Chapter 07 The expected value of x equals the mean of the population from which

CHAPTER SEVEN

SAMPLING AND SAMPLING DISTRIBUTIONS

MULTIPLE-CHOICE QUESTIONS

In the following multiple-choice questions, circle the correct answer.

1. The expected value of

x

equals the mean of the population from which the sample is

drawn

a. only if the sample size is 30 or greater

b. only if the sample size is 50 or greater

c. only if the sample size is 100 or greater

d. for any sample size

2. The basis for using a normal probability distribution to approximate the sampling

distribution of

and xp

is

a. Chebyshev’s theorem

b. the empirical rule

c. the central limit theorem

d. Bayes’ theorem

3. The standard deviation of

p

is referred to as the

a. standard proportion

b. sample proportion

c. average proportion

d. standard error of the proportion

4. The standard deviation of

x

is referred to as the

a. standard x

b. standard error of the mean

c. sample standard mean

d. sample mean deviation

5. The value of the ___________ is used to estimate the value of the population parameter.

a. population statistic

b. sample parameter

c. population estimate

d. sample statistic

6. The population being studied is usually considered ______ if it involves an ongoing

process that makes listing or counting every element in the population impossible.

a. finite

b. infinite

c. skewed

d. symmetric

7. A probability sampling method in which we randomly select one of the first k elements

and then select every kth element thereafter is

a. stratified random sampling

b. cluster sampling

c. systematic sampling

d. convenience sampling

8. The standard deviation of a point estimator is the

a. standard error

b. sample statistic

c. point estimate

d. sampling error

9. The finite correction factor should be used in the computation of

and

xp



when n/N is

greater than

a. .01

b. .025

c. .05

d. .10

10. The set of all elements of interest in a study is

a. set notation

b. a set of interest

c. a sample

d. a population

11. A subset of a population selected to represent the population is a

a. subset

b. sample

c. small population

d. None of the alternative answers is correct.

12. The purpose of statistical inference is to provide information about the

a. sample based upon information contained in the population

b. population based upon information contained in the sample

c. population based upon information contained in the population

d. mean of the sample based upon the mean of the population

13. A simple random sample of size n from a finite population of size N is a sample selected

such that each possible sample of size

a. N has the same probability of being selected

b. n has a probability of 0.5 of being selected

c. n has a probability of 0.1 of being selected

d. n has the same probability of being selected

14. The number of random samples (without replacement) of size 3 that can be drawn from a

population of size 5 is

a. 15

b. 10

c. 20

d. 125

15. number of simple random samples of size 2 (without replacement) which are possible

equals

a. 12

b. 15

c. 3

d. 16

16. How many different samples of size 3 (without replacement) can be taken from a finite

population of size 10?

a. 30

b. 1,000

c. 720

d. 120

17. A population consists of 8 items. The number of different simple random samples of size

3 (without replacement) that can be selected from this population is

a. 24

b. 56

c. 512

d. 128

18. A population consists of 500 elements. We want to draw a simple random sample of 50

elements from this population. On the first selection, the probability of an element being

selected is

a. 0.100

b. 0.010

c. 0.001

d. 0.002

19. Excel’s RAND function

a. determines sample size

b. selects a simple random sample

c. randomizes a population

d. generates random numbers

20. A simple random sample of size n from a finite population of size N is to be selected.

Each possible sample should have

a. the same probability of being selected

b. a probability of 1/n of being selected

c. a probability of 1/N of being selected

d. a probability of N/n of being selected

21. A simple random sample from a process (an infinite population) is a sample selected such

that

a. each element selected comes from the same population

b. each element is selected independently

c. each element selected comes from the same population and each element is

selected independently

d. the probability of being selected changes

22. A numerical measure from a population, such as a population mean, is called

a. a statistic

b. a parameter

c. a sample

d. the mean deviation

23. A numerical measure from a sample, such as a sample mean, is known as

a. a statistic

b. a parameter

c. the mean deviation

d. the central limit theorem

24. A sample statistic, such as

x

, that estimates the value of the corresponding population

parameter is known as a

a. point estimator

b. parameter

c. population parameter

d. Both a parameter and a population parameter are correct.

25. A single numerical value used as an estimate of a population parameter is known as

a. a parameter

b. a population parameter

c. both a parameter or a population parameter are correct

d. a point estimate

26. In point estimation, data from the

a. population is used to estimate the population parameter

b. sample is used to estimate the population parameter

c. sample is used to estimate the sample statistic

d. None of the alternative ANSWERS is correct.

27. The sample mean is the point estimator of

a. 

b. 

c.

x

d.

p

28. The sample statistic s is the point estimator of

a. 

b. 

c.

x

d.

p

29. Which of the following is(are) point estimator(s)?

a. 

b. 

c. s

d. All of these answers are correct.

30. A simple random sample of 5 observations from a population containing 400 elements

was taken, and the following values were obtained.

12

18

19

20

21

A point estimate of the population mean is

a. 5

b. 18

c. 19

d. 20

31. The sampling error is the

a. same as the standard error

b. absolute value of the difference between an unbiased point estimate and the

corresponding population parameter

c. error caused by selecting a bad sample

d. standard deviation multiplied by the sample size

32. A probability distribution for all possible values of a sample statistic is known as a

a. sample statistic

b. parameter

c. simple random sample

d. sampling distribution

33. A simple random sample of 28 observations was taken from a large population. The

sample mean equaled 50. Fifty is a

a. population parameter

b. point estimator

c. sample parameter

d. point estimate

Exhibit 7-1

The following data was collected from a simple random sample from a process (an infinite

population).

13

15

14

16

12

34. Refer to Exhibit 7-1. The point estimate of the population mean

a. is 5

b. is 14

c. is 4

d. cannot be determined because the population is infinite

35. Refer to Exhibit 7-1. The point estimate of the population standard deviation is

a. 2.500

b. 1.581

c. 2.000

d. 1.414

36. Refer to Exhibit 7-1. The mean of the population

a. is 14

b. is 15

c. is 15.1581

d. could be any value

Exhibit 7-2

Four hundred registered voters were randomly selected asked whether gun laws should be

changed. Three hundred said “yes,” and one hundred said “no.”

37. Refer to Exhibit 7-2. The point estimate of the proportion in the population who will

respond “yes” is

a. 300

b. approximately 300

c. 0.75

d. 0.25

38. Refer to Exhibit 7-2. The point estimate of the proportion in the population who will

respond “no” is

a. 75

b. 0.25

c. 0.75

d. 0.50

Exhibit 7-3

The following information was collected from a simple random sample of a population.

16

19

18

17

20

18

39. Refer to Exhibit 7-3. The point estimate of the mean of the population is

a. 18.0

b. 19.6

c. 108

d. sixteen, since 16 is the smallest value in the sample

40. Refer to Exhibit 7-3. The point estimate of the population standard deviation is

a. 2.000

b. 1.291

c. 1.414

d. 1.667

41. If we consider the simple random sampling process as an experiment, the sample mean is

a. always zero

b. always smaller than the population mean

c. a random variable

d. exactly equal to the population mean

42. The probability distribution of all possible values of the sample mean is called the

a. central probability distribution

b. sampling distribution of the sample mean

c. random variation

d. standard error

43. The sampling distribution of the sample mean

a. is the probability distribution showing all possible values of the sample mean

b. is used as a point estimator of the population mean 

c. is an unbiased estimator

d. shows the distribution of all possible values of 

44. Since the sample size is always smaller than the size of the population, the sample mean

must

a. always be smaller than the population mean

b. be larger than the population mean

c. be equal to the population mean

d. None of the alternative ANSWERS is correct.

45. The expected value of the random variable

x

is

a. 

b. the standard error

c. the sample size

d. 

46. The standard deviation of all possible

x

values is called the

a. standard error of proportion

b. standard error of the mean

c. mean deviation

d. central variation

47. In computing the standard error of the mean, the finite population correction factor is not

used when

a. n/N  0.05

b. N/n  0.05

c. n/N  0.05

d. n > 30

48. A finite population correction factor is needed in computing the standard deviation of the

sampling distribution of sample means

a. whenever the population is infinite

b. whenever the sample size is more than 5% of the population size

c. whenever the sample size is less than 5% of the population size

d. The correction factor is not necessary if the population has a normal distribution.

49. From a population of 200 elements, the standard deviation is known to be 14. A sample

of 49 elements is selected. It is determined that the sample mean is 56. The standard

error of the mean is

a. 3

b. 2

c. greater than 2

d. less than 2

50. From a population of 500 elements, a sample of 225 elements is selected. It is known

that the variance of the population is 900. The standard error of the mean is

approximately

a. 1.1022

b. 2

c. 30

d. 1.4847

51. A simple random sample of 64 observations was taken from a large population. The

population standard deviation is 120. The sample mean was determined to be 320. The

standard error of the mean is

a. 1.875

b. 40

c. 5

d. 15

52. As the sample size increases, the

a. standard deviation of the population decreases

b. population mean increases

c. standard error of the mean decreases

d. standard error of the mean increases

53. As the sample size increases, the variability among the sample means

a. increases

b. decreases

c. remains the same

d. depends upon the specific population being sampled

54. Doubling the size of the sample will

a. reduce the standard error of the mean to one-half its current value

b. reduce the standard error of the mean to approximately 70% of its current value

c. have no effect on the standard error of the mean

d. double the standard error of the mean

55. Random samples of size 49 are taken from a population that has 200 elements, a mean of

180, and a variance of 196. The distribution of the population is unknown. The mean

and the standard error of the distribution of sample means are

a. 180 and 24.39

b. 180 and 28

c. 180 and 1.74

d. 180 and 2

56. Random samples of size 81 are taken from a process (an infinite population) whose mean

and standard deviation are 200 and 18, respectively. The distribution of the population is

unknown. The mean and the standard error of the distribution of sample means are

a. 200 and 18

b. 81 and 18

c. 9 and 2

d. 200 and 2

57. Random samples of size 36 are taken from a process (an infinite population) whose mean

and standard deviation are 20 and 15, respectively. The distribution of the population is

unknown. The mean and the standard error of the distribution of sample mean are

a. 36 and 15

b. 20 and 15

c. 20 and 0.417

d. 20 and 2.5

58. 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

a. approximation theorem

b. normal probability theorem

c. central limit theorem

d. central normality theorem

59. The fact that the sampling distribution of the sample mean can be approximated by a

normal probability distribution whenever the sample size is large is based on the

a. central limit theorem

b. fact that there are tables of areas for the normal distribution

c. assumption that the population has a normal distribution

d. All of these answers are correct.

60. As the sample size becomes larger, the sampling distribution of the sample mean

approaches a

a. binomial distribution

b. Poisson distribution

c. hypergeometric distribution

d. None of the alternative answers is correct.

61. Whenever the population has a normal probability distribution, the sampling distribution

of

x

is a normal probability distribution for

a. only large sample sizes

b. only small sample sizes

c. any sample size

d. only samples of size thirty or greater

62. For a population with an unknown distribution, the form of the sampling distribution of

the sample mean is

a. approximately normal for all sample sizes

b. exactly normal for large sample sizes

c. exactly normal for all sample sizes

d. approximately normal for large sample sizes

63. A sample of 24 observations is taken from a population that has 150 elements. The

sampling distribution of

x

is

a. approximately normal because

x

is always approximately normally distributed

b. approximately normal because the sample size is large in comparison to the

population size

c. approximately normal because of the central limit theorem

d. normal if the population is normally distributed

64. A sample of 92 observations is taken from a process (an infinite population). The

sampling distribution of

x

is approximately normal because

a.

x

is always approximately normally distributed

b. the sample size is small in comparison to the population size

c. of the central limit theorem

d. None of the alternative answers is correct.

65. A population has a mean of 80 and a standard deviation of 7. A sample of 49

observations will be taken. The probability that the mean from that sample will be larger

than 82 is

a. 0.5228

b. 0.9772

c. 0.4772

d. 0.0228

66. A population has a mean of 180 and a standard deviation of 24. A sample of 64

observations will be taken. The probability that the mean from that sample will be

between 183 and 186 is

a. 0.1359

b. 0.8185

c. 0.3413

d. 0.4772

67. A population has a mean of 84 and a standard deviation of 12. A sample of 36

observations will be taken. The probability that the sample mean will be between 80.54

and 88.9 is

a. 0.0347

b. 0.7200

c. 0.9511

d. None of the alternative answers is correct.

68. A population has a mean of 53 and a standard deviation of 21. A sample of 49

observations will be taken. The probability that the sample mean will be greater than

57.95 is

a. 0

b. .0495

c. .4505

d. None of the alternative answers is correct.

Exhibit 7-4

A random sample of 121 bottles of cologne showed an average content of 4 ounces. It is known

that the standard deviation of the contents (i.e., of the population) is 0.22 ounces.

69. Refer to Exhibit 7-4. The standard error of the mean equals

a. 0.3636

b. 0.0331

c. 0.0200

d. 4.000

70. Refer to Exhibit 7-4. The point estimate of the mean content of all bottles is

a. 0.22

b. 4

c. 121

d. 0.02

71. Refer to Exhibit 7-4. In this problem the 0.22 is

a. a parameter

b. a statistic

c. the standard error of the mean

d. the average content of colognes in the long run

Exhibit 7-5

Random samples of size 17 are taken from a population that has 200 elements, a mean of 36, and

a standard deviation of 8.

72. Refer to Exhibit 7-5. The mean and the standard deviation of the sampling distribution of

the sample means are

a. 8.7 and 1.94

b. 36 and 1.94

c. 36 and 1.86

d. 36 and 8

73. Refer to Exhibit 7-5. Which of the following best describes the form of the sampling

distribution of the sample mean for this situation?

a. Approximately normal because the sample size is small relative to the population

size.

b. Approximately normal because of the central limit theorem.

c. exactly normal

d. None of the alternative answers is correct.

74. The probability distribution of all possible values of the sample proportion

p

is the

a. probability density function of

p

b. sampling distribution of

x

c. same as

p

, since it considers all possible values of the sample proportion

d. sampling distribution of

p

75. Random samples of size 525 are taken from a process (an infinite population) whose

population proportion is 0.3. The standard deviation of the sample proportions (i.e., the

standard error of the proportion) is

a. 0.0004

b. 0.2100

c. 0.3000

d. 0.0200

76. A random sample of 150 people was taken from a very large population. Ninety of the

people in the sample were females. The standard error of the proportion of females is

a. 0.0016

b. 0.2400

c. 0.1600

d. 0.0400

77. A population of size 1,000 has a proportion of 0.5. Therefore, the proportion and the

standard deviation of the sample proportion for samples of size 100 are

a. 500 and 0.047

b. 500 and 0.050

c. 0.5 and 0.047

d. 0.5 and 0.050

78. Random samples of size 100 are taken from a process (an infinite population) whose

population proportion is 0.2. The mean and standard deviation of the distribution of

sample proportions are

a. 0.2 and .04

b. 0.2 and 0.2

c. 20 and .04

d. None of the alternative answers is correct.

79. As a general rule, the sampling distribution of the sample proportions can be

approximated by a normal probability distribution whenever

a. np  5

b. n(1 – p)  5

c. n  30

d. Both np  5 and n(1 – p)  5 are true.

80. A sample of 25 observations is taken from a process (an infinite population). The

sampling distribution of

p

is

a. not normal since n  30

b. approximately normal because

p

is always normally distributed

c. approximately normal if np  5 and n(1–p)  5

d. approximately normal if np  30 and n(1–p)  30

81. A sample of 400 observations will be taken from a process (an infinite population). The

population proportion equals 0.8. The probability that the sample proportion will be

greater than 0.83 is

a. 0.4332

b. 0.9332

c. 0.0668

d. 0.5668

82. A sample of 66 observations will be taken from a process (an infinite population). The

population proportion equals 0.12. The probability that the sample proportion will be

less than 0.1768 is

a. 0.0568

b. 0.0778

c. 0.4222

d. 0.9222

83. A sample of 51 observations will be taken from a process (an infinite population). The

population proportion equals 0.85. The probability that the sample proportion will be

between 0.9115 and 0.946 is

a. 0.8633

b. 0.6900

c. 0.0819

d. 0.0345

84. Stratified random sampling is a method of selecting a sample in which

a. the sample is first divided into groups, and then random samples are taken from

each group

b. various strata are selected from the sample

c. the population is first divided into groups, and then random samples are drawn

from each group

d. None of the alternative answers is correct.

85. Cluster sampling is

a. a nonprobability sampling method