Chapter 08

Sampling Distributions and Estimation

True / False Questions

1. The expected value of an unbiased estimator is equal to the parameter
whose value is being estimated.
True

False

2. All estimators are biased since sampling errors always exist to some
extent.
True

False

3. An estimator must be unbiased if you are to use it for statistical

analysis.
True

False

4. The efficiency of an estimator depends on the variance of the

estimator's sampling distribution.
True

False

5. In comparing estimators, the more efficient estimator will have a

smaller standard error.
True

False

6. A 90 percent confidence interval will be wider than a 95 percent

confidence interval, ceteris paribus.
True

False

7. In constructing a confidence interval for the mean, the z distribution

provides a result nearly identical to the t distribution when n is large.
True

False

8. The Central Limit Theorem says that, if n exceeds 30, the population
will be normal.
True

False

9. The Central Limit Theorem says that a histogram of the sample means
will have a bell shape, even if the population is skewed and the sample
is small.
True

False

10 The confidence level refers to the procedure used to construct the

. confidence interval, rather than to the particular confidence interval we
have constructed.
True

False

11 The Central Limit Theorem guarantees an approximately normal

. sampling distribution when n is sufficiently large.
True

False

12 A sample of size 5 shows a mean of 45.2 and a sample standard

. deviation of 6.4. The standard error of the sample mean is
approximately 2.86.
True

False

13 As n increases, the width of the confidence interval will decrease,

. ceteris paribus.
True

False

14 As n increases, the standard error decreases.

.
True False

15 A higher confidence level leads to a narrower confidence interval,

. ceteris paribus.
True

False

16 When the sample standard deviation is used to construct a confidence

. interval for the mean, we would use the Student's t distribution instead
of the normal distribution.
True

False

17 As long as the sample is more than one item, the standard error of the
. sample mean will be smaller than the standard deviation of the
population.
True

False

18 For a sample size of 20, a 95 percent confidence interval using the t

. distribution would be wider than one constructed using the z
distribution.
True

False

19 In constructing a confidence interval for a mean, the width of the

. interval is dependent on the sample size, the confidence level, and the
population standard deviation.
True

False

20 In constructing confidence intervals, it is conservative to use the z

. distribution when n 30.
True

False

21 The Central Limit Theorem can be applied to the sample proportion.

.
True False
22 The distribution of the sample proportion p = x/n is normal when n
. 30.
True

False

23 The standard deviation of the sample proportion p = x/n increases as n

. increases.
True

False

24 A 95 percent confidence interval constructed around p will be wider

. than a 90 percent confidence interval.
True

False

25 The sample proportion is always the midpoint of a confidence interval

. for the population proportion.
True

False

26 The standard error of the sample proportion is largest when = .50.

.
True False
27 The standard error of the sample proportion does not depend on the
. confidence level.
True

False

28 To narrow the confidence interval for , we can either increase n or

. decrease the level of confidence.
True

False

29 Ceteris paribus, the narrowest confidence interval for is achieved

. when p = .50.
True

False

30 The statistic p = x/n may be assumed normally distributed when np

. 10 and n(1 - p) 10.
True

False

31 The Student's t distribution is always symmetric and bell-shaped, but

. its tails lie above the normal.
True

False

32 The confidence interval half-width when = .50 is called the margin of

. error.
True

False

33 Based on the Rule of Three, if no events occur in n independent trials

. we can set the upper 95 percent confidence bound at 3/n.
True

False

34 The sample standard deviation s is halfway between the lower and

. upper confidence limits for the population (i.e., the confidence
interval is symmetric around s).
True

False

35 In a sample size calculation, if the confidence level decreases, the size

. of the sample needed will increase.
True

False

36 To calculate the sample size needed for a survey to estimate a

. proportion, the population standard deviation must be known.
True

False

37 Assuming that = .50 is a quick and conservative approach to use in a

. sample size calculation for a proportion.
True

False

38 To estimate the required sample size for a proportion, one method is to

. take a small pilot sample to estimate and then apply the sample size
formula.
True

False

39 To estimate , you typically need a sample size equal to at least 5

. percent of your population.
True

False

40 To estimate a proportion with a 4 percent margin of error and a 95

. percent confidence level, the required sample size is over 800.
True

False

41 Approximately 95 percent of the population X values will lie within the

. 95 percent confidence interval for the mean.
True

False

42 A 99 percent confidence interval has more confidence but less

. precision than a 95 percent confidence interval.
True

False

43 Sampling variation is not controllable by the statistician.

.
True False
44 The sample mean is not a random variable when the population
. parameters are known.
True

False

45 The finite population correction factor (FPCF) can be ignored if n = 7

. and N = 700.
True

False

46 In constructing a confidence interval, the finite population correction

. factor (FPCF) can be ignored if samples of 12 items are drawn from a
population of 300 items.
True

False

47 The finite population correction factor (FPCF) can be ignored when the
. sample size is large relative to the population size.
True

False

Multiple Choice Questions

48 A sampling distribution describes the distribution of:
.
A.
B.
C.
D.

a parameter.
a statistic.
either a parameter or a statistic.
neither a parameter nor a statistic.

49 As the sample size increases, the standard error of the mean:

.
A.
B.
C.

increases.
decreases.
may increase or decrease.

50 Which statement is most nearly correct, other things being equal?

.
A. Doubling the sample size will cut the standard error of the mean in
half.
B. The standard error of the mean depends on the population size.
C. Quadrupling the sample size roughly halves the standard error of
the mean.
D. The standard error of the mean depends on the confidence level.

51 The width of a confidence interval for is not affected by:

.
A.
B.
C.
D.

the sample size.

the confidence level.
the standard deviation.
the sample mean.

52 The Central Limit Theorem (CLT) implies that:

.
A.
B.
C.
D.

the population will be approximately normal if n 30.

repeated samples must be taken to obtain normality.
the distribution of the mean is approximately normal for large n.
the mean follows the same distribution as the population.

53 The owner of Limp Pines Resort wanted to know the average age of its
. clients. A random sample of 25 tourists is taken. It shows a mean age
of 46 years with a standard deviation of 5 years. The width of a 98
percent CI for the true mean client age is approximately:

A.
B.
C.
D.

1.711 years.
2.326 years.
2.492 years.
2.797 years.

54 In constructing a confidence interval for a mean with unknown

. variance with a sample of 25 items, Bob used z instead of t. "Well, at
least my interval will be wider than necessary, so it was a conservative
error," said he. Is Bob's statement correct?

A.
B.
C.

Yes.
No.
It depends on .

55 A random sample of 16 ATM transactions at the Last National Bank of

. Flat Rock revealed a mean transaction time of 2.8 minutes with a
standard deviation of 1.2 minutes. The width (in minutes) of the 95
percent confidence interval for the true mean transaction time is:

A.
B.
C.
D.

0.639
0.588
0.300
2.131

56 We could narrow a 95 percent confidence interval by:

.
A.
B.
C.

using 99 percent confidence.

using a larger sample.
raising the standard error.

57 The owner of Torpid Oaks B&B wanted to know the average distance its
. guests had traveled. A random sample of 16 guests showed a mean
distance of 85 miles with a standard deviation of 32 miles. The 90
percent confidence interval (in miles) for the mean is approximately:

A.
B.
C.
D.

(71.0, 99.0)
(71.8, 98.2)
(74.3, 95.7)
(68.7, 103.2)

58 A highway inspector needs an estimate of the mean weight of trucks

. crossing a bridge on the interstate highway system. She selects a
random sample of 49 trucks and finds a mean of 15.8 tons with a
sample standard deviation of 3.85 tons. The 90 percent confidence
interval for the population mean is:

A.
B.
C.
D.

14.72 to 16.88 tons.

14.90 to 16.70 tons.
14.69 to 16.91 tons.
14.88 to 16.72 tons.

59 To determine a 72 percent level of confidence for a proportion, the

. value of z is approximately:

A.
B.
C.
D.

1.65
0.77
1.08
1.55

60 To estimate the average annual expenses of students on books and

. class materials a sample of size 36 is taken. The sample mean is $850
and the sample standard deviation is $54. A 99 percent confidence
interval for the population mean is:

A.
B.
C.
D.

$823.72 to $876.28
$832.36 to $867.64
$826.82 to $873.18
$825.48 to $874.52

61 In constructing a 95 percent confidence interval, if you increase n to

. 4n, the width of your confidence interval will (assuming other things
remain the same) be:

A.
B.
C.
D.

about 25 percent of its former width.

about two times wider.
about 50 percent of its former width.
about four times wider.

62 Which of the following is not a characteristic of the t distribution?

.
A.
It is a continuous distribution.
B.
It has a mean of 0.
C.
It is a symmetric distribution.
D. It approaches z as degrees of freedom decrease.

63 Which statement is incorrect? Explain.

.
A. If p = .50 and n = 100, the standard error of the sample proportion
is .05.
B. In a sample size calculation for estimating , it is conservative to
assume = .50.
C. If n = 250 and p = .06, we cannot assume normality in a confidence
interval for .
64 What is the approximate width of a 90 percent confidence interval for
. the true population proportion if there are 12 successes in a sample of
25?

A.
B.
C.
D.

.196
.164
.480
.206

65 A poll showed that 48 out of 120 randomly chosen graduates of

. California medical schools last year intended to specialize in family
practice. What is the width of a 90 percent confidence interval for the
proportion that plan to specialize in family practice?

A.
B.
C.
D.

.0447
.0736
.0876
.0894

66 What is the approximate width of an 80 percent confidence interval for

. the true population proportion if there are 12 successes in a sample of
80?

A.
B.
C.
D.

.078
.066
.051
.094

67 A random sample of 160 commercial customers of PayMor Lumber

. revealed that 32 had paid their accounts within a month of billing. The
95 percent confidence interval for the true proportion of customers
who pay within a month would be:

A.
B.
C.
D.

0.148 to 0.252
0.138 to 0.262
0.144 to 0.256
0.153 to 0.247

68 A random sample of 160 commercial customers of PayMor Lumber

. revealed that 32 had paid their accounts within a month of billing. Can
normality be assumed for the sample proportion?

A.
B.
C.

Yes.
No.
Need more information to say.

69 The conservative sample size required for a 95 percent confidence

. interval for with an error of 0.04 is:

A.
B.
C.
D.

271.
423.
385.
601.

70 Last week, 108 cars received parking violations in the main university
. parking lot. Of these, 27 had unpaid parking tickets from a previous
violation. Assuming that last week was a random sample of all parking
violators, find the 95 percent confidence interval for the percentage of
parking violators that have prior unpaid parking tickets.

A.
B.
C.
D.

18.1 to 31.9 percent.

16.8 to 33.2 percent.
15.3 to 34.7 percent.
19.5 to 30.5 percent.

71 In a random sample of 810 women employees, it is found that 81

. would prefer working for a female boss. The width of the 95 percent
confidence interval for the proportion of women who prefer a female
boss is:

A.
B.
C.
D.

.0288
.0105
.0207
.0196

72 Jolly Blue Giant Health Insurance (JBGHI) is concerned about rising lab
. test costs and would like to know what proportion of the positive lab
tests for prostate cancer are actually proven correct through
subsequent biopsy. JBGHI demands a sample large enough to ensure
an error of 2 percent with 90 percent confidence. What is the
necessary sample size?

A.
B.
C.
D.

4,148
2,401
1,692
1,604

73 A university wants to estimate the average distance that commuter

. students travel to get to class with an error of 3 miles and 90
percent confidence. What sample size would be needed, assuming that
travel distances are normally distributed with a range of X = 0 to X =
50 miles, using the Empirical Rule 3 to estimate .

A.
B.
C.
D.

About 28 students
About 47 students
About 30 students
About 21 students

74 A financial institution wishes to estimate the mean balances owed by

. its credit card customers. The population standard deviation is $300. If
a 99 percent confidence interval is used and an interval of $75 is
desired, how many cardholders should be sampled?

A.
B.
C.
D.

3382
629
87
107

75 A company wants to estimate the time its trucks take to drive from city
. A to city B. The standard deviation is known to be 12 minutes. What
sample size is required in order that error will not exceed 2 minutes,
with 95 percent confidence?

A.
B.
C.
D.

12 observations
139 observations
36 observations
129 observations

76 In a large lecture class, the professor announced that the scores on a

. recent exam were normally distributed with a range from 51 to 87.
Using the Empirical Rule 3 to estimate , how many students
would you need to sample to estimate the true mean score for the
class with 90 percent confidence and an error of 2?

A.
B.
C.
D.

About 17 students
About 35 students
About 188 students
About 25 students

77 Using the conventional polling definition, find the margin of error for a
. customer satisfaction survey of 225 customers who have recently
dined at Applebee's.

A.
B.
C.
D.

5.0 percent
4.2 percent
7.1 percent
6.5 percent

78 A marketing firm is asked to estimate the percent of existing customers

. who would purchase a "digital upgrade" to their basic cable TV service.
The firm wants 99 percent confidence and an error of 5 percent.
What is the required sample size (to the next higher integer)?

A.
B.
C.
D.

664
625
801
957

79 An airport traffic analyst wants to estimate the proportion of daily

. takeoffs by small business jets (as opposed to commercial passenger
jets or other aircraft) with an error of 4 percent with 90 percent
confidence. What sample size should the analyst use?

A.
B.
C.
D.

385
601
410
423

80 Ersatz Beneficial Insurance wants to estimate the cost of damage to

. cars due to accidents. The standard deviation of the cost is known to
be $200. They want to estimate the mean cost using a 95 percent
confidence interval within $10. What is the minimum sample size n?

A.
B.
C.
D.

1083
4002
1537
2301

81 Professor York randomly surveyed 240 students at Oxnard University

. and found that 150 of the students surveyed watch more than 10
hours of television weekly. Develop a 95 percent confidence interval to
estimate the true proportion of students who watch more than 10
hours of television each week. The confidence interval is:

A.
B.
C.
D.

.533 to .717
.564 to .686
.552 to .698
.551 to .739

82 Professor York randomly surveyed 240 students at Oxnard University

. and found that 150 of the students surveyed watch more than 10
hours of television weekly. How many additional students would
Professor York have to sample to estimate the proportion of all Oxnard
University students who watch more than 10 hours of television each
week within 3 percent with 99 percent confidence?

A.
B.
C.
D.

761
1001
1489
1728

83 The sample proportion is in the middle of the confidence interval for

. the population proportion:

A.
B.
C.

in any sample.
only if the samples are large.
only if is not too far from .50.

84 For a sample of size 16, the critical values of chi-square for a 95

. percent confidence interval for the population variance are:

A.
B.
C.
D.

6.262, 27.49
6.908, 28.85
5.629, 26.12
7.261, 25.00

85 For a sample of size 11, the critical values of chi-square for a 90

. percent confidence interval for the population variance are:

A.
B.
C.
D.

6.262, 27.49
6.908, 28.85
3.940, 18.31
3.247, 20.48

86 For a sample of size 18, the critical values of chi-square for a 99

. percent confidence interval for the population variance are:

A.
B.
C.
D.

6.262, 27.49
5.697, 35.72
5.629, 26.12
7.261, 25.00

87 Which of the following statements is most nearly correct, other things

. being equal?

A. Using Student's t instead of z makes a confidence interval narrower.

B. The table values of z and t are about the same when the mean is
large.
C. For a given confidence level, the z value is always smaller then the t
value.
D. Student's t is rarely used because it is more conservative to use z.
88 The Central Limit Theorem (CLT):
.
A. applies only to samples from normal populations.
B.
applies to any population.
C. applies best to populations that are skewed.
D. applies only when and are known.
89 In which situation may the sample proportion safely be assumed to
. follow a normal distribution?

A.
B.
C.
D.

12 successes in a sample of 72 items

8 successes in a sample of 40 items
6 successes in a sample of 200 items
4 successes in a sample of 500 items

90 In which situation may the sample proportion safely be assumed to

. follow a normal distribution?

A.
B.
C.
D.

n = 100, = .06
n = 250, = .02
n = 30, = .50
n = 500, = .01

91 If = 12, find the sample size to estimate the mean with an error of
. 4 and 95 percent confidence (rounded to the next higher integer).

A.
B.
C.
D.

75
35
58
113

92 If = 25, find the sample size to estimate the mean with an error of
. 3 and 90 percent confidence (rounded to the next higher integer).

A.
B.
C.
D.

426
512
267
188

93 Sampling error can be avoided:

.
A.
by using an unbiased estimator.
B. by eliminating nonresponses (e.g., older people).
C. by no method under the statistician's control.
D. either by using an unbiased estimator or by eliminating
nonresponse.
94 A consistent estimator for the mean:
.
A. converges on the true parameter as the variance increases.
B. converges on the true parameter as the sample size increases.
C. consistently follows a normal distribution.
D. is impossible to obtain using real sample data.

95 Concerning confidence intervals, which statement is most nearly

. correct?

A. We should use z instead of t when n is large.

B. We use the Student's t distribution when is unknown.
C. We use the Student's t distribution to narrow the confidence
interval.
96 The standard error of the mean decreases when:
.
A.
the sample size decreases.
B.
the standard deviation increases.
C. the standard deviation decreases or n increases.
D.
the population size decreases.
97 For a given sample size, the higher the confidence level, the:
.
A.
B.
C.
D.

more accurate the point estimate.

smaller the standard error.
smaller the interval width.
greater the interval width.

98 A sample is taken and a confidence interval is constructed for the

. mean of the distribution. At the center of the interval is always which
value?

A.
The sample mean
B.
The population mean
C. Neither nor since with a sample anything can happen
D. Both and as long as there are not too many outliers

99 If a normal population has parameters = 40 and = 8, then for a

. sample size n = 4:

A.
B.
C.
D.

the standard error of the sample mean is approximately 2.

the standard error of the sample mean is approximately 4.
the standard error of the sample mean is approximately 8.
the standard error of the sample mean is approximately 10.

Short Answer Questions

100 On the basis of a survey of 545 television viewers, a statistician has
.
constructed a confidence interval and estimated that the proportion of
people who watched the season premiere of Glee is between .16 and .
24. What level of confidence did the statistician use in constructing
this interval? Explain carefully, showing all steps in your reasoning.

101 Read the news story below. Using the 95 percent confidence level,
.
what sample size would be needed to estimate the true proportion of
stores selling cigarettes to minors with an error of 3 percent?
Explain carefully, showing all steps in your reasoning.

102 In a survey, 858 out of 2600 homeowners said they expected good
.
economic conditions to continue for the next 12 months. Construct a
95 percent confidence interval for "good times" in the next 12
months.

103 Fulsome University has 16,059 students. In a sample of 200 students,

.
12 were born outside the United States. Construct a 95 percent
confidence interval for the true population proportion. How large a
sample is needed to estimate the true proportion of Fulsome students
who were born outside the United States with an error of 2.5
percent and 95 percent confidence? Show your work and explain fully.

104 List differences and similarities between Student's t and the standard
.
normal distribution.

105 Why does pose a problem for sample size calculation for a mean?
.
How can be approximated when it is unknown?

Chapter 08 Sampling Distributions and Estimation Answer Key

True / False Questions

1.

The expected value of an unbiased estimator is equal to the

parameter whose value is being estimated.
TRUE
An unbiased estimator's expected value is the true parameter value.
AACSB: Analytic
Blooms: Remember
Difficulty: 1 Easy
Learning Objective: 08-02 Explain the desirable properties of estimators.
Topic: Estimators and Sampling Error

2.

All estimators are biased since sampling errors always exist to some
extent.
FALSE
Some estimators are systematically biased, regardless of sampling
error.
AACSB: Analytic
Blooms: Remember
Difficulty: 1 Easy
Learning Objective: 08-02 Explain the desirable properties of estimators.
Topic: Estimators and Sampling Error

3.

An estimator must be unbiased if you are to use it for statistical

analysis.
FALSE
An estimator can be useful as long as its bias is known.
AACSB: Analytic
Blooms: Remember
Difficulty: 2 Medium
Learning Objective: 08-02 Explain the desirable properties of estimators.
Topic: Estimators and Sampling Error

4.

The efficiency of an estimator depends on the variance of the

estimator's sampling distribution.
TRUE
Efficiency is measured by the variance of the estimator's sampling
distribution.
AACSB: Analytic
Blooms: Remember
Difficulty: 1 Easy
Learning Objective: 08-02 Explain the desirable properties of estimators.
Topic: Estimators and Sampling Error

5.

In comparing estimators, the more efficient estimator will have a

smaller standard error.
TRUE
Efficiency is measured by the variance of the estimator's sampling
distribution.
AACSB: Analytic
Blooms: Remember
Difficulty: 1 Easy
Learning Objective: 08-02 Explain the desirable properties of estimators.
Topic: Estimators and Sampling Error

6.

A 90 percent confidence interval will be wider than a 95 percent

confidence interval, ceteris paribus.
FALSE
We can make a more precise statement about the true parameter if
we are willing to sacrifice some confidence. For example, z.025 =
1.960 (for 95 percent confidence) gives a wider interval than z.05 =
1.645 (for 90 percent confidence). The proffered statement would
also hold true for the Student's t distribution.
AACSB: Analytic
Blooms: Understand
Difficulty: 1 Easy
Learning Objective: 08-05 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Mean () with Known

7.

In constructing a confidence interval for the mean, the z distribution

provides a result nearly identical to the t distribution when n is
large.
TRUE
Student's t approaches z as sample size increases.
AACSB: Analytic
Blooms: Remember
Difficulty: 1 Easy
Learning Objective: 08-06 Know when to use Student's t instead of z to estimate .
Topic: Confidence Interval for a Mean () with Unknown

8.

The Central Limit Theorem says that, if n exceeds 30, the population
will be normal.
FALSE
The population cannot be changed.
AACSB: Analytic
Blooms: Remember
Difficulty: 2 Medium
Learning Objective: 08-03 State the Central Limit Theorem for a mean.
Topic: Sample Mean and the Central Limit Theorem

9.

The Central Limit Theorem says that a histogram of the sample

means will have a bell shape, even if the population is skewed and
the sample is small.
FALSE
A large sample size may be required if the population is skewed.
AACSB: Analytic
Blooms: Remember
Difficulty: 2 Medium
Learning Objective: 08-03 State the Central Limit Theorem for a mean.
Topic: Sample Mean and the Central Limit Theorem

10.

The confidence level refers to the procedure used to construct the

confidence interval, rather than to the particular confidence interval
we have constructed.
TRUE
A particular interval either does or does not contain the true
parameter.
AACSB: Analytic
Blooms: Understand
Difficulty: 2 Medium
Learning Objective: 08-05 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Mean () with Known

11.

The Central Limit Theorem guarantees an approximately normal

sampling distribution when n is sufficiently large.
TRUE
Yes, although a large sample size may be required if the population
is skewed.
AACSB: Analytic
Blooms: Understand
Difficulty: 2 Medium
Learning Objective: 08-03 State the Central Limit Theorem for a mean.
Topic: Sample Mean and the Central Limit Theorem

12.

A sample of size 5 shows a mean of 45.2 and a sample standard

deviation of 6.4. The standard error of the sample mean is
approximately 2.86.
TRUE
The standard error is the standard deviation divided by the square
root of the sample size.
AACSB: Analytic
Blooms: Apply
Difficulty: 1 Easy
Learning Objective: 08-04 Explain how sample size affects the standard error.
Topic: Confidence Interval for a Mean () with Unknown

13.

As n increases, the width of the confidence interval will decrease,

ceteris paribus.
TRUE
The standard error is the standard deviation divided by the square
root of the sample size.
AACSB: Analytic
Blooms: Understand
Difficulty: 1 Easy
Learning Objective: 08-05 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Mean () with Known

14.

As n increases, the standard error decreases.

TRUE
The standard error is the standard deviation divided by the square
root of the sample size.
AACSB: Analytic
Blooms: Remember
Difficulty: 2 Medium
Learning Objective: 08-04 Explain how sample size affects the standard error.
Topic: Sample Mean and the Central Limit Theorem

15.

A higher confidence level leads to a narrower confidence interval,

ceteris paribus.
FALSE
Higher confidence requires more uncertainty (a wider interval). For
example, z.025 = 1.960 (for 95 percent confidence) gives a wider
interval than z.05 = 1.645 (for 90 percent confidence). The proffered
statement would also hold true for the Student's t distribution.
AACSB: Analytic
Blooms: Understand
Difficulty: 2 Medium
Learning Objective: 08-05 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Mean () with Known

16.

When the sample standard deviation is used to construct a

confidence interval for the mean, we would use the Student's t
distribution instead of the normal distribution.
TRUE
We should use t when the population variance is unknown.
AACSB: Analytic
Blooms: Remember
Difficulty: 1 Easy
Learning Objective: 08-06 Know when to use Student's t instead of z to estimate .
Topic: Confidence Interval for a Mean () with Unknown

17.

As long as the sample is more than one item, the standard error of
the sample mean will be smaller than the standard deviation of the
population.
TRUE
The standard error is the standard deviation divided by the square
root of the sample size.
AACSB: Analytic
Blooms: Understand
Difficulty: 1 Easy
Learning Objective: 08-04 Explain how sample size affects the standard error.
Topic: Sample Mean and the Central Limit Theorem

18.

For a sample size of 20, a 95 percent confidence interval using the t

distribution would be wider than one constructed using the z
distribution.
TRUE
Student's t is always larger than z for the same level of confidence.
AACSB: Analytic
Blooms: Remember
Difficulty: 2 Medium
Learning Objective: 08-06 Know when to use Student's t instead of z to estimate .
Topic: Confidence Interval for a Mean () with Unknown

19.

In constructing a confidence interval for a mean, the width of the

interval is dependent on the sample size, the confidence level, and
the population standard deviation.
TRUE
The confidence interval depends on all of these.
AACSB: Analytic
Blooms: Remember
Difficulty: 2 Medium
Learning Objective: 08-05 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Mean () with Known

20.

In constructing confidence intervals, it is conservative to use the z

distribution when n 30.
FALSE
While t and z may be similar for large samples, it is more
conservative to use t.
AACSB: Analytic
Blooms: Remember
Difficulty: 1 Easy
Learning Objective: 08-06 Know when to use Student's t instead of z to estimate .
Topic: Confidence Interval for a Mean () with Unknown

21.

The Central Limit Theorem can be applied to the sample proportion.

TRUE
We are sampling a Bernoulli population, but the CLT still applies.
AACSB: Analytic
Blooms: Remember
Difficulty: 1 Easy
Learning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Proportion ()

22.

The distribution of the sample proportion p = x/n is normal when n

30.
FALSE
We want at least 10 successes and 10 failures to assume that p is
normally distributed.
AACSB: Analytic
Blooms: Understand
Difficulty: 1 Easy
Learning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Proportion ()

23.

The standard deviation of the sample proportion p = x/n increases as

n increases.
FALSE
The proffered statement is backwards because n is in the
denominator of [p(1 - p)/n]1/2.
AACSB: Analytic
Blooms: Remember
Difficulty: 1 Easy
Learning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Proportion ()

24.

A 95 percent confidence interval constructed around p will be wider

than a 90 percent confidence interval.
TRUE
Higher confidence requires more uncertainty (a wider interval).
AACSB: Analytic
Blooms: Understand
Difficulty: 1 Easy
Learning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Proportion ()

25.

The sample proportion is always the midpoint of a confidence

interval for the population proportion.
TRUE
The interval is p z[p(1 - p)/n]1/2.
AACSB: Analytic
Blooms: Remember
Difficulty: 2 Medium
Learning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Proportion ()

26.

The standard error of the sample proportion is largest when = .50.

TRUE
The value of [(1 - )/n]1/2 is smaller for any value less than = .50.
AACSB: Analytic
Blooms: Understand
Difficulty: 2 Medium
Learning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Proportion ()

27.

The standard error of the sample proportion does not depend on the
confidence level.
TRUE
The standard error of p is [(1 - )/n]1/2.
AACSB: Analytic
Blooms: Understand
Difficulty: 2 Medium
Learning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Proportion ()

28.

To narrow the confidence interval for , we can either increase n or

decrease the level of confidence.
TRUE
The interval is p z[p(1 - p)/n]1/2.
AACSB: Analytic
Blooms: Understand
Difficulty: 2 Medium
Learning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Proportion ()

29.

Ceteris paribus, the narrowest confidence interval for is achieved

when p = .50.
FALSE
The value of [p(1 - p)/n]1/2 is smaller for any value less than = .50.
AACSB: Analytic
Blooms: Understand
Difficulty: 2 Medium
Learning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Proportion ()

30.

The statistic p = x/n may be assumed normally distributed when np

10 and n(1 - p) 10.
TRUE
We want at least 10 successes and 10 failures in the sample to
assume normality of p.
AACSB: Analytic
Blooms: Remember
Difficulty: 1 Easy
Learning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Proportion ()

31.

The Student's t distribution is always symmetric and bell-shaped, but

its tails lie above the normal.
TRUE
Student's t resembles a normal, but its PDF is above the normal PDF
in the tails.
AACSB: Analytic
Blooms: Remember
Difficulty: 2 Medium
Learning Objective: 08-06 Know when to use Student's t instead of z to estimate .
Topic: Confidence Interval for a Mean () with Unknown

32.

The confidence interval half-width when = .50 is called the margin

of error.
TRUE
Pollsters use this definition.
AACSB: Analytic
Blooms: Remember
Difficulty: 1 Easy
Learning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Proportion ()

33.

Based on the Rule of Three, if no events occur in n independent trials

we can set the upper 95 percent confidence bound at 3/n.
TRUE
We need a special rule because when p = 0 we can't apply the usual
formula p z[p(1 - p)/n]1/2.
AACSB: Analytic
Blooms: Remember
Difficulty: 2 Medium
Learning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Proportion ()

34.

The sample standard deviation s is halfway between the lower and

upper confidence limits for the population (i.e., the confidence
interval is symmetric around s).
FALSE
The chi-square distribution is not symmetric.
AACSB: Analytic
Blooms: Apply
Difficulty: 3 Hard
Learning Objective: 08-10 Construct a confidence interval for a variance (optional).
Topic: Confidence Interval for a Population Variance, 2 (Optional)

35.

In a sample size calculation, if the confidence level decreases, the

size of the sample needed will increase.
FALSE
Reduced confidence allows a smaller sample.
AACSB: Analytic
Blooms: Understand
Difficulty: 2 Medium
Learning Objective: 08-09 Calculate sample size to estimate a mean or proportion.
Topic: Sample Size Determination for a Mean

36.

To calculate the sample size needed for a survey to estimate a

proportion, the population standard deviation must be known.
FALSE
For a proportion, the sample size formula requires not .
AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-09 Calculate sample size to estimate a mean or proportion.
Topic: Sample Size Determination for a Proportion

37.

Assuming that = .50 is a quick and conservative approach to use in

a sample size calculation for a proportion.
TRUE
Assuming that = .50 is quick and safe (but may give a larger
sample than is needed).
AACSB: Analytic
Blooms: Remember
Difficulty: 2 Medium
Learning Objective: 08-09 Calculate sample size to estimate a mean or proportion.
Topic: Sample Size Determination for a Proportion

38.

To estimate the required sample size for a proportion, one method is

to take a small pilot sample to estimate and then apply the sample
size formula.
TRUE
This is a common method, but assuming that = .50 is quicker and
safer.
AACSB: Analytic
Blooms: Remember
Difficulty: 2 Medium
Learning Objective: 08-09 Calculate sample size to estimate a mean or proportion.
Topic: Sample Size Determination for a Proportion

39.

To estimate , you typically need a sample size equal to at least 5

percent of your population.
FALSE
The sample size n bears no necessary relation to N.
AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-09 Calculate sample size to estimate a mean or proportion.
Topic: Sample Size Determination for a Proportion

40.

To estimate a proportion with a 4 percent margin of error and a 95

percent confidence level, the required sample size is over 800.
FALSE
n = (z/E)2()(1 - ) = (1.96/.04)2(.50)(1 - .50) = 600.25.
AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-09 Calculate sample size to estimate a mean or proportion.
Topic: Sample Size Determination for a Proportion

41.

Approximately 95 percent of the population X values will lie within

the 95 percent confidence interval for the mean.
FALSE
The confidence interval is for the true mean, not for individual X
values.
AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-05 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Mean () with Known

42.

A 99 percent confidence interval has more confidence but less

precision than a 95 percent confidence interval.
TRUE
The higher confidence level widens the interval so it is less precise.
AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-05 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Mean () with Known

43.

Sampling variation is not controllable by the statistician.

TRUE
Sampling variation is inevitable.
AACSB: Analytic
Blooms: Remember
Difficulty: 1 Easy
Learning Objective: 08-01 Define sampling error; parameter; and estimator.
Topic: Sampling Variation

44.

The sample mean is not a random variable when the population

parameters are known.
FALSE
The sample mean is a random variable regardless of what we know
about the population.
AACSB: Analytic
Blooms: Remember
Difficulty: 1 Easy
Learning Objective: 08-01 Define sampling error; parameter; and estimator.
Topic: Sampling Variation

45.

The finite population correction factor (FPCF) can be ignored if n = 7

and N = 700.
TRUE
The FPCF has a negligible effect when the sample is less than 5
percent of the population.
AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-08 Construct confidence intervals for finite populations.
Topic: Estimating from Finite Populations

46.

In constructing a confidence interval, the finite population correction

factor (FPCF) can be ignored if samples of 12 items are drawn from a
population of 300 items.
TRUE
The FPCF has a negligible effect when the sample is less than 5
percent of the population.
AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-08 Construct confidence intervals for finite populations.
Topic: Estimating from Finite Populations

47.

The finite population correction factor (FPCF) can be ignored when

the sample size is large relative to the population size.
TRUE
The FPCF has a negligible effect when n is small relative to N.
AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-08 Construct confidence intervals for finite populations.
Topic: Estimating from Finite Populations

Multiple Choice Questions

48.

A sampling distribution describes the distribution of:

A.
B.
C.
D.

a parameter.
a statistic.
either a parameter or a statistic.
neither a parameter nor a statistic.

A statistic has a sampling distribution.

AACSB: Analytic
Blooms: Remember
Difficulty: 1 Easy
Learning Objective: 08-03 State the Central Limit Theorem for a mean.
Topic: Sample Mean and the Central Limit Theorem

49.

As the sample size increases, the standard error of the mean:

A.
B.
C.

increases.
decreases.
may increase or decrease.

The standard error of the mean is /(n)1/2.

AACSB: Analytic
Blooms: Remember
Difficulty: 1 Easy
Learning Objective: 08-03 State the Central Limit Theorem for a mean.
Topic: Sample Mean and the Central Limit Theorem

50.

Which statement is most nearly correct, other things being equal?

A. Doubling the sample size will cut the standard error of the mean
in half.
B. The standard error of the mean depends on the population size.
C. Quadrupling the sample size roughly halves the standard error of
the mean.
D. The standard error of the mean depends on the confidence level.
The standard error of the mean is /(n)1/2 so replacing n by 4n would
cut the SEM in half.
AACSB: Analytic
Blooms: Understand
Difficulty: 2 Medium
Learning Objective: 08-04 Explain how sample size affects the standard error.
Topic: Sample Mean and the Central Limit Theorem

51.

The width of a confidence interval for is not affected by:

A.
B.
C.
D.

the sample size.

the confidence level.
the standard deviation.
the sample mean.

The mean is not used in calculating the width of the confidence

interval z/(n)1/2.
AACSB: Analytic
Blooms: Remember
Difficulty: 2 Medium
Learning Objective: 08-05 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Mean () with Known

52.

The Central Limit Theorem (CLT) implies that:

A.
B.
C.
D.

the population will be approximately normal if n 30.

repeated samples must be taken to obtain normality.
the distribution of the mean is approximately normal for large n.
the mean follows the same distribution as the population.

The sampling distribution of the mean is asymptotically normal for

any population.
AACSB: Analytic
Blooms: Understand
Difficulty: 2 Medium
Learning Objective: 08-03 State the Central Limit Theorem for a mean.
Topic: Sample Mean and the Central Limit Theorem

53.

The owner of Limp Pines Resort wanted to know the average age of
its clients. A random sample of 25 tourists is taken. It shows a mean
age of 46 years with a standard deviation of 5 years. The width of a
98 percent CI for the true mean client age is approximately:

A.
B.
C.
D.

1.711
2.326
2.492
2.797

years.
years.
years.
years.

The width is ts/(n)1/2 = (2.492)(5)/(25)1/2 = 2.492.

AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-06 Know when to use Student's t instead of z to estimate .
Topic: Confidence Interval for a Mean () with Unknown

54.

In constructing a confidence interval for a mean with unknown

variance with a sample of 25 items, Bob used z instead of t. "Well, at
least my interval will be wider than necessary, so it was a
conservative error," said he. Is Bob's statement correct?

A.
B.
C.

Yes.
No.
It depends on .

z is always smaller than t (ceteris paribus) so the interval would be

narrower than is justified.
AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-06 Know when to use Student's t instead of z to estimate .
Topic: Confidence Interval for a Mean () with Unknown

55.

A random sample of 16 ATM transactions at the Last National Bank of

Flat Rock revealed a mean transaction time of 2.8 minutes with a
standard deviation of 1.2 minutes. The width (in minutes) of the 95
percent confidence interval for the true mean transaction time is:

A.
B.
C.
D.

0.639
0.588
0.300
2.131

The width is ts/(n)1/2 = (2.131)(1.2)/(16)1/2 = 0.639.

AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-06 Know when to use Student's t instead of z to estimate .
Topic: Confidence Interval for a Mean () with Unknown

56.

We could narrow a 95 percent confidence interval by:

A.
B.
C.

using 99 percent confidence.

using a larger sample.
raising the standard error.

A larger sample would narrow the interval width z/(n)1/2.

AACSB: Analytic
Blooms: Understand
Difficulty: 1 Easy
Learning Objective: 08-05 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Mean () with Known

57.

The owner of Torpid Oaks B&B wanted to know the average distance
its guests had traveled. A random sample of 16 guests showed a
mean distance of 85 miles with a standard deviation of 32 miles. The
90 percent confidence interval (in miles) for the mean is
approximately:

A.
B.
C.
D.

(71.0, 99.0)
(71.8, 98.2)
(74.3, 95.7)
(68.7, 103.2)

The interval is 85 ts/(n)1/2 or 85 (1.753)(32)/(16)1/2 with d.f = 15

(don't use z).
AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-06 Know when to use Student's t instead of z to estimate .
Topic: Confidence Interval for a Mean () with Unknown

58.

A highway inspector needs an estimate of the mean weight of trucks

crossing a bridge on the interstate highway system. She selects a
random sample of 49 trucks and finds a mean of 15.8 tons with a
sample standard deviation of 3.85 tons. The 90 percent confidence
interval for the population mean is:

A.
B.
C.
D.

14.72
14.90
14.69
14.88

to
to
to
to

16.88
16.70
16.91
16.72

tons.
tons.
tons.
tons.

The interval is 15.8 ts/(n)1/2 or 15.8 (1.677)(3.85)/(49)1/2 using

d.f. = 48 (don't use z).
AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-06 Know when to use Student's t instead of z to estimate .
Topic: Confidence Interval for a Mean () with Unknown

59.

To determine a 72 percent level of confidence for a proportion, the

value of z is approximately:

A.
B.
C.
D.

1.65
0.77
1.08
1.55

Look up the z value that puts 14 percent in each tail.

AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Proportion ()

60.

To estimate the average annual expenses of students on books and

class materials a sample of size 36 is taken. The sample mean is
$850 and the sample standard deviation is $54. A 99 percent
confidence interval for the population mean is:

A.
B.
C.
D.

$823.72
$832.36
$826.82
$825.48

to
to
to
to

$876.28
$867.64
$873.18
$874.52

The interval is 850 ts/(n)1/2 or 850 (2.724)(54)/(36)1/2 with d.f =

35 (don't use z).
AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-05 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Mean () with Unknown

61.

In constructing a 95 percent confidence interval, if you increase n to

4n, the width of your confidence interval will (assuming other things
remain the same) be:

A.
B.
C.
D.

about 25 percent of its former width.

about two times wider.
about 50 percent of its former width.
about four times wider.

The standard error of the mean is /(n)1/2 so replacing n by 4n would

cut the SEM in half.
AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-04 Explain how sample size affects the standard error.
Topic: Confidence Interval for a Mean () with Known

62.

Which of the following is not a characteristic of the t distribution?

A.
It is a continuous distribution.
B.
It has a mean of 0.
C.
It is a symmetric distribution.
D. It approaches z as degrees of freedom decrease.
It approaches z as degrees of freedom increase.
AACSB: Analytic
Blooms: Apply
Difficulty: 1 Easy
Learning Objective: 08-06 Know when to use Student's t instead of z to estimate .
Topic: Confidence Interval for a Mean () with Unknown

63.

Which statement is incorrect? Explain.

A. If p = .50 and n = 100, the standard error of the sample

proportion is .05.
B. In a sample size calculation for estimating , it is conservative to
assume = .50.
C. If n = 250 and p = .06, we cannot assume normality in a
confidence interval for .
Normality of p may be assumed because np = 15 and n(1 - p) = 235.
AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Proportion ()

64.

What is the approximate width of a 90 percent confidence interval

for the true population proportion if there are 12 successes in a
sample of 25?

A.
B.
C.
D.

.196
.164
.480
.206

The interval width is z[p(1 - p)/n]1/2 = (1.645)[(.48)(.52)/25]1/2.

AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Proportion ()

65.

A poll showed that 48 out of 120 randomly chosen graduates of

California medical schools last year intended to specialize in family
practice. What is the width of a 90 percent confidence interval for
the proportion that plan to specialize in family practice?

A.
B.
C.
D.

.0447
.0736
.0876
.0894

The interval width is z[p(1 - p)/n]1/2 = (1.645)[(.40)(.60)/120]1/2.

AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Proportion ()

66.

What is the approximate width of an 80 percent confidence interval

for the true population proportion if there are 12 successes in a
sample of 80?

A.
B.
C.
D.

.078
.066
.051
.094

The interval width is z[p(1 - p)/n]1/2 = (1.282)[(.15)(.85)/80]1/2.

AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Proportion ()

67.

A random sample of 160 commercial customers of PayMor Lumber

revealed that 32 had paid their accounts within a month of billing.
The 95 percent confidence interval for the true proportion of
customers who pay within a month would be:

A.
B.
C.
D.

0.148
0.138
0.144
0.153

to
to
to
to

0.252
0.262
0.256
0.247

The interval is p z[p(1 - p)/n]1/2 = .20 (1.960)[(.20)(.80)/160]1/2.

AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Proportion ()

68.

A random sample of 160 commercial customers of PayMor Lumber

revealed that 32 had paid their accounts within a month of billing.
Can normality be assumed for the sample proportion?

A.
B.
C.

Yes.
No.
Need more information to say.

Yes, because there were at least 10 "successes" and at least 10

"failures" in the sample.
AACSB: Analytic
Blooms: Apply
Difficulty: 1 Easy
Learning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Proportion ()

69.

The conservative sample size required for a 95 percent confidence

interval for with an error of 0.04 is:

A.
B.
C.
D.

271.
423.
385.
601.

n = (z/E)2()(1 - ) = (1.96/.04)2(.50)(1 - .50) = 600.25 (round up).

AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-09 Calculate sample size to estimate a mean or proportion.
Topic: Sample Size Determination for a Proportion

70.

Last week, 108 cars received parking violations in the main

university parking lot. Of these, 27 had unpaid parking tickets from a
previous violation. Assuming that last week was a random sample of
all parking violators, find the 95 percent confidence interval for the
percentage of parking violators that have prior unpaid parking
tickets.

A.
B.
C.
D.

18.1
16.8
15.3
19.5

to
to
to
to

31.9
33.2
34.7
30.5

percent.
percent.
percent.
percent.

The interval is p z[p(1 - p)/n]1/2 = .25 (1.960)[(.25)(.75)/108]1/2.

AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Proportion ()

71.

In a random sample of 810 women employees, it is found that 81

would prefer working for a female boss. The width of the 95 percent
confidence interval for the proportion of women who prefer a female
boss is:

A.
B.
C.
D.

.0288
.0105
.0207
.0196

The width is z[p(1 - p)/n]1/2 or (1.960)[(.10)(.90)/810]1/2 or .

0207.
AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Proportion ()

72.

Jolly Blue Giant Health Insurance (JBGHI) is concerned about rising

lab test costs and would like to know what proportion of the positive
lab tests for prostate cancer are actually proven correct through
subsequent biopsy. JBGHI demands a sample large enough to ensure
an error of 2 percent with 90 percent confidence. What is the
necessary sample size?

A.
B.
C.
D.

4,148
2,401
1,692
1,604

n = (z/E)2()(1 - ) = (1.645/.02)2(.50)(1 - .50) = 1691.3 (round up).

AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-09 Calculate sample size to estimate a mean or proportion.
Topic: Sample Size Determination for a Proportion

73.

A university wants to estimate the average distance that commuter

students travel to get to class with an error of 3 miles and 90
percent confidence. What sample size would be needed, assuming
that travel distances are normally distributed with a range of X = 0
to X = 50 miles, using the Empirical Rule 3 to estimate .

A.
B.
C.
D.

About
About
About
About

28
47
30
21

students
students
students
students

Using = (50 - 0)/6 = 8.333, we get n = [z/E]2 = [(1.645)(8.333)/3]2

= 20.9 (round up).
AACSB: Analytic
Blooms: Apply
Difficulty: 3 Hard
Learning Objective: 08-09 Calculate sample size to estimate a mean or proportion.
Topic: Sample Size Determination for a Mean

74.

A financial institution wishes to estimate the mean balances owed by

its credit card customers. The population standard deviation is $300.
If a 99 percent confidence interval is used and an interval of $75 is
desired, how many cardholders should be sampled?

A.
B.
C.
D.

3382
629
87
107

n = [z/E]2 = [(2.576)(300)/75]2 = 106.2 (round up).

AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-09 Calculate sample size to estimate a mean or proportion.
Topic: Sample Size Determination for a Mean

75.

A company wants to estimate the time its trucks take to drive from
city A to city B. The standard deviation is known to be 12 minutes.
What sample size is required in order that error will not exceed 2
minutes, with 95 percent confidence?

A.
B.
C.
D.

12 observations
139 observations
36 observations
129 observations

n = [z/E]2 = [(1.960)(12)/2]2 = 138.3 (round up).

AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-09 Calculate sample size to estimate a mean or proportion.
Topic: Sample Size Determination for a Mean

76.

In a large lecture class, the professor announced that the scores on a

recent exam were normally distributed with a range from 51 to 87.
Using the Empirical Rule 3 to estimate , how many students
would you need to sample to estimate the true mean score for the
class with 90 percent confidence and an error of 2?

A.
B.
C.
D.

About 17 students
About 35 students
About 188 students
About 25 students

Using = (87 - 51)/6 = 6, we get n = [z/E]2 = [(1.645)(6)/2]2 =

24.35 (round up).
AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-09 Calculate sample size to estimate a mean or proportion.
Topic: Sample Size Determination for a Mean

77.

Using the conventional polling definition, find the margin of error for
a customer satisfaction survey of 225 customers who have recently
dined at Applebee's.

A.
B.
C.
D.

5.0
4.2
7.1
6.5

percent
percent
percent
percent

The margin of error is z[(1 - )/n]1/2 or (1.960)[(.50)(.50)/225]1/2

or .065.
AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-09 Calculate sample size to estimate a mean or proportion.
Topic: Sample Size Determination for a Proportion

78.

A marketing firm is asked to estimate the percent of existing

customers who would purchase a "digital upgrade" to their basic
cable TV service. The firm wants 99 percent confidence and an error
of 5 percent. What is the required sample size (to the next higher
integer)?

A.
B.
C.
D.

664
625
801
957

n = (z/E)2()(1 - ) = (2.576/.05)2(.50)(1 - .50) = 663.6 (round up).

AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-09 Calculate sample size to estimate a mean or proportion.
Topic: Sample Size Determination for a Proportion

79.

An airport traffic analyst wants to estimate the proportion of daily

takeoffs by small business jets (as opposed to commercial passenger
jets or other aircraft) with an error of 4 percent with 90 percent
confidence. What sample size should the analyst use?

A.
B.
C.
D.

385
601
410
423

n = (z/E)2()(1 - ) = (1.645/.04)2(.50)(1 - .50) = 422.8 (round up).

AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-09 Calculate sample size to estimate a mean or proportion.
Topic: Sample Size Determination for a Proportion

80.

Ersatz Beneficial Insurance wants to estimate the cost of damage to

cars due to accidents. The standard deviation of the cost is known to
be $200. They want to estimate the mean cost using a 95 percent
confidence interval within $10. What is the minimum sample size
n?

A.
B.
C.
D.

1083
4002
1537
2301

n = [z/E]2 = [(1.960)(200)/10]2 = 1536.6 (round up).

AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-09 Calculate sample size to estimate a mean or proportion.
Topic: Sample Size Determination for a Mean

81.

Professor York randomly surveyed 240 students at Oxnard University

and found that 150 of the students surveyed watch more than 10
hours of television weekly. Develop a 95 percent confidence interval
to estimate the true proportion of students who watch more than 10
hours of television each week. The confidence interval is:

A.
B.
C.
D.

.533
.564
.552
.551

to
to
to
to

.717
.686
.698
.739

The interval is p z[p(1 - p)/n]1/2 = .625 (1.960)[(.625)

(.375)/240]1/2.
AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Proportion ()

82.

Professor York randomly surveyed 240 students at Oxnard University

and found that 150 of the students surveyed watch more than 10
hours of television weekly. How many additional students would
Professor York have to sample to estimate the proportion of all
Oxnard University students who watch more than 10 hours of
television each week within 3 percent with 99 percent
confidence?

A.
B.
C.
D.

761
1001
1489
1728

Using p = .625 we get n = (z/E)2()(1 - ) = (2.576/.03)2(.625)(.375)

= 1728.06 (round up).
AACSB: Analytic
Blooms: Apply
Difficulty: 3 Hard
Learning Objective: 08-09 Calculate sample size to estimate a mean or proportion.
Topic: Sample Size Determination for a Proportion

83.

The sample proportion is in the middle of the confidence interval for

the population proportion:

A.
B.
C.

in any sample.
only if the samples are large.
only if is not too far from .50.

The interval is p z[p(1 - p)/n]1/2.

AACSB: Analytic
Blooms: Remember
Difficulty: 2 Medium
Learning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Proportion ()

84.

For a sample of size 16, the critical values of chi-square for a 95

percent confidence interval for the population variance are:

A.
B.
C.
D.

6.262,
6.908,
5.629,
7.261,

27.49
28.85
26.12
25.00

Using d.f. = n - 1 = 15, we get 2L = 6.262 and 2U = 27.49 from

Appendix E.
AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-10 Construct a confidence interval for a variance (optional).
Topic: Confidence Interval for a Population Variance, 2 (Optional)

85.

For a sample of size 11, the critical values of chi-square for a 90

percent confidence interval for the population variance are:

A.
B.
C.
D.

6.262,
6.908,
3.940,
3.247,

27.49
28.85
18.31
20.48

d.f. = n - 1 = 10, we get 2L = 3.940 and 2U = 18.31 from Appendix

E.
AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-10 Construct a confidence interval for a variance (optional).
Topic: Confidence Interval for a Population Variance, 2 (Optional)

86.

For a sample of size 18, the critical values of chi-square for a 99

percent confidence interval for the population variance are:

A.
B.
C.
D.

6.262,
5.697,
5.629,
7.261,

27.49
35.72
26.12
25.00

d.f. = n - 1 = 17, we get 2L = 5.697 and 2U = 35.72 from Appendix

E.
AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-10 Construct a confidence interval for a variance (optional).
Topic: Confidence Interval for a Population Variance, 2 (Optional)

87.

Which of the following statements is most nearly correct, other

things being equal?

A. Using Student's t instead of z makes a confidence interval

narrower.
B. The table values of z and t are about the same when the mean is
large.
C. For a given confidence level, the z value is always smaller then
the t value.
D. Student's t is rarely used because it is more conservative to use z.
As n increases, t approaches z, but t is always larger than z.
AACSB: Analytic
Blooms: Understand
Difficulty: 2 Medium
Learning Objective: 08-04 Explain how sample size affects the standard error.
Topic: Confidence Interval for a Mean () with Unknown

88.

The Central Limit Theorem (CLT):

A. applies only to samples from normal populations.

B.
applies to any population.
C. applies best to populations that are skewed.
D. applies only when and are known.
The appeal of the CLT is that is applies to populations of any shape.
AACSB: Analytic
Blooms: Remember
Difficulty: 2 Medium
Learning Objective: 08-03 State the Central Limit Theorem for a mean.
Topic: Sample Mean and the Central Limit Theorem

89.

In which situation may the sample proportion safely be assumed to

follow a normal distribution?

A.
B.
C.
D.

12 successes in a sample of 72 items

8 successes in a sample of 40 items
6 successes in a sample of 200 items
4 successes in a sample of 500 items

We prefer at least 10 "successes" and at least 10 "failures" to

assume that p is normal.
AACSB: Analytic
Blooms: Remember
Difficulty: 2 Medium
Learning Objective: 08-03 State the Central Limit Theorem for a mean.
Topic: Confidence Interval for a Proportion ()

90.

In which situation may the sample proportion safely be assumed to

follow a normal distribution?

A.
B.
C.
D.

n = 100, = .06
n = 250, = .02
n = 30, = .50
n = 500, = .01

We want n 10 and n(1 - ) 10 to assume that p is normal.

AACSB: Analytic
Blooms: Remember
Difficulty: 2 Medium
Learning Objective: 08-03 State the Central Limit Theorem for a mean.
Topic: Confidence Interval for a Proportion ()

91.

If = 12, find the sample size to estimate the mean with an error of
4 and 95 percent confidence (rounded to the next higher integer).

A.
B.
C.
D.

75
35
58
113

n = [z/E]2 = [(1.960)(12)/4]2 = 34.6 (round up).

AACSB: Analytic
Blooms: Remember
Difficulty: 2 Medium
Learning Objective: 08-03 State the Central Limit Theorem for a mean.
Topic: Sample Size Determination for a Mean

92.

If = 25, find the sample size to estimate the mean with an error of
3 and 90 percent confidence (rounded to the next higher integer).

A.
B.
C.
D.

426
512
267
188

n = [z/E]2 = [(1.645)(25)/3]2 = 187.9 (round up).

AACSB: Analytic
Blooms: Remember
Difficulty: 2 Medium
Learning Objective: 08-03 State the Central Limit Theorem for a mean.
Topic: Sample Size Determination for a Mean

93.

Sampling error can be avoided:

A.
by using an unbiased estimator.
B. by eliminating nonresponses (e.g., older people).
C. by no method under the statistician's control.
D. either by using an unbiased estimator or by eliminating
nonresponse.
Sampling error occurs in any random sample used to estimate an
unknown parameter.
AACSB: Analytic
Blooms: Remember
Difficulty: 1 Easy
Learning Objective: 08-02 Explain the desirable properties of estimators.
Topic: Estimators and Sampling Error

94.

A consistent estimator for the mean:

A. converges on the true parameter as the variance increases.

B. converges on the true parameter as the sample size increases.
C. consistently follows a normal distribution.
D. is impossible to obtain using real sample data.
The variance becomes smaller and the estimator approaches the
parameter as n increases.
AACSB: Analytic
Blooms: Remember
Difficulty: 2 Medium
Learning Objective: 08-02 Explain the desirable properties of estimators.
Topic: Estimators and Sampling Error

95.

Concerning confidence intervals, which statement is most nearly

correct?

A. We should use z instead of t when n is large.

B. We use the Student's t distribution when is unknown.
C. We use the Student's t distribution to narrow the confidence
interval.
Student's t distribution widens the confidence interval when is
unknown.
AACSB: Analytic
Blooms: Understand
Difficulty: 2 Medium
Learning Objective: 08-06 Know when to use Student's t instead of z to estimate .
Topic: Confidence Interval for a Mean () with Unknown

96.

The standard error of the mean decreases when:

A.
the sample size decreases.
B.
the standard deviation increases.
C. the standard deviation decreases or n increases.
D.
the population size decreases.
The standard error of the mean /(n1/2) depends on n and .
AACSB: Analytic
Blooms: Understand
Difficulty: 2 Medium
Learning Objective: 08-04 Explain how sample size affects the standard error.
Topic: Confidence Interval for a Mean () with Known

97.

For a given sample size, the higher the confidence level, the:

A.
B.
C.
D.

more accurate the point estimate.

smaller the standard error.
smaller the interval width.
greater the interval width.

To have more confidence, we must widen the interval. For example,

z.025 = 1.960 (for 95 percent confidence) gives a wider interval than
z.05 = 1.645 (for 90 percent confidence). The proffered statement
would also be true for the Student's t distribution.
AACSB: Analytic
Blooms: Understand
Difficulty: 2 Medium
Learning Objective: 08-05 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Mean () with Known

98.

A sample is taken and a confidence interval is constructed for the

mean of the distribution. At the center of the interval is always which
value?

A.
The sample mean
B.
The population mean
C. Neither nor since with a sample anything can happen
D. Both and as long as there are not too many outliers
The confidence interval for the mean is symmetric around the
sample mean.
AACSB: Analytic
Blooms: Remember
Difficulty: 2 Medium
Learning Objective: 08-05 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Mean () with Known

99.

If a normal population has parameters = 40 and = 8, then for a

sample size n = 4:

A.
B.
C.
D.

the standard error of the sample mean is approximately 2.

the standard error of the sample mean is approximately 4.
the standard error of the sample mean is approximately 8.
the standard error of the sample mean is approximately 10.

The standard error is /(n1/2) = (8)/(41/2) = 4.

AACSB: Analytic
Blooms: Apply
Difficulty: 1 Easy
Learning Objective: 08-04 Explain how sample size affects the standard error.
Topic: Confidence Interval for a Mean () with Known

Short Answer Questions

100. On the basis of a survey of 545 television viewers, a statistician has

constructed a confidence interval and estimated that the proportion
of people who watched the season premiere of Glee is between .16
and .24. What level of confidence did the statistician use in
constructing this interval? Explain carefully, showing all steps in your
reasoning.

We solve to get z = 2.33, which corresponds approximately to a 98

percent confidence level.
Feedback: The confidence interval is

and the interval half-width is .04 so we set

and p = .20 (the midpoint of the interval) to solve for

= 2.33 which corresponds approximately to a 98 percent confidence

level.
AACSB: Analytic
Blooms: Evaluate
Difficulty: 3 Hard
Learning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Proportion ()

101. Read the news story below. Using the 95 percent confidence level,
what sample size would be needed to estimate the true proportion of
stores selling cigarettes to minors with an error of 3 percent?
Explain carefully, showing all steps in your reasoning.

=
= 813.5, or 814 (rounded up), using the sample proportion because
it is available (instead of assuming that = .50).

Feedback:

=
= 813.5, or 814 (rounded up). We use the sample proportion
because it is available, instead of assuming that = .50.
AACSB: Analytic
Blooms: Apply

Difficulty: 2 Medium
Learning Objective: 08-09 Calculate sample size to estimate a mean or proportion.
Topic: Sample Size Determination for a Proportion

102. In a survey, 858 out of 2600 homeowners said they expected good
economic conditions to continue for the next 12 months. Construct a
95 percent confidence interval for "good times" in the next 12
months.

The confidence interval is .3119 < < .3481.

Feedback:

or
or
or .33 .0181, so the confidence interval is .3119 < < .3481.
AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-07 Construct a 90; 95; or 99 percent confidence interval for .
Topic: Confidence Interval for a Proportion ()

103. Fulsome University has 16,059 students. In a sample of 200

students, 12 were born outside the United States. Construct a 95
percent confidence interval for the true population proportion. How
large a sample is needed to estimate the true proportion of Fulsome
students who were born outside the United States with an error of
2.5 percent and 95 percent confidence? Show your work and explain
fully.

We have sampled less than 5 percent of the population, so the FPCF

is unnecessary (i.e., we can ignore the population size. The 95
percent confidence interval is p z.025[p(1 - p)/n]1/2 = .06 (1.960)
[(.06)(.94)/200]1/2 or .06 .032914 or .027 < < .093. To reduce the
error to .025, the required sample size is

or
= 346.7, or n = 347 (rounded up). We can use the sample value for
p so we do not need to assume that = .50.
Feedback: The 95 percent confidence interval is p z.025[p(1 - p)/n]1/2
= .06 (1.960)[(.06)(.94)/200]1/2 or .06 .032914 or .027 < < .
093. To reduce the error to .025, the required sample size is

or
= 346.7, or n = 347 (rounded up). We have a sample value for p so
we do not need to assume that = .50. If you did assume = .50,

you would get an unnecessarily large required sample since the

preliminary sample indicates that is not .50. The sample does not
exceed 5 percent of the population size, so the finite population
correction would make little difference.
AACSB: Analytic
Blooms: Apply
Difficulty: 3 Hard
Learning Objective: 08-09 Calculate sample size to estimate a mean or proportion.
Topic: Sample Size Determination for a Proportion

104. List differences and similarities between Student's t and the standard
normal distribution.

Both are bell-shaped and symmetric, but the Student's t distribution

lies below the standard normal in the middle, and its tails are above
the standard normal.
Feedback: They are both bell-shaped and symmetric. However, the
Student's t distribution lies below the standard normal in the middle,
and its tails are above the standard normal ("thicker" or "heavier"
tails). Therefore, the value of Student's t for a given tail area will
always be greater than the corresponding z value. We use the
Student's t whenever the standard deviation is estimated from a
sample, which is to say, most of the time.
AACSB: Analytic
Blooms: Apply
Difficulty: 1 Easy
Learning Objective: 08-06 Know when to use Student's t instead of z to estimate .
Topic: Confidence Interval for a Mean () with Unknown

105. Why does pose a problem for sample size calculation for a mean?
How can be approximated when it is unknown?

Truehe formula for the sample size to estimate requires knowing .

But because is unknown (we are trying to estimate it), then
probably is unknown as well. There are several ways to estimate :
(1) take a small preliminary sample and calculate the sample
standard deviation s as an estimate of ; or (2) if the range is known,
we can estimate = Range/6 because from the Empirical Rule
3 contains almost all of the data in a normal distribution (a
sometimes doubtful assumption if there are outliers or a skewed
population); or (3) we might have some value for from prior
experience (e.g., a previous sample or historical data).
Feedback: The formula for the sample size to estimate requires
knowing . But because is unknown (we are trying to estimate it),
then probably is unknown as well. There are several ways to
estimate : (1) take a small preliminary sample and calculate the
sample standard deviation s as an estimate of ; or (2) if the range is
known, we can estimate = Range/6 because from the Empirical
Rule 3 contains almost all of the data in a normal distribution (a
sometimes doubtful assumption if there are outliers or a skewed
population); or (3) we might have some value for from prior
experience (e.g., a previous sample or historical data).
AACSB: Analytic
Blooms: Apply
Difficulty: 2 Medium
Learning Objective: 08-09 Calculate sample size to estimate a mean or proportion.
Topic: Sample Size Determination for a Mean