GM533 Chapter 6 Study Guide

Chapter 6Sampling Distributions

True/False

1. If we have a sample size of 100 and the estimate of the population proportion is .10, we can estimate the sampling distribution of p̂ with a normal distribution. Answer: True Difficulty: Easy

2. A sample size of 500 is sufficiently large enough to conclude that the sampling distribution of p̂ is a normal distribution, when the estimate of the population proportion is .995. Answer: False Difficulty: Medium

3. The sampling distribution of X must be a normal distribution with a mean 0 and standard deviation 1. Answer: False Difficulty: Medium (REF)

4. For any sampled population, the population of all sample means is approximately normally distributed. Answer: False Difficulty: Medium

5. The sampling distribution of a sample statistic is the probability distribution of the population of all possible values of the sample statistic. Answer: True Difficulty: Easy

6. A sample statistic is an unbiased point estimate of a population parameter if the mean of the populations of all possible values of the sample statistic equals the population parameter. Answer: True Difficulty: Medium

7. A minimum variance, unbiased point estimate has a variance that is as small or smaller than the variances of any other unbiased point estimate. Answer: True Difficulty: Medium

8. We can randomly select a sample from an infinite population of potential process measurements by sampling the process at different and equally spaced time points. Answer: True Difficulty: Medium (REF)

Bowerman, Essentials of Business Statistics, 2/e 171

Bowerman, Essentials of Business Statistics, 2/e172

9. The reason sample variance has a divisor of n-1 rather than n is that it makes the variance an unbiased estimate of the population variance. Answer: True Difficulty: Hard

10. The standard deviation of all possible sample proportions increases as the sample size increases. Answer: False Difficulty: Medium (REF)

11. The central limit theorem states that as sample size increases, the population distribution more closely approximates a normal distribution. Answer: False Difficulty: Medium (REF)

12. If a population is known to be normally distributed, then it follows that the sample standard deviation must equal Answer: False Difficulty: Medium

13. If the sampled population is exactly normal distribution, then the sampling distribution of X is also expected to be normal regardless of the sample size. Answer: True Difficulty: Medium

14. If a population is known to be normally distributed, then it follows that the sample mean must equal the population mean. Answer: False Difficulty: Medium

15. If p = .8 and n = 50, then we can conclude that the sampling distribution of p̂ is approximately a normal distribution. Answer: True Difficulty: Easy

16. If p = .9 and n = 40, then we can conclude that the sampling distribution of p̂ is approximately a normal distribution. Answer: False Difficulty: Medium

17. If the sampled population distribution is skewed, then in most cases the sampling distribution of the mean can be approximated by the normal distribution if the sample size n is at least 30. Answer: True Difficulty: Easy


18. The sampling distribution of a sample statistic is the probability distribution of the population of all possible values of the sample statistic. Answer: True Difficulty: Medium

19. As the sample size increases, the standard deviation of the sampling distribution increases. Answer: False Difficulty: Easy (REF)

20. The standard deviation of the sampling distribution of the sample mean is . Answer: False Difficulty: Medium

21. The mean of the sampling distribution of X is always equal to the mean of the sampled population. Answer: True Difficulty: Medium

22. The sampling distribution of the sample mean is developed by repeatedly taking samples of size n and computing the sample means and reporting the resulting sample means in the form of a probability distribution. Answer: True Difficulty: Medium

23. If the sample size n is infinitely large, then is an unbiased estimator of . Answer: True Difficulty: Medium

24. The sample standard deviation s is an unbiased estimator of the population standard deviation . Answer: False Difficulty: Medium

Multiple Choice

25. The central limit theorem states that as the sample size increases the distribution of the sample ________ approach the normal distribution. A) medians B) means C) standard deviations D) variances Answer: B Difficulty: Medium


26. Consider two population distributions labeled A and B. Distribution A is highly skewed and non-normal, while the distribution B is slightly skewed and near normal. In order for the sampling distributions of A and B to achieve the same degree of normality: A) Population A will require a larger sample size. B) Population B will require a larger sample size. C) Population A and B will require the same sample size. D) None of the above Answer: A Difficulty: Medium

27. As the sample size ______________ the variation of the sampling distribution of X ___________. A) Decreases, decreases B) Increases, remains the same C) Decreases, remains the same D) Increases, decreases E) None of the above Answer: D Difficulty: Medium (REF)

28. Consider a sampling distribution formed based on n = 3. The standard deviation of the population of all sample means X is ______________less than the standard deviation of the population of individual measurements .A) Always B) Sometimes C) Never Answer: A Difficulty: Easy

29. If the sampled population has a mean 48 and standard deviation 16, then the mean and the standard deviation for the sampling distribution of X for n = 16 are: A) 4 and 1 B) 12 and 4 C) 48 and 4 D) 48 and 1 E) 48 and 16 Answer: C Difficulty: Medium

Use the following information to answer questions 30-31:A manufacturing company measures the weight of boxes before shipping them to the customers. If the box weights have a population mean and standard deviation of 90 lbs. and 24 lbs. respectively, then based on a sample size of 36 boxes, the probability that the average weight of the boxes will:


30. Exceed 94 lbs. is: A) 34.13% B) 84.13% C) 15.87% D) 56.36% E) 16.87% Answer: C Difficulty: Medium

31. Be less than 84 lbs. is: A) 16.87% B) 93.32% C) 43.32% D) 6.678% E) 84.13% Answer: D Difficulty: Medium

32. The population of all sample proportions has a normal distribution if the sample size (n) is sufficiently large. The rule of thumb for ensuring that n is sufficiently large is: A) np 5 B) n(1 – p) 5 C) np 5 D) n(1 – p) .5 and np 5 E) np .5 and n(1 – p) 5 Answer: E Difficulty: Medium

33. If we have a sample size of 100 and the estimate of the population proportion is .10, the standard deviation of the sampling distribution of the sample proportion is: A) 10 B) 9 C) 3 D) 3.162 E) 90 Answer: C Difficulty: Medium

34. Whenever the population has a normal distribution, the sampling distribution of X is normal or near normal distribution: A) For only large sample sizes B) For only small sample sizes C) For any sample size D) For only samples of size 30 or more Answer: C Difficulty: Medium


35. If a population distribution is known to be normal, then it follows that: A) the sample mean must equal the population mean. B) the sample mean must equal the population mean for large samples. C) the sample standard deviation must equal the population standard deviation. D) All of the above. E) None of the above. Answer: E Difficulty: Easy

36. If we have a sample size of 100 and the estimate of the population proportion is .10, the standard deviation of the sampling distribution of the sample proportion is:A) .0009B) .03C) 3D) 9E) .10Answer: B Difficulty: Medium

37. The internal auditing staff of a local manufacturing company performs a sample audit each quarter to estimate the proportion of accounts that are delinquent more than 90 days overdue. The historical records of the company show that over the past 8 years 13 percent of the accounts are delinquent. For this quarter, the auditing staff randomly selected 250 customer accounts. What is the probability that no more than 40 accounts will be classified as delinquent? A) 42.07 % B) 92.07 % C) 7.93 % D) 40.15 % E) 90.15% Answer: B Difficulty: Hard

Use the following information to answer questions 38-39:The internal auditing staff of a local manufacturing company performs a sample audit each quarter to estimate the proportion of accounts that are delinquent more than 90 days overdue. The historical records of the company show that over the past 8 years 14 percent of the accounts are delinquent. For this quarter, the auditing staff randomly selected 250 customer accounts.

38. What is the probability that at least 30 accounts will be classified as delinquent? A) 31.86 % B) 18.14 % C) 81.86 % D) 63.72 % E) 75.84 % Answer: C Difficulty: Hard


39. What are the mean and the standard deviation of the sampling distribution of p̂ ? A) 14 and 250 B) .14 and .1204 C) 35 and 5.486 D) .14 and .0219 E) 35 and 30.1 Answer: D Difficulty: Medium

Use the following information to answer questions 40-42:According to a hospital administrator, historical records over the past 10 years have shown that 20% of the major surgery patients are dissatisfied with after-surgery care in the hospital. A scientific poll based on 400 hospital patients has just been conducted.

40. What is the probability that less than 64 patients will not be satisfied with the after-surgery care? A) 47.72%B) 2.28%C) 97.72%D) 95.44%E) 4.56%Answer: B Difficulty: Hard (AS)

41. What is the probability that at least 70 patients will not be satisfied with the after-surgery care? A) 89.44%B) 39.44%C) 10.56%D) 78.88%E) 84.49%Answer: A Difficulty: Hard (AS)

42. Sixty-four (64) patients indicated that they were dissatisfied with the after surgery care. What are the mean and the standard deviation of the sampling distribution of p̂ ? A) 16% and .034% B) 20% and 1.83% C) 20% and 2% D) 20% and .034% E) 20% and 16% Answer: C Difficulty: Hard (AS)


43. For non-normal populations, as the sample size (n) ___________________, the distribution of sample means approaches a(n) ________________ distribution. A) Decreases, Uniform B) Increases, Normal C) Decreases, Normal D) Increases, Uniform E) Increases, Exponential Answer: B Difficulty: Hard

Use the following information to answer questions 44-46:In a manufacturing process a machine produces bolts that have an average length of 3 inches with a variance of .03. If we randomly select three bolts from this process:

44. What is the standard deviation of the sample mean? A) .03 B) .01 C) .1732 D) .0577 E) .10 Answer: E Difficulty: Medium

45. What is the probability the mean length of the bolt is at least 3.16 inches? A) 97.72 % B) 5.48 % C) 94.52 % D) 44.52 % E) 2.28 % Answer: B Difficulty: Medium

46. What is the probability the mean length of the bolt is at most 3.1 inches? A) 84.13% B) 100 % C) 71.57 % D) 28.43 % E) 15.87 % Answer: A Difficulty: Medium


Fill-in-the-Blank

47. The spread of the sampling distribution of X is ____________ than the spread of the corresponding population distribution sampling distribution Answer: Smaller Difficulty: Medium

48. For large samples, the sampling distribution of X is approximately normal with a mean of _____. Answer: Difficulty: Medium

49. The _____ says that if the sample size is sufficiently large, then the sample means are approximately normally distributed. Answer: Central Limit Theorem Difficulty: Hard

50. If a sample size is at least _____, then for any sampled population, we can conclude that the sample means are approximately normal. Answer: 30 Difficulty: Medium

51. If we wish to estimate a population parameter by using a sample statistic, we are using _____ estimation. Answer: Point Difficulty: Hard

52. The mean of the sampling distribution of the sample proportion is equal to _____ when the sample size is sufficiently large. Answer: p Difficulty: Medium

53. If the sampled population is finite and at least _____ times larger than the sample size, the standard deviation of p̂ decreases as the sample size increases. Answer: 20 Difficulty: Medium

54. The notation for the standard deviation of the sample mean is __________. Answer: X Xor s Difficulty: Medium

55. The sampling distribution of the sample mean is a normal distribution for very ________ sample sizes regardless of the shape of the corresponding population distribution. Answer: Large Difficulty: Medium


56. A theorem that allows us to use the normal probability distribution to approximate the sampling distribution of sample means and sample proportions whenever the sample size is large is known as the _______________. Answer: Central limit theorem Difficulty: Medium

57. As the sample size increases the variability of the sampling distribution of the mean ______________. Answer: decreases Difficulty: Medium

58. As the sample size ___________, the standard deviation of the population of all sample proportions increases Answer: decreases Difficulty: Medium

59. The population of all _________________ proportions is described by sampling distribution of p̂ .Answer: sample Difficulty: Medium

60. The _______________________ of a sample statistic is the probability distribution of the population of all possible values of the sample statistic. Answer: sampling distribution Difficulty: Medium

61. The ___________ is a minimum-variance unbiased point estimate of the mean of a normally distributed population. Answer: sample mean Difficulty: Medium

Essay

62. Find P( X < 35) if = 40, σx = 16, n = 16. Answer: .1056

Difficulty: Medium


63. A PGA (Professional Golf Association) tournament organizer is attempting to determine whether hole (pin) placement has a significant impact on the average number of strokes for the 13th hole on a given golf course. Historically, the pin has been placed in the front right corner of the green, and the historical mean number of strokes for the hole has been 4.25, with a standard deviation of 1.6 strokes. On a particular day during the most recent golf tournament, the organizer placed the hole (pin) in the back left corner of the green. 64 golfers played the hole with the new placement on that day. Determine the probability of the sample average number of strokes exceeding 4.75. Answer:

P( 4.75) = .5 - .4938 = .0062Difficulty: Hard (AS)

64. Packages of sugar bags for Sweeter Sugar Inc. have an average weight of 16 ounces and a standard deviation of 0.2 ounces. The weights of the sugar bags are normally distributed. What is the probability that 16 randomly selected packages will have a weight in excess of 16.075 ounces? Answer: 0.0668

16.075 16( 1.5) .5 .4332 .0668

.2

16

P Z P Z

Difficulty: Medium

65. The population of lengths of aluminum-coated steel sheets is normally distributed with a mean of 30.05 inches and a standard deviation of 0.3 inches. What is the probability that average length of a steel sheet from a sample of 9 units is more than 29.95 inches long? Answer: 0.8413

29.95 30.05( 1) .3413 .5 .8413

.3

9

P Z P Z

Difficulty: Medium


66. Find P ( X > 172), if = 175 and 2X = 9.

Answer: .8413172 175

19

( 172) .5 .3413 .8413

Z

P X

Difficulty: Medium

67. Find P ( X < 25) if = 16 and X = 4. Answer: .9878

25 162.25

4( 25) .5 .4878 .9878

Z

P X

Difficulty: Medium

68. Find P ( X > 2,510) if = 2,500 and X = 7. Answer: .0764

2510 25001.43

7( 2510) .5 .4236 .0764

Z

P X

Difficulty: Medium

69. Find , if = 400, P( X < 396) =.0228, and n = 100. Answer: 20

396 400 420

2 .2100

X

XZ

n

Difficulty: Hard

70. Find P ( X < 402), if = 400, = 200, and n = 100. Answer: .5398

402 400 2.1

200 20100

( 402) .5 .0398 .5398

Z

P X

Difficulty: Medium


71. Find P(395.4 < X < 404.6), if the population mean = 400, = 20, and n = 100. Answer: .9786

1

2

395.4 400 4.62.3

20 2400

404.6 400 4.62.3

20 2400

(395.4 404.6) .4893 .4893 .9786

Z

Z

P X

Difficulty: Medium

72. The number of defectives in the samples of 50 observations each are the following: 5,1,1,2,3,3,1,4,2,3. What is the estimate of the population proportion of defectives? Answer: .05

5 1 1 2 3 3 1 4 2 3 25.05

(10)(50) 500p

Difficulty: Medium

73. The number of defectives in the samples of 100 observations each are the following: 1,2,1,0,2,3,1,4,2,1. What is the estimate of the population proportion of defectives? Answer: .017

1 2 1 0 2 3 1 4 2 1 17.017

(10)(100) 1,000p

Difficulty: Medium

Use the following information to answer questions 74-75:Suppose that 60 percent of the voters in a particular region support a candidate. Find the probability that a sample of 1,000 voters would yield a sample proportion in favor of the candidate within:

74. 4 percentage points of the actual proportion. Answer: .9902

(.6)(.4).0155

1000.04

2.58.0155

.4951 .4951 .9902

p

Z

Difficulty: Hard (AS)


75. 2 percentage points. Answer: .8030

(.6)(.4).0155

1000.02

1.29.0155

.4015 .4015 .803

p

Z


76. An unbiased estimate of is _____. Answer: Difficulty: Medium

Use the following information to answer questions 77-83:A random sample of size 30 is taken from a population with mean 50 and standard deviation 5.

77. Describe the sampling distribution of X . Answer: Normal/approximately normal Difficulty: Medium

78. What is ? Answer: 50 Difficulty: Medium

79. What is X ? Answer: .913

5.913

30X

Difficulty: Medium

80. Find P ( X > 49). Answer: .8643

49 50 11.0953

5 .91330

( 49) .5 .3643 .8643

Z

P X

Difficulty: Medium


81. Find P ( X < 48). Answer: .0143

48 50 2.02.19

5 .91330

( 48) .5 .4857 .0143

Z

P X

Difficulty: Medium

82. Find P ( X > 50.5). Answer: .2912

50.5 50 0.5.548

5 .91330

( 50.5) .5 .2088 .2912

Z

P X

Difficulty: Medium

83. Find P ( X < 51.5). Answer: .95

51.5 50 1.51.643

5 .91330

( 51.5) .5 .45 .95

Z

P X

Difficulty: Medium

Use the following information to answer questions 84-90:A random sample of size 1,000 is taken from a population where p = .20.

84. Describe the sampling distribution of p̂ Answer: Normal/approximately normal Difficulty: Medium

85. What is p̂ ? Answer: .20 Difficulty: Medium


86. What is ( p̂ )? Answer: .01265

ˆ

(.2)(.8).01265

1000p

Difficulty: Medium

87. Find P( p̂ < .18). Answer: .0571

.18 .20 .21.581

.01265(.2)(.8)1000

ˆ( .18) .5 .4429 ..0571

Z

P p

Difficulty: Medium

88. Find P( p̂ > .175). Answer: .9761

.175 .20 .251.976

.01265(.2)(.8)1000

ˆ( .175) .5 .4761 .9761

Z

P p

Difficulty: Medium

89. Find P( p̂ > .21). Answer: .2148

.21 .20 .01.79

.01265(.2)(.8)1000

ˆ( .21) .5 .2852 .2148

Z

P p

Difficulty: Medium

90. Find P( p̂ < .22). Answer: .9429

.22 .20 .021.58

.01265(.2)(.8)1000

ˆ( .22) .5 .4429 .9429

Z

P p

Difficulty: Medium


91. Packages of sugar bags for Sweeter Sugar Inc. have an average weight of 16 ounces and a standard deviation of 0.3 ounces. The weights of the sugar bags are normally distributed. What is the probability that 9 randomly selected packages will have a weight in excess of 16.025 ounces? Answer: 0.0062

16.025 16( 2.5) .5 .4938 .0062

.3

9

P Z P Z

Difficulty: Medium

Use the following information to answer questions 92-94:The population of lengths of aluminum-coated steel sheets is normally distributed with a mean of 30.05 inches and a standard deviation of 0.2 inches.

92. What is the probability that randomly selected 4 sheets will have an average length of less than 29.9 inches long? Answer: 0.668

29.90 30.05( 1.5) .5 .4332 .0668

.2

4

P Z P Z

Difficulty: Medium

93. A sample of four metal sheets is randomly selected from a batch. What is the probability that the average length of a sheet is between 30.25 and 30.35 inches long? Answer: 0.0215

30.25 30.05 30.35 30.05(2 3) .4987 ..4772 .0215

0.2 0.2

4 4

P Z P Z

Difficulty: Medium


94. The population of lengths of aluminum-coated steel sheets is normally distributed with a mean of 30.05 inches and a standard deviation of 0.3 inches. What is the probability that average length of a steel sheet from a sample of 9 units is more than 29.95 inches long? Answer: 0.8413

29.95 30.05( 1) .3413 .5 .8413

.3

9

P Z P Z

Difficulty: Medium

Use the following information to answer questions 95-96:The chief chemist for a major oil/gasoline production company claims that the regular unleaded gasoline produced by the company contains on average 4 ounces of a certain ingredient. The chemist further states that the distribution of this ingredient per gallon of regular unleaded gasoline is normal and has a standard deviation of 1.2 ounces.

95. What is the probability of finding an average in excess of 4.3 ounces of this ingredient from randomly inspected 100 gallons of regular unleaded gasoline? Answer: .0062

4.3 4 .32.5

1.2 .12100

( 4.3) .5 .4938 .0062

Z

P X

Difficulty: Medium

96. What is the probability of finding an average less than 3.85 ounces of this ingredient from randomly inspected 64 gallons of regular unleaded gasoline? Answer: .1587

3.85 4 0.151.0

1.2 .0.1564

( 3.85) .5 .3413 .1587

Z

P X

Difficulty: Medium

Use the following information to answer questions 97-99:In the upcoming governor's election, the most recent poll based on 900 respondents predicts that the incumbent will be reelected with 55% of the votes.


97. From the 900 respondents, how many indicated that they would not vote for the current governor or indicated that they were undecided? Answer: 405(.45)(900) = 405 Difficulty: Easy

98. What is ? Answer: .0166

Difficulty: Medium

99. For the sake of argument, assume that 51% of the actual voters in the state support the incumbent governor (p = 0.51). Calculate the probability of observing a sample proportion of voters 0.55 or higher supporting the incumbent governor. Answer: .0082

Difficulty: Medium

Use the following information to answer questions 100-104:It has been reported that the average time to download the home page from a government website was 0.9 seconds. Suppose that the download times were normally distributed with a standard deviation of 0.3 seconds. If random samples of 36 download times are selected:

100. Describe the shape of the sampling distribution. How was this determined?Answer: Normal – either because the original population was normal or size of sample > 30Difficulty: Easy (REF)

101. Calculated the mean of the sampling distribution of the sampling mean.Answer: 0.9

= = 0.9Difficulty: Easy (AS)


102. Calculate the standard deviation of the sampling distribution of the sample mean.Answer:0.05

=

Difficulty: Medium (AS)

103. 80% of the sample means will be between what two values symmetrically distributed arount the population mean.Answer: .83575, .96425Z = ±1.285

1.285= , -1.285=


104. What is the probability that the sample mean will be less than 0.80 seconds?Answer: .0228

Difficulty: Hard

Use the following information to answer questions 105-108:The College Student Journal (December 1992) investigated differences in traditional and nontraditional students, where nontraditional students are defined as 25 years or older and working. Based on the study results, we can assume the population mean and standard deviation for the GPA of nontraditional students is µ=3.5 and σ=0.5. Suppose a random sample of 100 nontraditional students is selected and each student’s GPA is calculated.

105.What is ? Answer: =3.5Difficulty: Easy

106. Calculated .Answer: .05

.=

Difficulty: Medium


107. Find the interval that contain 95.44% of the sample means.Answer:3.4, 3.6[3.5 ±2(.05)] = (3.5±.1)= [3.4, 3.6]Difficulty: Medium

108. What is the probability that the random sample of 100 nontraditional students have a mean GPA greater than 3.65?Answer: .0013

, .5-.4987=.0013

Difficulty: Medium

Use the following information to answer questions 109-112:The diameter of small Nerf balls manufactured at a factory in Chine is expected to be approximately normally distributed with a mean of 5.2 inches and a standard deviation of .08 inches. Suppose a random sample of 20 balls are selected.

109. Calculate the mean of the sampling distribution of the sample mean.Answer: Difficulty: Easy (AS)

110. Calculate the standard deviation of the sampling distribution of the sample mean.Answer: .018

Difficulty: Medium (AS)

111. Find the interval that contains 95.44% of the sample means/Answer: 5.164, 5.236[5.2±2(.018)]=[5.2±.036]=5.164, 5.236Difficulty: Medium (AS)

112. What percentage of sample means will be less than 5.00 inches?Answer: less than .01%

less than .01%



GM533 Chapter 6 Study Guide

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