepen d e n tly from two populations with means 1µ and 2µ , then the mean of the sampling distribution of the sample mean difference, 21 XX − , equals:a. 1µ + 2µb. 1µ 2µc. 1µ / 2µd. 1µ 2µANSWER: b

24. If two rando m samples of sizes 1n and 2n are selected indepen d e n tly from

two populations with variances 21σ and 2

2σ , then the stand ard error of the

sampling distribution of the sample mean difference, 21 XX − , equals:

a. 2122

21 /)( nnσσ −

b. 2122

21 /)( nnσσ +

c.2

22

1

21

nn

σσ−

d.2

22

1

21

nn

σσ+

ANSWER: d

25. Suppose that the probability p of a success on any trail of a binomial distribution equals 0.90. Then for which of the following number of trials, n,

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would the normal distribution provide a good approxima tion to the binomial distribution?a. 25b. 35c. 45d. 55ANSWER: d

26. If two rando m samples of sizes 1n and 2n are selected indepen d e n tly from two non normally distributed populations , then the sampling distribution of the sample mean difference , 21 XX − , isa. always non normalb. always normalc. approxima t ely normal only if 1n and 2n are both larger than 30d. approxima t ely normal regardles s of 1n and 2n ANSWER: c

27. Given that X is a binomial rando m variable, the binomial probability P(X= x ) is approxima t e d by the area under a normal curve betwe ena. x 0.5 and 0.0b. 0.0 and x +0.5c. 1 x and 1+ xd. x 0.5 and x +0.5ANSWER: d

28. The Central Limit Theore m state s that, if a random sample of size n is drawn from a population, then the sampling distribution of the sample mean X :a. is approxima t ely normal if n > 30b. is approxima t ely normal if n < 30c. is approxima t ely normal if the underlying population is normald. has the same variance as the populationANSWER: a

29. The expecte d value of the sampling distribution of the sample mean X equals the population mean µ :a. when the population is normally distributedb. when the population is symm e t ricc. when the population size N > 30d. for all populationsANSWER: d


30. If all possible samples of size n are drawn from an infinite population with a mean of µ and a standard deviation of σ , then the standard error of the sample mean is inversely proportional to:a. µb. σc. nd. n

ANSWER: d

31. Given that X is a binomial rando m variable, the binomial probability P(X ≤x ) is approxima t e d by the area under a normal curve to the left ofa. xb. –xc. x +0.5d. x 0.5ANSWER: c

32. The stand ard deviation of the sampling distribution of the sample mean is also called the:a. central limit theore mb. stand ard error of the meanc. finite population correction factord. population stand ard deviationANSWER: b

33. If a random sample of size n is drawn from a normal population, then the sampling distribution of the sample mean X will be:a. normal for all values of nb. normal only for n > 30c. approxima t ely normal for all values of nd. approxima t ely normal only for n > 30ANSWER: a

34. If all possible samples of size n are drawn from a population, the probability distribution of the sample mean X is called the:a. stand ard error of Xb. expecte d value of Xc. sampling distribution of Xd. normal distributionANSWER: c

TRUE/FALSE QUESTIONS

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35. In an effort to identify the true proportion of college freshm a n who are under 18 years of age, a rando m sample of 500 freshm a n was taken. Only fifty of them were under the age of 18. The value 0.10 would be used as a point estima t e to the true proportion of under age 18 freshm a n .ANSWER: T

36. The central limit theore m is basic to the concept of statistical inference, becaus e it permits us to draw conclusions about the population based strictly on sample data , and without having any knowledge about the distribution of the underlying population.ANSWER: T

37. When a grea t many simple random samples of size n are drawn from a population that is normally distributed , the sampling distribution of the sample means will be normal regardless of sample size n .ANSWER: T

38. The mean of the sampling distribution of the sample proportion p̂ , when the sample size n = 100 and the population proportion p = 0.92, is 92.0.ANSWER: F

39. The stand ard error of the sampling distribution of the sample proportion p̂ , when the sample size n = 100 and the population proportion p = 0.30, is 0.0021.ANSWER: F

40. Recall the rule of thumb used to indicate when the normal distribution is a good approximation of the sampling distribution for the sample proportion p̂ . For the combina tion n = 50; p = 0.05, the rule is satisfied.ANSWER: F

41. The stand ard error of the mean is the standard deviation of the sampling distribution of X .ANSWER: T

42. The stand ard deviation of the sampling distribution of the sample mean is also called the central limit theore m.ANSWER: F

43. Consider an infinite population with a mean of 100 and a standard deviation of 20. A random sample of size 64 is taken from this population. The stand ard deviation of the sample mean equals 2.5.ANSWER: T

44. If all possible samples of size n are drawn from an infinite population with a mean of 60 and a standard deviation of 8, then the stand ard error of the sample mean equals 1.0 only for samples of size 64.ANSWER: T


45. If all possible samples of size n are drawn from a population, the probability distribution of the sample mean X is referred to as the normal distribution.ANSWER: F

46. As a general rule, the normal distribution is used to approxima t e the sampling distribution of the sample proportion only if the sample size n is great e r than or equal to 30. ANSWER: F

47. A sample of size n is selected at rando m from an infinite population. As n increas e s , the standard error of the sample mean decrea s e s .ANSWER: T

48. If the stand ard error of the sampling distribution of the sample proportion is 0.0245 for samples of size 400, then the population proportion must be 0.40.ANSWER: F

49. A sample of size 25 is selected at random from a finite population. If the finite population correction factor is 0.822, the population size must be 75.ANSWER: T

50. If a simple random sample of 300 observa tions is taken from a population whose proportion p = 0.6, then the expecte d value of the sample proportion p̂ is 0.40.

ANSWER: F

51. The normal approxima tion to the binomial distribution works best when the numb er of trials is large, and when the binomial distribution is symm e trical (like the normal).ANSWER: T

52. If two rando m samples of size 36 each are selected indepen d e n tly from two populations with variances 42 and 50, then the standard error of the sampling distribution of the sample mean difference, 21 XX − , equals 2.5556.ANSWER: F

53. If two rando m samples of sizes 30 and 32 are selected independ e n t ly from two populations with means 109 and 121, then the mean of the sampling distribution of the sample mean difference, 21 XX − , equals 12.ANSWER: T

54. As a general rule, the normal distribution is used to approxima t e the sampling distribution of the sample proportion only if the sample size n is great e r than 30.ANSWER: F

TEST QUESTIONS

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55. A researche r conducte d a survey on a university campus for a sample of 64 seniors and reported that seniors read an averag e of 3.12 books in the prior acade mic seme s t e r , with a standard deviation of 2.15 books. Determine the probability that the sample mean is:a. above 3.45b. betwe en 3.38 and 3.58c. below 2.94

ANSWERS:a. 0.1093b. 0.1224c. 0.2514

56. An infinite population has a mean of 150 and a stand ard deviation of 40. A sample of 100 observa tions will be selected at rando m from the population.a. What is the expecte d value of the sample mean?b. What is the stand ard deviation of the sample mean?c. What is the shape of the sampling distribution of the sample meand. What does the sampling distribution of the sample mean show?

ANSWERS:a. xµ =150b. xσ = 4c. Approxima tely normal with a mean of 150 and a standard deviation of 4.d. It shows the probability distribution of all possible sample means that can

be observed with rando m samples of size 100. This distribution can be used to calculate the mean and the standard deviation of the sample mean. It can also be used to comput e the probability that the sample mean is within a specified range from the population mean.

57. If the weekly dema n d for cases of soda at a store is normally distributed with a mean of 47.6 cases and a stand ard deviation of 5.8 cases , what is the probability that the averag e dema n d for a sample of 10 stores will exceed 50 cases in a given week?

ANSWER:0 .0952

58. The probability of a success on any trial of a binomial experime n t is 20%. Find the probability that the proportion of success in a sample of 400 is less than 18%.

ANSWER:0.1587


59. Suppose that the time need ed to complet e a final exam is normally distributed with a mean of 85 minutes and a standard deviation of 18 minutes . .a. What is the probability that the total time taken by a group of 100

studen t s will not exceed 8,200 minutes?b. What assum p tion did you have to make in your computa tions in part (a)?

ANSWERS:a. 0.0475b. The studen t s’ times neede d to complet e the exam are independ e n t of one

another .

60. Heights of 10 year old children are normally distributed with a mean of 52 inches and a stand ard deviation of 4 inches.a. Find the probability that one rando mly selected 10 year old child is taller

than 54 inches.b. Find the probability that two randomly select ed 10 year old children are

both taller than 54 inches.c. Find the probability that the mean height of two randomly select ed 10

year old children is grea t e r than 54 inches.

ANSWERS:a. 0.3085b. 0.0952c. 0.2389

61. Find the sampling distribution of the sample mean X if samples of size 2 are drawn from the following population:

x 2 0 2p (x ) 0.2 0.6 0.2

ANSWER:

x 2 1 0 1 2p ( x ) 0.04 0.24 0.44 0.24 0.04

62. Let X be a binomial random variable with n = 25 and p = 0.6. Approximat e the following probabilities, using the normal distribution.a. P(X ≥ 20)b. P(X ≤ 15)c. P(X = 10)

ANSWERS:a. 0.0329b. 0.5793c. 0.0207

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63. An infinite population has a mean of 100 and a stand ard deviation of 20. Suppose that the population is not normally distributed. What does the central limit theore m say about the sampling distribution of the mean if samples of size 64 are drawn at rando m from this population?

ANSWER:The sampling distribution of X is approxima t ely normal with a mean of 100 and a stand ard deviation of 2.5

64. Suppose that the averag e annual income of a defense attorney is $150,000 with a standard deviation of $40,000. Assume that the income distribution is normal.a. What is the probability that the average annual income of a sample of 5

defens e attorneys is more than $120,000?b. What is the probability that the average annual income of a sample of 15

defens e attorneys is more than $120,000?

ANSWERS:a. 0.9535b. 0.9981

65. Assume the time need ed by a worker to perform a mainten a n c e operation is normally distributed with a mean of 70 minutes and a stand ard deviation of 6 minutes . What is the probability that the averag e time need ed by a sample of 5 workers to perform the mainten a nc e in betwe e n 63 minutes and 68 minutes?

ANSWER:0.221

66. In order to estimat e the mean salary for a population of 500 employees , the president of a certain company selected at rando m a sample of 40 employee s .a. Would you use the finite population correction factor in calculating the

stand ard error of the sample mean? Explain.b. If the population stand ard deviation is $800, comput e the standard error

both with and without using the finite population correction factor.c. What is the probability that the sample mean salary of the employe es will

be within ±$200 of the population mean salary:

ANSWERS:a. n/N = 0.08 > 0.01; therefore , the finite population correction factor is

necess ary.b. xσ =121.448 and 126.491 with and without the finite population

correction factor, respec tively.c. 0.901


67. A sample of 50 observa tions is drawn at rando m from a normal population whose mean and standard deviation are 75 and 6, respec tively.a. What does the central limit theore m say about the sampling distribution

of the sample mean? Why?b. Find the mean and standard error of the sampling distribution of the

sample mean.c. Find P( X >73)d. Find P( X <74)

ANSWERS:a. X is normal becaus e the parent population is normal.b. xµ =75 and xσ = .8485 c. 0.9909d. 0.1190

68. Suppose it is known that 60% of studen t s at a particular college are smokers . A sample of 500 student s from the college is select ed at random. Approxima te the probability that at least 280 of these student s are smokers .

ANSWER:0 .9664

69. Al Gore, the former Vice President of the USA, believes that the proportion of voters who will vote for a democra t candida t e in the year 2004 presidential elections is 0.65. A sample of 500 voters is selected at rando m.a. Assume that Gore is correct and p = 0.65. What is the sampling

distribution of the sample proportion p̂ ? Explain.b. Find the expecte d value and the stand ard deviation of the sample

proportion p̂ .c. What is the probability that the numb er of voters in the sample who will

vote for a democra t presidential candida t e in the year 2004 will be betwe en 340 and 350?

ANSWERS:a. Approxima tely normal, since np = 25 and n( 1 p )= 175 are both grea t e r

than 5.b. E( p̂ ) = 0.65, and p̂σ = 0.0213c. 0.0699

70. Find the sampling distribution of the sample mean X if samples of size 2 are drawn from the following population:

x 3 5p (x ) 0.6 0.4

ANSWER:

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x 3 4 5p ( x ) 0.36 0.48 0.16

71 Suppose that the starting salaries of male math professors are normally distributed with a mean of $56,000 and a stand ard deviation of $12,000. The starting salaries of female math professors are normally distributed with a mean of $50,000 and a standard deviation of $10,000. A rando m sample of 50 male math professors and a random sample of 40 female math professors are selected .a. What is the sampling distribution of the sample mean difference 21 XX − ?

Explain.b. Find the expecte d value and the stand ard error of the sample mean

difference.c. What is the probability that the sample mean salary of female math

professors will not exceed that of the male math professors?

ANSWERS:a. 21 XX − is normally distributed, since the parent populations are normally

distributed.b. E( 21 XX − ) = 6,000, and

2xxi −σ = 2319.483

c. 0.9952

72. Let X be a binomial rando m variable with n = 100 and p = 0.7. Approximat e the following probabilities, using the normal distribution. a. P(X =75)b. P(X ≤ 70)c. P(X > 60)

ANSWERS:a. 0.0484b. 0.5438c. 0.9808

73. A sample of size 400 is drawn from a population whose mean and variance are 5,000 and 10,000, respec tively. Find the following probabilities:a. P( X < 4,990)b. P(4,995 < X < 5,010)c. P( X = 5,000)

ANSWERS:a. 0.0228b. 0.8185c. 0.0

74. A fair coin is tossed 500 times. Approxima te the probability that the number of heads observed is betwee n 240 and 270 (inclusive).


ANSWER:0 .7928

75. Historical data collected at First of America bank in Michigan revealed that 80% of all custom ers applying for a loan are accept ed . Suppose that 50 new loan applications are selected at rando m.a. Find the expecte d value and the stand ard deviation of the numb er of

loans that will be accepte d by the bank.b. What is the probability that at least 42 loans will be accept ed?c. What is the probability that the numb er of loans rejected is betwe en 10

and 15, inclusive?

ANSWERS:a. Let X be the number of loans out of 50 that are accept ed . Then X is a

binomial rando m variable with n = 50, and p = 0.80. Therefore , E(X)=40, and σ = 2.828.

b. 0.2981c. 0.5452

76. A sample of 25 observa tions is drawn from a normal population with mean of 900 and a standard deviation of 300. Suppose the population size is 600. a. Find the expecte d value of the sample mean X .b. Find the stand ard error of the sample mean X .c. Find P( X > 1000)d. Find P( X < 960)e. Find P(980 < X < 1050)

ANSWERS:a. xµ µ= = 900

b.1−

−=N

nN

nx

σσ = 58.7857

c. 0.0436d. 0.8461e. 0.0815

77. An auditor knows from past history that the averag e account s receivable for a compa ny is $521.72 with a standard deviation of $584.64. If the auditor takes a simple random sample of 100 account s , what is the probability that the mean of the sample will be within $120 of the population mean?

ANSWER:0.9596

78. In a given year, the averag e annual salary of a NFL football player was $205,000 with a stand ard deviation of $24,500. If a simple rando m sample of 50 players was taken, what is the probability that the sample mean will exceed $210,000?

ANSWER:

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0.0749

QUESTIONS 79 THROUGH 82 ARE BASED ON THE FOLLOWING INFORMATION:

The heights of men in the USA are normally distributed with a mean of 68 inches and a stand ard deviation of 4 inches .

79. What is the probability that a rando mly selected man is taller than 70 inches?

ANSWER:0.3085

80. A random sample of five men is selected . What is the probability that the sample mean is grea t e r than 70 inches?

ANSWER:0.1314

81. What is the probability that the mean height of a rando m sample of 36 men is grea t e r than 70 inches?

ANSWER:0.0013

82. If the population of men’s heights is not normally distributed, which, if any, of the ques tions 59 through 61 can you answer?

ANSWER:We can answer Questions 80 and 81. We cannot answer Question 79.


The amount of time spent by American adults playing sports per day is normally distributed with a mean of 4 hours and standard deviation of 1.25 hours.

83. Find the probability that a rando mly selected American adult plays sports for more than 5 hours per day

ANSWER:0.2119

84. Find the probability that if four American adults are randomly selected, their averag e number of hours spent playing sports is more than 5 hours per day.

ANSWER:0.0548


85. Find the probability that if four American adults are rando mly selected, all four play sports for more than 5 hours per day.

ANSWER:0.0020


The following data give the number of pets owned for a population of 4 families

Family A B C DNumber of Pets Owned 2 1 4 3

86. Find the mean and the standard deviation for the population.

ANSWER:µ =2.5 and σ =1.118

87. Samples of size 2 will be drawn at rando m from the population. Use the answers to Question 86 to calculat e the mean and the standard deviation of the sampling distribution of the sample means .

ANSWER:

5.2== µµx and 1−

−=N

nN

nx

σσ = 0.6455

88. List all possible samples of 2 families that can be selected without replace m e n t from this population, and compute the sample mean X for each sample.

ANSWER:

Sample AB AC AD BC BD CDx 1.5 3.0 2.5 2.5 2.0 3.5

89. Find the sampling distribution of X .

ANSWER:

x 1.5 2.0 2.5 3.0 3.5p ( x ) 1/6 1/6 1/6 1/6 1/6

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90. Use the sampling distribution in Question 89 and directly recalculat e the mean and standard deviation of X . Compare the answers to that of Question 87. What do you conclude?

ANSWER:xµ = ∑ )(xpx = 2.50, and

∑ −= 22 )()( xx xpx µσ = 0.4167 = 0.6455The same answers .

QUESTIONS 91 AND 92 ARE BASED ON THE FOLLOWING INFORMATION:

Two independ e n t random samples of 25 observa tions each are drawn from two normal populations. The param e t e r s of these populations are:

Population 1: µ =150 σ =20Population 2: µ =130 σ =16

91. Find the probability that the mean of sample 1 will exceed the mean of sample 2.

ANSWER:0.7823

92. Find the probability that the mean of sample 1 is great e r than the mean of sample 2 by more than 15.

ANSWER: 0.0158


A videocass e t t e rental store wants to know what proportion of its custom ers are under age 21. A simple rando m sample of 500 custom er s was taken, and 375 of them were under age 21. Presum e that the true population proportion of custom ers under age 21 is 0.68.

93. Describe the sampling distribution of proportion of custom ers who are under age 21.

ANSWER:Since 5, and (1 ) 5np n p≥ − ≥ , the sampling distribution of p̂ is approxima t ely

normal.

94. Find the mean and standard deviation of p̂

ANSWER:


p̂ pµ = = 0.68, and ˆ (1 ) /p p p nσ = − = 0.02086

95. What is the probability that the sample proportion p̂ will be within 0.03 of the true proportion of custom ers who are under age 21?

ANSWER:0.8502

96. Suppose that the starting salaries of finance gradua t e s from university A are normally distributed with a mean of $36,750 and a stand ard deviation of $5,320. The starting salaries of finance gradua t e s from university B are normally distributed with a mean of $34,625 and a stand ard deviation of $6,540. If simple rando m samples of 50 finance gradua t e s are select ed from each university, what is the probability that the sample mean of university A gradua t e s will exceed that of university B gradua t e s?

ANSWER:

1 2 1.78( 0) ( )P X X P Z− > = > − = 0.9625

97. The publisher of a daily newspap er claims that 90% of its subscribers are under the age of 30. Suppose that a sample of 300 subscribers is selected at random. Assuming the claim is correct , approxima t e the probability of finding at least 240 subscribers in the sample under the age of 30.

ANSWER:0 .9783

98. A local newspa p er sells an averag e of 2100 papers per day, with a standard deviation of 500 papers . Consider a sample of 60 days of operation.a. What is the shape of the sampling distribution of the sample mean

numb er of papers sold per day? Why?b. Find the expecte d value and the stand ard error of the sample mean.c. What is the probability that the sample mean will be betwe en 2000 and

2300 papers .?

ANSWERS:a. Approxima tely normal since n >30.b. xµ = 2100 and xσ = 77.46c. 0.8966

99. Given a binomial random variable with n =15 and p =.40, find the exact probabilities of the following events and their normal approximations.a. X = 6b. X ≥ 9c. X ≤ 10

ANSWERS:a. Exact and approxima t e d probabilities are 0.207 and 0.2052, respec tively.b. Exact and approxima t e d probabilities are 0.095 and 0.0934, respec tively.

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c. Exact and approxima t e d probabilities are 0.991 and 0.9911, respec tively.100. A simple random sample of 300 observa tions is taken from a population.

Assume that the population proportion p = 0.6.a. What is the expecte d value of the sample proportion p̂ ?b. What is the stand ard error of the sample proportion p̂ ?c. What is the probability that the sample proportion p̂ will be within ±0.02

of the population pro portion p ?

ANSWERS:a. 0.6000b. 0.0283c. 0.5222

101. The chairman of a statistics depar t m e n t in a certain college believes that 70% of the depar t m e n t’s gradua t e assistan t ships are given to interna tional studen t s . A random sample of 50 gradua t e assistan t s is taken.a. Assume that the chairman is correct and p = 0.70. What is the sampling

distribution of the sample proportion p̂ ? Explain.b. Find the expecte d value and the stand ard error of the sampling

distribution of p̂ .c. What is the probability that the sample proportion p̂ will be betwee n 0.65

and 0.73?d. What is the probability that the sample proportion p̂ will be within ±.05

of the population proportion p ?

ANSWERS:a. Approxima tely normal, since np =35 and n( 1 p )=15 are both great e r than

5.b. E( p̂ ) = 0.70, and p̂σ = 0.0648c. 0.4566d. 0.5588

CHAPTER 9 SAMPLING DISTRIBUTIONS MULTIPLE CHOICE...

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