CHAPTER 10
INTRODUCTION TO ESTIMATION
SECTIONS 1
MULTIPLE CHOICE QUESTIONS
In the following multiple-choice questions, please circle the correct answer.
1.
Which of the following is not a characteristic for a good estimator?
a. Biasedness
b. Consistency
c. Relative efficiency
d. Unbiasedness
ANSWER: a
2.
A point estimate is defined as:
a. the average of the sample values
b. the average of the population values
c. a single value that is the best estimate of an unknown population parameter
d. a single value that is the best estimate of an unknown sample statistic
ANSWER: c
3.
An unbiased estimator of a population parameter is defined as:
a. an estimator whose expected value is equal to the parameter
b. an estimator whose variance is equal to one
c. an estimator whose expected value is equal to zero
d. an estimator whose variance goes to zero as the sample size goes to infinity
ANSWER: a
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Chapter Ten
4.
An estimator is said to be consistent if:
a. it is an unbiased estimator
b. the variance of the estimator is close to one
c. the expected value of the estimator is known and positive
d. it is an unbiased estimator and the difference between the estimator and the
population parameter grows smaller as the sample size grows larger
ANSWER: d
5.
If there are two unbiased estimators of a population parameter, the one whose variance is
smaller is said to be:
a. a biased estimator
b. relatively efficient
c. consistent
d. relatively unbiased
ANSWER: b
6.
Which of the following statements is correct?
a. The sample mean is an unbiased estimator of the population mean
b. The sample proportion is an unbiased estimator of the population proportion
c. The difference between two sample means is an unbiased estimator of the difference
between two population means
d. All of the above
ANSWER: d
7.
Which of the following statements is true?
a. The sample mean is relatively more efficient than the sample median
b. The sample median is relatively more efficient than the sample mean
c. The sample variance is relatively more efficient than the sample variance
d. All of the above
ANSWER: a
8.
The problem with relying on a point estimate of a population parameter is that:
a. it has no variance
b. it might be unbiased
c. it might not be relatively efficient
d. it does not tell us how close or far the point estimate might be from the parameter
ANSWER: d
9.
As its name suggests, the objective of estimation is to determine the approximate value
of:
a. a population parameter on the basis of a sample statistic
b. a sample statistic on the basis of a population parameter
c. the sample mean
d. the sample variance
ANSWER: a
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10.
The sample variance s 2 is an unbiased estimator of the population variance  2 when the
denominator of s 2 is
a. n + 1
b. n
c. n - 1
d. n - 2
ANSWER: c
11.
Private colleges and universities rely on money contributed by individuals and
corporations for their operating expenses. Much of this money is put into a fund called an
endowment, and the college spends only the interest earned by the fund. A recent survey
of 8 private colleges in the United States revealed the following endorsements (in
millions of dollars): 79.225, 66.025, 254.125, 509.025, 141.625, 196.525, 114.425, and
239.025. What value will be used as the point estimate for the mean endowment of all
private colleges in the United States?
a. $1466.0
b. $200.0
c. $162.0
d. $27.0
ANSWER: b
12.
The librarian at the Library of congress has asked her assistant for an interval estimate of
the mean number of books checked out each day. The assistant provides the following
interval estimate: from 790 to 970 books per day. An efficient, unbiased point estimate
of the number of books checked out each day at the Library of Congress is
a. 790
b. 880
c. 970
d. 1,760
ANSWER: b
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Chapter Ten
TRUE / FALSE QUESTIONS
13.
An interval estimate is a range of values within which the actual value of the population
parameter, such as  , may fall.
ANSWER: T
14.
An interval estimate is an estimate of the range for a sample statistic.
ANSWER: F
15.
An unbiased estimator of a population parameter is an estimator whose standard deviation
is the same as the actual value of the population standard deviation.
ANSWER: F
16.
The sample proportion is an unbiased estimator of the population proportion, and that the
difference between two sample means is an unbiased estimator of the difference between
two population means.
ANSWER: T
17.
The sample standard deviation is an unbiased estimator of the population standard
deviation.
ANSWER: F
18.
Knowing that an estimator is unbiased only assures us that its expected value equals the
parameter, but it does not tell us how close the estimator is to the parameter.
ANSWER: T
19.
An unbiased estimator is said to be consistent if the difference between the estimator and
the parameter grows smaller as the sample size grows larger.
ANSWER: T
20.
The sample mean X is a consistent estimator of the population mean  .
ANSWER: T
21.
The sample proportion p̂ is a consistent estimator of the population proportion p because
it is unbiased and the variance of p̂ is p(1-p)/n, which grows smaller as n grows larger.
ANSWER: T
22.
An unbiased estimator is said to be consistent if the difference between the estimator and
the parameter grows larger as the sample size grows larger.
ANSWER: F
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23.
An unbiased estimator is a sample statistic, which has an expected value equal to the
value of the population parameter.
ANSWER: T
24.
The sample variance s 2 is an unbiased estimator of the population variance  2 when the
denominator of s 2 is n.
ANSWER: F
25.
If there are two unbiased estimators of a parameter, the one whose variance is smaller is
said to be relatively efficient.
ANSWER: T
26.
The sample mean is an unbiased estimator of the population mean  , and (when sampling
from a normal population) the sample median is also an unbiased estimator of  .
However, the sample median is relatively more efficient than the sample mean.
ANSWER: F
27.
An unbiased estimator will have a value, on average across samples, equal to the
population parameter value.
ANSWER: T
28.
The sample variance is a point estimate of the population variance.
ANSWER: T
29.
The confidence interval estimate of the population mean is constructed around the sample
mean.
ANSWER: T
30.
A point estimate consists of a single sample statistic that is used to estimate the true
population parameter.
ANSWER: T
31.
The confidence interval obtained will always correctly estimate the population parameter.
ANSWER: F
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Chapter Ten
STATISTICAL CONCEPTS & APPLIED QUESTIONS
32.
Define unbiasedness.
ANSWER:
An unbiased estimator of a parameter is an estimator whose expected value equals the
parameter.
33.
Define consistency
ANSWER:
An unbiased estimator is said to be consistent if the difference between the estimator and
the parameter grows smaller as the sample size grows larger.
34.
Is the sample mean a consistent estimator of the population mean? Explain
ANSWER:
Yes, the mean is consistent because it is unbiased and the standard error grows smaller as
the sample size n increases.
35.
Draw a sampling distribution of an unbiased estimator.
ANSWER:
36.
Draw a sampling distribution of a biased estimator.
ANSWER:
Introduction to Estimation
37.
327
Draw diagrams representing what happens to the sampling distribution of a consistent
estimator when the sample size increases.
ANSWER:
38.
Define relative efficiency.
ANSWER:
If there are two unbiased estimators of a parameter, the one whose variance is smaller is
relatively efficient.
39.
Draw a diagram representing the sampling distribution representing two unbiased
estimators, one of which is relatively efficient.
ANSWER:
40.
Explain briefly why interval estimators are preferred to point estimators.
ANSWER:
In drawing inferences about a population, it is intuitively reasonable to expect that a large
sample will produce more accurate results, because it contains more information than a
smaller sample does. The interval estimator is affected by the sample size; because it
possesses this feature, but point estimators don’t have the capacity to reflect the effects of
larger sample sizes
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SECTION 2
MULTIPLE CHOICE QUESTIONS
In the following multiple-choice questions, please circle the correct answer.
41.
The term 1-  refers to:
a. the probability that a confidence interval does not contain the population parameter
b. the confidence level
c. the level of unbiasedness
d. the level of consistency
ANSWER: b
42.
The letter  in the formula for constructing a confidence interval estimate of the
population mean is
a. the level of confidence
b. the probability that a confidence interval will contain the population mean
c. the probability that a confidence interval will not contain the population mean
d. the area in the lower tail of the sampling distribution of the sample mean
ANSWER: c
43.
A 90% confidence interval estimate of the population mean  can be interpreted to mean
that:
a. if we repeatedly draw samples of the same size from the same population, 90% of the
values of the sample means x will result in a confidence interval that includes the
population mean  .
b. there is a 90% probability that the population mean  will lie between the lower
confidence limit (LCL) and the upper confidence limit (UCL).
c. we are 90% confident that we have selected a sample whose range of values does not
contain the population mean  .
d. We are 90% confident that 10% the values of the sample means x will result in a
confidence interval that includes the population mean  .
ANSWER: a
44.
The width of a confidence interval estimate of the population mean widens when the:
a. level of confidence increases
b. sample size decreases
c. value of the population standard deviation increases
d. All of the above
ANSWER: d
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45.
Which of the following statements is false?
a. The width of a confidence interval estimate of the population mean narrows when the
sample size increases
b. The width of a confidence interval estimate of the population mean narrows when the
value of the sample mean increases
c. The width of a confidence interval estimate of the population mean widens when the
confidence level increases
d. All of the above
ANSWER: b
46.
A 98% confidence interval estimate for a population mean  is determined to be 75.38 to
86.52. If the confidence level is reduced to 90%, the confidence interval for 
a. becomes wider
b. remains the same
c. becomes narrower
d. None of the above.
ANSWER: c
47.
The z value for a 96.6% confidence interval estimate for a population mean  is
a. 2.12
b. 1.82
c. 2.00
d. 1.96
ANSWER: a
48.
In developing an interval estimate for a population mean, the population standard
deviation  was assumed to be 10. The interval estimate was 50.92  2.14. Had 
equaled 20, the interval estimate would be
a. 60.92  2.14
b. 50.92  12.14
c. 101.84  4.28
d. 50.92  4.28
ANSWER: d
49.
In developing an interval estimate for a population mean, a sample of 50 observations
was used. The interval estimate was 19.76  1.32. Had the sample size been 200 instead
of 50, the interval estimate would have been
a. 19.76  .33
b. 19.76  .66
c. 9.88  1.32
d. 4.94  1.32
ANSWER: b
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Chapter Ten
50.
After constructing a confidence interval estimate for a population mean, you believe that
the interval is useless because it is too wide. In order to correct this problem, you need to:
a. increase the population standard deviation
b. increase the sample size
c. increase the level of confidence
d. increase the sample mean
ANSWER: b
51.
In developing an 87.4% confidence interval estimate for a population mean, the value of z
to use is
a. 1.15
b. 0.32
c. 1.53
d. 0.16
ANSWER: c
52.
A random sample of 64 observations has a mean of 30. The population variance is
assumed to be 9. The 85.3% confidence interval estimate for the population mean (to the
third decimal place) is
a. 28.369  31.631
b. 29.456  30.544
c. 28.560  31.440
d. 29.383  30.617
ANSWER: b
53.
In developing an interval estimate for a population mean, the interval estimate was 62.84
to 69.46. The population standard deviation was assumed to be 6.50, and a sample of 100
observations was used. The mean of the sample was
a. 56.34
b. 62.96
c. 13.24
d. 66.15
ANSWER: d
54.
A confidence interval is defined as:
a. a point estimate plus or minus a specific confidence level
b. a lower and upper confidence limit associated with a specific level of confidence
c. an interval that has a 95% probability of containing the population parameter
d. a lower and upper confidence limit that has a 95% probability of containing the
population parameter
ANSWER: b
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55.
Which of the following assumptions must be true in order to use the formula
x  z / 2 / n to find a confidence interval estimate of the population mean?
a. The population variance is known
b. The population mean is known
c. The population is normally distributed
d. The confidence level is greater than 90%
ANSWER: a
56.
In the formula x  z / 2 / n , the  / 2 refers to:
a. the probability that the confidence interval will contain the population mean
b. the probability that the confidence interval will not contain the population mean
c. the area in the lower tail or upper tail of the sampling distribution of the sample mean
d. the level of confidence
ANSWER: c
57.
Which of the following is not a part of the formula for constructing a confidence interval
estimate of the population mean?
a. A point estimate of the population mean
b. The standard error of the sampling distribution of the sample mean
c. The confidence level
d. The value of the population mean
ANSWER: d
58.
The larger the confidence level used in constructing a confidence interval estimate of the
population mean, the:
a. smaller the probability that the confidence interval will contain the population mean
b. smaller the value of z / 2
c. wider the confidence interval
d. narrower the confidence interval
ANSWER: c
59.
A 99% confidence interval estimate of the population mean  can be interpreted to mean
a. if all possible sample are taken and confidence interval estimates are developed, 99%
of them would include the true population mean somewhere within their interval
b. we have 99% confidence that we have selected a sample whose interval does include
the population
c. we estimate that the population mean falls between the lower and upper confidence
limits, and this type of estimator is correct 99% of the time
d. All of the above
ANSWER: d
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Chapter Ten
60.
It is desired to estimate the average total compensation of CEOs in the service industry.
Data were randomly collected from 18 CEOs and 95% confidence interval was calculated
to be ($2,190,000, $4,720,000). Based on the interval above, do you believe the average
total compensation of CEOs in the service industry is more than $3,000,000?
a. Yes, and I am 95% confident of it
b. Yes, and I am 80% confident of it
c. I am 95% confident that the average compensation is $3,000,000
d. I cannot conclude that the average exceeds $3,000,000 at the 95% confidence level
ANSWER: d
61.
Suppose a 95% confidence interval for  turns out to be (1,000, 2,100). Give a
definition of what it means to be “95% confident” in an inference.
a. In repeated sampling, the population parameter would fall in the given interval 95%
of the time.
b. In repeated sampling, 95% of the intervals constructed would contain the population
mean
c. 95% of the observations in the entire population fall in the given interval
d. 95% of the observations in the sample fall in the given interval
ANSWER: b
62.
Suppose a 95% confidence interval for  turns out to be (1,000, 2,100). To make more
useful inferences from the data, it is desired to reduce the width of the confidence
interval. Which of the following will result in a reduced interval width?
a. Increase the sample size
b. Decrease the confidence level
c. Increase the sample size and decrease the confidence level
d. Increase the confidence level and decrease the sample size
ANSWER: c
63.
Suppose a 95% confidence interval for  has been constructed. If it is decided to take a
larger sample and to decrease the confidence level of the interval, assume that the sample
statistics gathered would not change very much for the new sample. Then the resulting
interval width would be
a. larger than the current interval width
b. narrower than the current interval width
c. the same as the current interval width
d. unknown until actual sample sizes and reliability levels were determined
ANSWER: b
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64.
In the construction of confidence intervals of  , if all other quantities are unchanged, an
increase in the sample size will lead to a
a. narrower interval
b. wider interval
c. less significant interval
d. biased interval
ANSWER: a
65.
The director of a hospital wishes to estimate the mean number of people who are
admitted to the emergency room during a 24-hour period. The director randomly selects
64 different 24-hour periods and determines the number of admissions for each. For this
sample, X  20.0. If the population standard deviation is known to be 5.0, which of the
following assumptions is necessary in order for a confidence interval to be valid?
a. The population sampled from has an approximate normal distribution
b. The population sampled from has an approximate t distribution
c. The mean of the sample equals the mean of the population
d. None of these assumptions are necessary
ANSWER: d
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Chapter Ten
TRUE / FALSE QUESTIONS
66.
A confidence interval is an interval estimate for which there is a specified degree of
certainty that the actual value of the population parameter will fall within the interval.
ANSWER: T
67.
One can reduce the width of a confidence interval by taking a smaller sample size.
ANSWER: F
68.
The range of a confidence interval is a measure of the expected sampling error.
ANSWER: T
69.
The difference between the sample statistic and actual value of the population parameter
is the confidence level of the estimate.
ANSWER: F
70.
The term 1-  refers to the probability that a confidence interval does not contain the
population parameter.
ANSWER: F
71.
In the formula x  z / 2 / n , the subscript  / 2 refers to the area in the lower tail or upper
tail of the sampling distribution of the sample mean.
ANSWER: T
72.
The larger the confidence level used in constructing a confidence interval estimate of the
population mean, the narrower the confidence interval.
ANSWER: F
73.
In order to construct a confidence interval estimate of the population mean, the value of
the population mean is needed.
ANSWER: F
74.
In developing an interval estimate for a population mean, the population standard
deviation  was assumed to be 8. The interval estimate was 50.0  2.50. Had  equaled
16, the interval estimate would be 100  5.0.
ANSWER: F
75.
A 95% confidence interval estimate for a population mean  is determined to be 75 to 85.
If the confidence level is reduced to 80%, the confidence interval for  becomes
narrower.
ANSWER: T
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76.
When constructing confidence interval for a parameter, we generally set the confidence
level 1   close to 1 (usually between 0.90 and 0.99) because it is the probability that the
interval includes the actual value of the population parameter.
ANSWER: T
77.
Suppose that a 90% confidence interval for  is given by x  0.75 . This notation means
that we are 90% confident that  falls between x  0.75 and x  0.75 .
ANSWER: F
78.
We cannot interpret the confidence interval estimate of  as a probability statement
about  , simply because the population mean is a fixed but unknown quantity.
ANSWER: T
79.
The width of the confidence interval estimate of the population mean  is a function of
only two quantities: the population standard deviation  and the sample size n.
ANSWER: F
80.
Doubling the population standard deviation  has the effect of doubling the width of the
confidence interval estimate of  .
ANSWER: T
81.
In general, increasing the confidence level 1   will narrow the interval, and decreasing it
widens the interval.
ANSWER: F
82.
Suppose that a 95% confidence interval for  is given by x  3.25 . This notation means
that, if we repeatedly draw samples of the same size from the same population, 95% of
the values of x will be such that  would lie somewhere between x  3.25 and x  3.25 .
ANSWER: T
83.
When constructing confidence interval estimate of  , doubling the sample size n,
decreases the width of the interval by half.
ANSWER: F
84.
In this chapter you need four values to construct the confidence interval estimate of  .
They are the sample mean, the sample size, the population standard deviation, and the
confidence level.
ANSWER: T
85.
Given a mean of 2.1 and a standard deviation of 0.7, a 90% confidence interval will have
a width of 2.36.
ANSWER: F
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Chapter Ten
86.
A sample size of 5 provides a sample mean of 9.6. If the population variance is known to
be 5 and the population distribution is assumed to be normal, the lower limit for a 92%
confidence interval is 7.85.
ANSWER: T
87.
Other things being equal, as the confidence level for a confidence interval of  increases,
the width of the interval increases.
ANSWER: T
88.
Other things being equal, the confidence interval for the mean will be wider for 99%
confidence than for 95% confidence.
ANSWER: T
89.
The upper limit of the 90% confidence interval for  , given that n = 64; x = 70; and  =
20 is 65.89.
ANSWER: F
90.
The lower and upper limits of the 68.26% confidence interval for the population mean  ,
given that n = 64; x = 110; and  = 8 are 109 and 111, respectively.
ANSWER: T
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STATISTICAL CONCEPTS & APPLIED QUESTIONS
91.
A survey of 100 retailers revealed that the mean after-tax profit was $80,000. If we
assume that the population standard deviation is $15,000, determine the 95% confidence
interval estimate of the mean after-tax profit for all retailers.
ANSWER:
LCL = $77,060 and UCL = $82,940
92.
The temperature readings for 20 winter days in Grand Rapids, Michigan are normally
distributed with a mean of 5.5 degrees and a standard deviation of 1.5. Determine the
90% confidence interval estimate for the winter mean temperature.
ANSWER:
LCL = 4.9483 and UCL = 6.0517
93.
A sample of 49 measurements of tensile strength (roof hanger) are calculated to have a
mean of 2.45 and a standard deviation of 0.25. Determine the 95% confidence interval for
the measurements of all hangers.
ANSWER:
LCL = 2.38 and UCL = 2.52
94.
A random sample of 10 waitresses in Iowa City, Iowa revealed the following hourly
earnings (including tips): $19, 18, 15, 16, 18, 17, 16, 18, 20, and 14.
If the hourly earnings are normally distributed with a standard deviation of $4.5, estimate
with 95% confidence the mean hourly earnings for all waitresses in Iowa City.
ANSWER:
LCL = $14.3109 and UCL = $19.8891
FOR QUESTIONS 95 THROUGH 105, USE THE FOLLOWING NARRATIVE:
Narrative: Time Spent on Internet
Suppose that the amount of time teenagers spend on the internet is normally distributed with a
standard deviation of 1.5 hours. A sample of 100 teenagers is selected at random, and the sample
mean computed as 6.5 hours.
95.
{Time Spent on Internet Narrative} Determine the 95% confidence interval estimate of
the population mean.
ANSWER:
LCL = 6.206 and UCL = 6.794
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96.
Chapter Ten
{Time Spent on Internet Narrative} Interpret what the confidence interval estimate tells
you.
ANSWER:
If we repeatedly draw samples of size 100 from the population of teenagers, 95% of the
values of sample means x will be such that the population mean amount of time
teenagers spend on the internet (  ) would be somewhere between 6.206 hours and 6.794,
and 5% of the values of x will produce intervals that would not include  .
97.
{Time Spent on Internet Narrative} Determine the 99% confidence interval estimate of
the population mean.
ANSWER:
LCL = 6.1137 and UCL = 6.8863
98.
{Time Spent on Internet Narrative} Determine the 90% confidence interval estimate of
the population mean.
ANSWER:
LCL = 6.2532 and UCL = 6.7468
99.
{Time Spent on Internet Narrative} Determine the 95% confidence interval estimate of
the population mean if the sample size is changed to 300.
ANSWER:
LCL = 6.3303 and UCL = 6.6697
100.
{Time Spent on Internet Narrative} Determine the 95% confidence interval estimate of
the population mean if the sample size is changed to 36.
ANSWER:
LCL = 6.01 and UCL = 6.99
101.
{Time Spent on Internet Narrative}Determine the 95% confidence interval estimate of the
population mean if the population standard deviation is changed to 2.
ANSWER:
LCL = 6.108 and UCL = 6.892
102.
{Time Spent on Internet Narrative} Determine the 95% confidence interval estimate of
the population mean if the population standard deviation is changed to 1.2.
ANSWER:
LCL = 6.2648 and UCL = 6.7352
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103.
339
{Time Spent on Internet Narrative} Determine the 95% confidence interval estimate of
the population mean if the sample mean is changed to 5.0 hours.
ANSWER:
LCL = 4.706 and UCL = 5.294
104.
{Time Spent on Internet Narrative} Determine the 95% confidence interval estimate of
the population mean if the sample mean is changed to 8.5 hours.
ANSWER:
LCL = 8.206 and UCL = 8.794
105.
{Time Spent on Internet Narrative} Based on your answers to the previous questions
describe what happens to the width of the confidence interval estimate when each of the
following happens:
a. The confidence level increases
b. The confidence level decreases
c. The sample size increases
d. The sample size decreases
e. The value of the population standard deviation increases
f. The value of the population standard deviation decreases
g. The value of the sample mean increases
h. The value of the sample mean decreases
ANSWER:
a. Widens
b. Narrows
c. Narrows
d. Widens
e. Widens
f. Narrows
g. No change
h. No change
106.
A random sample of 10 university students was surveyed to determine the amount of time
spent weekly using a personal computer. The times are: 13, 14, 5, 6, 8, 10, 7, 12, 15,
and 3. If the times are normally distributed with a standard deviation of 5.2 hours,
estimate with 90% confidence the mean weekly time spent using a personal computer by
all university students.
ANSWER:
LCL = 6.5952 and UCL = 12.0048
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107.
Chapter Ten
A financial analyst wanted to determine the mean annual return on mutual funds. A
random sample of 60 returns shows a mean of 12%. If the population standard deviation
is assumed to be 4%, estimate with 95% confidence the mean annual return on all mutual
funds.
ANSWER:
LCL = 10.988% and UCL = 13.012%
108.
An economist is interested in studying the incomes of consumers in a particular region.
The population standard deviation is known to be $1,000. A random sample of 50
individuals resulted in an average income of $15,000. What is the upper end point in a
99% confidence interval for the average income?
ANSWER:
UCL = $15,364.16
109.
An economist is interested in studying the incomes of consumers in a particular region.
The population standard deviation is known to be $1,000. A random sample of 50
individuals resulted in an average income of $15,000. What is the width of the 90%
confidence interval?
ANSWER:
Width = $465.28
110.
A quality control engineer is interested in the mean length of sheet insulation being cut
automatically by machine. The desired length of the insulation is 12 feet. It is known
that the standard deviation in the cutting length is 0.15 feet. A sample of 60 cut sheets
yields a mean length of 12.15 feet. This sample will be used to obtain a 99% confidence
interval for the mean length cut by machine.
a. What is the critical value to use in obtaining the confidence interval?
b. Develop the 99% confidence interval for  .
ANSWER:
a. z = 2.575
b. LCL = 12.10, and UCL = 12.20
FOR QUESTIONS 111 AND 112, USE THE FOLLOWING NARRATIVE:
Narrative: Hotel Rooms Rented
A hotel chain wants to estimate the average number of rooms rented daily in each month. The
population of rooms rented daily is assumed to be normally distributed for each with a standard
deviation of 24 rooms.
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111.
341
{Hotel Rooms Rented Narrative} During January, a sample of 16 days has a sample mean
of 48 rooms. This information is used to calculate an interval estimate for the population
mean to be from 40 to 56 rooms. What is the level of confidence of this interval?
ANSWER:
81.64%
112.
{Hotel Rooms Rented Narrative} During February, a sample of 25 days has a sample
mean of 37 rooms. Use this information to calculate a 92% confidence interval for the
population mean.
ANSWER:
LCL = 28.6 and UCL = 45.4
342
Chapter Ten
SECTION 3
MULTIPLE CHOICE QUESTIONS
In the following multiple-choice questions, please circle the correct answer.
113.
Which of the following statements is false regarding the sample size needed to estimate a
population mean?
a. It is directly proportional to the population variance
b. It is directly proportional to the square of the standard normal cutoff value z / 2
c. It is directly proportional to the square of the maximum allowable error W
d. None of the above
ANSWER: c
114.
The sample size needed to estimate a population mean within 2 units with a 95%
confidence when the population standard deviation equals 8 is
a. 9
b. 61
c. 62
d. 8
ANSWER: c
115.
When determining the sample size necessary for estimating the true population mean,
which factor is not considered when sampling with replacement?
a. The population size
b. The population standard deviation
c. The level of confidence desired in the estimate
d. The allowable or tolerable sampling error
ANSWER: a
116.
The head librarian at the Library of Congress has asked her assistant for an interval
estimate of the mean number of books checked out each day. The assistant provides the
following interval estimate: from 790 to 970 books per day. If the head librarian knows
that the population standard deviation is 150 books checked out per day, approximately
how large a sample did her assistant use to determine the interval estimate?
a. 52
b. 53
c. 65
d. It cannot be determined from the information given
ANSWER: d
Introduction to Estimation
343
117.
The head librarian at the Library of Congress has asked her assistant for an interval
estimate of the mean number of books checked out each day. The assistant provides the
following interval estimate: from 790 to 970 books per day. If the head librarian knows
that the population standard deviation is 150 books checked out per day, and she asked
her assistant for a 95% confidence interval, approximately how large a sample did her
assistant use to determine the interval estimate?
a. 52
b. 53
c. 65
d. It cannot be determined from the information given
ANSWER: d
118.
Which of the following formulas is used to determine the sample size needed to estimate
a population mean?
2
a. n   z / 2  w /  
b. n   z / 2   / w
c. n  z / 2  w / 
d. n  z / 2   / w
ANSWER: b
2
119.
To estimate with 99% confidence the mean of a normal population, whose standard
deviation is assumed to be 6 and the maximum allowable sampling error is assumed to be
1.2, requires a random sample of size
a. 166
b. 165
c. 164
d. 163
ANSWER: a
120.
The head librarian at the Library of Congress has asked her assistant for an interval
estimate of the mean number of books checked out each day. The assistant provides the
following interval estimate: from 790 to 970 books per day. If the head librarian knows
that the population standard deviation is 150 books checked out per day, and she asked
her assistant to use 25 days of data to construct the interval estimate, what confidence
level can she attach to the interval estimate?
a. 99.7%
b. 99.0%
c. 98.0%
d. 95.4%
ANSWER: a
344
Chapter Ten
121.
The sample size needed to estimate a population mean to within 10 units was found to be
68. If the population standard deviation was 50, then the confidence level used was
a. 99%
b. 95%
c. 90%
d. 80%
ANSWER: c
122.
The sample size needed to estimate a population mean to within 50 units was found to be
97. If the population standard deviation was 250, then the confidence level used was
a. 99%
b. 95%
c. 90%
d. 80%
ANSWER: b
Introduction to Estimation
345
TRUE / FALSE QUESTIONS
123.
In determining the sample size n needed to estimate the population mean, n decreases as
the width of the confidence interval decreases.
ANSWER: F
124.
In determining the sample size n needed to estimate the population mean, n decreases as
the population standard deviation  decreases.
ANSWER: T
125.
In determining the size n needed to estimate the population mean, n increases as the
confidence level decreases.
ANSWER: F
126.
To estimate with 95% confidence the mean of a normal population, whose standard
deviation is assumed to be 4 and the maximum allowable sampling error is assumed to
be 1, requires a random sample of size 62.
ANSWER: T
127.
The sample size needed to estimate the population mean was found to be 865. If the
value of the population standard deviation was 75, and the maximum allowable error
was 5, then the confidence level used was 95%
ANSWER: T
128.
The sample size needed to estimate a population mean to within 1 unit with 90%
confidence given that the population standard deviation is 10 is 239.
ANSWER: F
129.
In the formula n   z / 2 / w , the letter w represents the width of the confidence interval
estimator.
ANSWER: F
130.
To use the formula n   z / 2 / w , it is frequently necessary to “guesstimate” the value
of the population standard deviation  .
ANSWER: T
2
2
346
Chapter Ten
STATISTICAL CONCEPTS & APPLIED QUESTIONS
131.
Determine the sample size that is required to estimate a population mean to within 0.4
units with a 99% confidence when the population standard deviation is 1.75.
ANSWER:
127
FOR QUESTIONS 132 THROUGH 139, USE THE FOLLOWING NARRATIVE:
Narrative: Statistics Professor
A statistics professor would like to estimate a population mean to within 40 units with 99%
confidence given that the population standard deviation is 200.
132.
{Statistics Professor Narrative} What sample size should be used?
ANSWER:
n = 166
133.
{Statistics Professor Narrative} What sample size should be used if the standard
deviation is changed to 100?
ANSWER:
n = 41
134.
{Statistics Professor Narrative} What sample size should be used if the standard
deviation is changed to 50?
ANSWER:
n =10
135.
{Statistics Professor Narrative} What sample size should be used if using a 95%
confidence level?
ANSWER:
n = 96
136.
{Statistics Professor Narrative}What sample size should be used if using a 90%
confidence level?
ANSWER:
n = 68
Introduction to Estimation
137.
347
{Statistics Professor Narrative}What sample size should be used if wherein we wish to
estimate the population mean to within 20 units?
ANSWER:
n = 663
138.
{Statistics Professor Narrative} What sample size should be used if wherein we wish to
estimate the population mean to within 10 units?
ANSWER:
n = 2,652
139.
{Statistics Professor Narrative} Based on your answers to the previous questions,
describe what happens to the sample size when each of the following happens:
a. The population standard deviation decreases
b. The confidence level decreases
c. The width of the interval decreases
ANSWER:
a. Decreases
b. Decreases
c. Increases
140.
A normal population has a standard deviation of 15. How large a sample should be drawn
to estimate with 95% confidence the population mean to within 1.5?
ANSWER:
n = 385
141.
A statistician wants to estimate the mean weekly family expenditure on clothes. He
believes that the standard deviation of the weekly expenditure is $125. Determine with
99% confidence the number of families that must be sampled to estimate the mean
weekly family expenditure on clothes to within $15.
ANSWER:
n = 461
142.
How large a sample of state employees should be taken if we want to estimate with 98%
confidence the mean salary to within $2,000. The population standard deviation is
assumed to be $10,500.
ANSWER:
n = 150
348
143.
Chapter Ten
The director of a hospital wishes to estimate the mean number of people who are
admitted to the emergency room during a 24-hour period. The director randomly selects
64 different 24-hour periods and determines the number of admissions for each. For this
sample, X  20.0. If the director wishes to estimate the mean number of admissions per
24-hour period to within 1 admission with 99% reliability, what size sample should she
choose if the population standard deviation is known to be 5.0?
ANSWER:
n = 166
144.
An economist is interested in studying the incomes of consumers in a particular region.
The population standard deviation is known to be $1,000. a random sample of 50
individuals resulted in an average income of $15,000. What sample size would the
economist need to use for a 95% confidence interval if the width of the interval should
not be more than $100?
ANSWER:
n = 1537
145.
A quality control engineer is interested in the mean length of sheet insulation being cut
automatically by machine. The desired length of the insulation is 12 feet. It is known
that the standard deviation in the cutting length is 0.15 feet. Suppose the engineer
decided to estimate the mean length to within 0.025 with 99% confidence. What sample
size would be needed?
ANSWER:
n = 239
146.
Suppose a department store wants to estimate the average age of its customers, correct to
within 2 years, with level of confidence equal to 0.95. Management believes that the
standard deviation is 8 years. Determine the sample size they should take.
ANSWER:
n = 62