Chapter 8 – Interval Estimation
a.
Determine the standard error of the mean.
b.
With a 0.95 probability, determine the margin of error.
c.
What is the 95% confidence interval of the population mean?
a.
4.00
b.
7.84
c.
112.16 to 127.84
69. In order to determine the average weight of carry-on luggage by passengers in airplanes, a sample of 36 pieces of
carry-on luggage was weighed. The average weight was 20 pounds. Assume that we know the standard deviation of the
population to be 8 pounds.
a.
Determine a 97% confidence interval estimate for the mean weight of the carry-on luggage.
b.
Determine a 95% confidence interval estimate for the mean weight of the carry-on luggage.
a.
17.11 to 22.89
b.
17.39 to 22.61
70. A small stock brokerage firm wants to determine the average daily sales (in dollars) of stocks to their clients. A
sample of the sales for 36 days revealed average daily sales of $200,000. Assume that the standard deviation of the
population is known to be $18,000.
a.
Provide a 95% confidence interval estimate for the average daily sale.
b.
Provide a 97% confidence interval estimate for the average daily sale.
a.
$194,120 to $205,880
b.
$193,490 to $206,510
71. A random sample of 121 checking accounts at a bank showed an average daily balance of $280. The population
standard deviation is known to be $60.
a.
Is it necessary to know anything about the shape of the distribution of the account balances in
order to make an interval estimate of the mean of all the account balances? Explain.
b.
Find the standard error of the mean.
c.
Give a point estimate of the population mean.
d.
Construct a 95% confidence interval estimate for the population mean.
e.
Interpret the confidence interval estimate that you constructed in part d.
a.
No, since the sample means will be normally distributed by the central limit theorem.
b.
5.4545
c.
280
d.
269.31 to 290.69
e.
accounts at this bank is between $269.31 and $290.69.
72. A random sample of 49 lunch customers was taken at a restaurant. The average amount of time the customers in the
sample stayed in the restaurant was 33 minutes. From past experience, it is known that the population standard deviation
equals 10 minutes.
a.
Compute the standard error of the mean.
b.
What can be said about the sampling distribution for the average amount of time customers
spent in the restaurant? Be sure to explain your answer.
c.
With a .95 probability, what statement can be made about the size of the margin of error?
d.
Construct a 95% confidence interval for the true average amount of time customers spent in the
restaurant.
Chapter 8 – Interval Estimation
e.
With a .95 probability, how large of a sample would have to be taken to provide a margin of
error of 2.5 minutes or less?
b.
Normal by the central limit theorem
There is a .95 probability that the sample mean will provide a margin of error of 2.80 or less.
d.
30.20 to 35.80
62
73. A simple random sample of 144 items resulted in a sample mean of 1080. The population standard deviation is known
to be 240. Develop a 95% confidence interval for the population mean.
1040.8 to 1119.2
74. A random sample of 26 checking accounts at a bank showed an average daily balance of $300 and a standard
deviation of $45. The balances of all checking accounts at the bank are normally distributed. Develop a 95% confidence
interval estimate for the mean of the population.
75. A random sample of 81 students at a local university showed that they work an average of 100 hours per month. The
population standard deviation is known to be 27 hours. Compute a 95% confidence interval for the mean hours per month
all students at the university work.
94.12 to 105.88
76. A random sample of 81 children with working mothers showed that they were absent from school an average of 6
days per term. The population standard deviation is known to be 1.8 days. Provide a 90% confidence interval for the
average number of days absent per term for all the children.
5.631 to 6.329
77. The Highway Safety Department wants to study the driving habits of individuals. A sample of 41 cars traveling on the
highway revealed an average speed of 60 miles per hour and a standard deviation of 7 miles per hour. The population of
car speeds is approximately normally distributed. Determine a 90% confidence interval estimate for the speed of all cars.
58.16 to 61.84
78. Computer Services, Inc. wants to determine a confidence interval for the average CPU time of their teleprocessing
transactions. A sample of 196 transactions yielded a mean of 5 seconds. The population standard deviation is 1.4 seconds.
Determine a 97% confidence interval for the average CPU time.
4.783 to 5.217
79. The average monthly electric bill of a random sample of 256 residents of a city is $90. The population standard
deviation is assumed to be $24.
a.
Construct a 90% confidence interval for the mean monthly electric bills of all residents.
b.
Construct a 95% confidence interval for the mean monthly electric bills of all residents.
87.5325 to 92.4675
80. A sample of 100 cans of coffee showed an average weight of 13 ounces. The population standard deviation is 0.8
ounces.
a.
Construct a 95% confidence interval for the mean of the population.
b.
Construct a 95.44% confidence interval for the mean of the population.
c.
Discuss why the answers in parts a and b are different.
a.
12.8432 to 13.1568
b.
12.84 to 13.16
c.
1
81. In order to determine how many hours per week freshmen college students watch television, a random sample of 256
students was selected. It was determined that the students in the sample spent an average of 14 hours. The standard
deviation is 3.2 hours per week for all freshman college students.
a.
Provide a 95% confidence interval estimate for the average number of hours that all college
freshmen spend watching TV per week.
b.
Suppose the sample mean came from a sample of 25 students. Provide a 95% confidence
interval estimate for the average number of hours that all college freshmen spend watching TV
per week. Assume that the hours are normally distributed.
a.
13.608 to 14.392
b.
12.679 to 15.321
1
82. A random sample of 36 magazine subscribers is taken to estimate the mean age of all subscribers. The data follow.
Use Excel to construct a 90% confidence interval estimate of the mean age of all of this magazine’s subscribers.
Subscriber
Age
Subscriber
Age
Subscriber
Age
1
39
13
40
25
38
2
27
14
35
26
51
3
38
15
35
27
26
4
33
16
41
28
39
5
40
17
34
29
35
6
35
18
46
30
37
7
51
19
44
31
33
8
36
20
44
32
41
9
47
21
43
33
36
10
28
22
32
34
33
11
33
23
29
35
46
12
35
24
33
36
37
b.
87.06 to 92.94
1
83. A simple random sample of 25 items from a normally distributed population resulted in a sample mean of 28 and a
standard deviation of 7.5. Construct a 95% confidence interval for the population mean.
24.904 to 31.096
84. A sample of 25 patients in a doctor’s office showed that they had to wait an average of 35 minutes with a standard
deviation of 10 minutes before they could see the doctor. Provide a 98% confidence interval estimate for the average
waiting time of all the patients who visit this doctor. Assume the population of waiting times is normally distributed.
30.016 to 39.984
85. A sample of 16 students from a large university is taken. The average age in the sample was 22 years with a standard
deviation of 6 years. Construct a 95% confidence interval for the average age of the population. Assume the population of
student ages is normally distributed.
86. The proprietor of a boutique in New York wanted to determine the average age of his customers. A random sample of
25 customers revealed an average age of 28 years with a standard deviation of 10 years. Determine a 95% confidence
interval estimate for the average age of all his customers. Assume the population of customer ages is normally distributed.
23.872 to 32.128
87. A statistician selected a sample of 16 accounts receivable and determined the mean of the sample to be $5,000 with a
standard deviation of $400. She reported that the sample information indicated the mean of the population ranges from
$4,739.80 to $5,260.20. She did not report what confidence coefficient she had used. Based on the above information,
determine the confidence coefficient that was used.
88. The makers of a soft drink want to identify the average age of its consumers. A sample of 16 consumers is taken. The
average age in the sample was 22.5 years with a standard deviation of 5 years. Assume the population of consumer ages is
normally distributed.
a.
Construct a 95% confidence interval for the average age of all the consumers.
b.
Construct an 80% confidence interval for the average age of all the consumers.
c.
Discuss why the 95% and 80% confidence intervals are different.
a.
19.836 to 25.164
b.
20.824 to 24.176
Interval is 35.6905 to 39.3095
89. A random sample of 25 observations was taken from a normally distributed population. The average in the sample was
84.6 with a variance of 400.
a.
Construct a 90% confidence interval for μ.
b.
Construct a 99% confidence interval for μ.
c.
Discuss why the 90% and 99% confidence intervals are different.
d.
What would you expect to happen to the confidence interval in part a if the sample size was
increased? Be sure to explain your answer.
a.
77.756 to 91.444
b.
73.412 to 95.788
c.
d.
Decrease in width since the margin of error decreased.
90. You are given the following information obtained from a random sample of 4 observations taken from a large,
normally distributed population.
25
47
32
56
Construct a 95% confidence interval for the mean of the population.
17.613 ≤ μ ≤ 62.387
91. You are given the following information obtained from a random sample of 4 observations from a large, normally
distributed population.
25
47
32
56
a.
What is the point estimate of μ?
b.
Construct a 95% confidence interval for μ.
c.
Construct a 90% confidence interval for μ.
d.
Discuss why the 90% and 95% confidence intervals are different.
a.
40
b.
17.613 to 62.387
c.
23.445 to 56.555
d.
92. The monthly incomes from a random sample of faculty at a university are shown below.
Monthly Income ($1000s)
3.0
4.0
6.0
3.0
5.0
5.0
6.0
8.0
Compute a 90% confidence interval for the mean of the population. The population of all faculty incomes is known to be
c.
Chapter 8 – Interval Estimation
normally distributed. Give your answer in dollars.
$3,867.52 to $6,132.48
93. Fifty students are enrolled in an Economics class. After the first examination, a random sample of 5 papers was
selected. The grades were 60, 75, 80, 70, and 90.
a.
Calculate the estimate of the standard error of the mean.
b.
What assumption must be made before we can determine an interval for the mean grade of all
the students in the class? Explain why.
c.
Assume the assumption of Part b is met. Provide a 90% confidence interval for the mean grade
of all the students in the class.
d.
If there were 200 students in the class, what would be the 90% confidence interval for the mean
grade of all the students in the class?
a.
4.79
b.
Since the sample is small (n < 30) and σ is estimated from s, we must assume the distribution
c.
64.783 to 85.217
d.
64.34 to 85.66
94. A local university administers a comprehensive examination to the recipients of a B.S. degree in Business
Administration. A sample of 5 examinations is selected at random and scored. The scores are shown below.
Grade
56
85
65
86
93
Use Excel to determine an interval estimate for the mean of the population at a 98% confidence level. Interpret your
results.
Chapter 8 – Interval Estimation
95. Below you are given ages that were obtained by taking a random sample of 9 undergraduate students.
19
22
23
19
21
22
19
23
21
Use Excel to determine an interval estimate for the mean of the population at a 99% confidence level. Interpret your
results.
96. The monthly starting salaries of students who receive an MBA degree have a standard deviation of $110. What size
sample should be selected to obtain a 0.95 probability of estimating the mean monthly income within $20 or less?
117
97. A coal company wants to determine a 95% confidence interval estimate for the average daily tonnage of coal that they
mine. Assuming that the company reports that the standard deviation of daily output is 200 tons, how many days should
they sample so that the margin of error will be 39.2 tons or less?
100
98. If the standard deviation of the lifetimes of vacuum cleaners is estimated to be 300 hours, how large of a sample must
be taken in order to be 97% confident that the margin of error will not exceed 40 hours?
Chapter 8 – Interval Estimation
99. A researcher is interested in determining the average number of years employees of a company stay with the company.
If past information shows a standard deviation of 7 months, what size sample should be taken so that at 95% confidence
the margin of error will be 2 months or less?
100. If the standard deviation for the lifetimes of washing machines is estimated to be 800 hours, how large a sample must
be taken in order to be 97% confident that the margin of error will not exceed 50 hours?
101. A real estate agent wants to estimate the mean selling price of two-bedroom homes in a particular area. She wants to
estimate the mean selling price to within $10,000 with an 89.9% level of confidence. The standard deviation of selling
prices is unknown but the agent estimates that the highest selling price is $1,000,000 and the lowest is $50,000. How
many homes should be sampled?
102. For inventory purposes, a grocery store manager wants to estimate the mean number of pounds of cat food sold per
month. The estimate is desired to be within 10 pounds with a 95% level of confidence. A pilot study provided a standard
deviation of 27.6 pounds. How many months should be sampled?
103. It is known that the variance of a population equals 484. A random sample of 81 observations is going to be taken
from the population.
a.
With a .80 probability, what statement can be made about the size of the margin of error?
b.
With a .80 probability, how large of a sample would have to be taken to provide a margin of
error of 3 or less?
a.
There is a .80 probability that the sample mean will provide a margin of error of 3.129 or less.
104. In a random sample of 400 registered voters, 120 indicated they plan to vote for Candidate A. Determine a 95%
confidence interval for the proportion of all the registered voters who will vote for Candidate A.
0.255 to 0.345
105. In a random sample of 200 registered voters, 120 indicated they are Democrats. Develop a 95% confidence interval
for the proportion of registered voters in the population who are Democrats.
.5321 to .6679
106. In a random sample of 500 college students, 23% say that they read or watch the news every day. Develop a 90%
confidence interval for the population proportion. Interpret your results.
0.199 to 0.261
107. Six hundred consumers were asked whether they would like to purchase a domestic or a foreign automobile. Their
responses are given below.
Preference
Frequency
Domestic
240
Foreign
360
Develop a 95% confidence interval for the proportion of all consumers who prefer to purchase domestic automobiles.
0.3608 to 0.4392
108. A university planner wants to determine the proportion of spring semester students who will attend summer school.
She surveys 32 current students discovering that 12 will return for summer school.
a.
Construct a 90% confidence interval estimate for the proportion of current spring students who
will return for summer school.
b.
With a 0.95 probability, how large of a sample would have to be taken to provide a margin of
error of 3% or less?
a.
0.234 to 0.516
b.
1001
109. A new brand of breakfast cereal is being market tested. One hundred boxes of the cereal were given to consumers to
try. The consumers were asked whether they liked or disliked the cereal. You are given their responses below.
Response
Frequency
Liked
60
Disliked
40
100
a.
What is the point estimate of the proportion of people who will like the cereal?
b.
Construct a 95% confidence interval for the proportion of all consumers who will like the
cereal.
c.
What is the margin of error for the 95% confidence interval that you constructed in part b?
d.
With a .95 probability, how large of a sample needs to be taken to provide a margin of error
of .09 or less?
a.
0.6
b.
0.504 to 0.696
c.
.096
d.
114
110. A marketing firm is developing a new television advertisement for a large discount retail chain. A sample of 30
people is shown two potential ads and asked their preference. The results for ad #1 follow. Use Excel to develop a 95%
confidence interval estimate of the proportion of people in the population who will prefer ad #1.
Prefer Advertisement #1
yes
no
no
yes
yes
no
no
no
no
yes
no
yes
With a 90% level of confidence we can state that the proportion of all college students who read or watch the
Chapter 8 – Interval Estimation
no
no
yes
yes
yes
no
yes
yes
no
no
no
yes
yes
no
yes
yes
no
no
111. A survey of 40 students at a local college asks, “Where do you buy the majority of your books?” The responses fell
into three categories: “at the campus bookstore,” “on the Internet,” and “other.” The results follow. Use Excel to estimate
the proportion of all of the college students who buy their books on the Internet.
Where Most Books Bought
bookstore
bookstore
internet
other
internet
other
bookstore
other
bookstore
bookstore
bookstore
bookstore
bookstore
other
bookstore
bookstore
bookstore
internet
internet
other
other
other
other
other
other
other
internet
bookstore
other
other
internet
other
bookstore
bookstore
other
bookstore
internet
internet
other
bookstore
112. A health club annually surveys its members. Last year, 33% of the members said they use the treadmill at least 4
times a week. How large of sample should be taken this year to estimate the percentage of members who use the treadmill
at least 4 times a week? The estimate is desired to have a margin of error of 5% with a 95% level of confidence.
340
113. A local hotel wants to estimate the proportion of its guests that are from out–of-state. Preliminary estimates are that
45% of the hotel guests are from out–of-state. How large a sample should be taken to estimate the proportion of out-of–
state guests with a margin of error no larger than 5% and with a 95% level of confidence?
381
114. The manager of a department store wants to determine what proportion of people who enter the store use the store’s
credit card for their purchases. What size sample should he take so that at 99% confidence the error will not be more than
8%?
260
115. The manager of Hudson Auto Repair wants to advertise one price for an engine tune–up, with parts included. Before
he decides the price to advertise, he needs a good estimate of the average cost of tune-up parts. A sample of 20 customer
invoices for tune-ups has been taken and the costs of parts, rounded to the nearest dollar, are listed below.
91
78
93
57
75
52
99
80
105
62
104
74
62
68
97
73
77
65
80
109
Provide a 90% confidence interval estimate of the mean cost of parts per tune–up for all of the tune-ups performed at
Hudson Auto Repair.
116. The manager of University Credit Union (UCU) is concerned about checking account transaction discrepancies.
Customers are bringing transaction errors to the attention of the bank’s staff several months after they occur. The manager
would like to know what proportion of his customers balance their checking accounts within 30 days of receiving a
transaction statement from the bank.
Using random sampling, 400 checking account customers are contacted by telephone and asked if they routinely balance
their accounts within 30 days of receiving a statement. 271 of the 400 customers respond Yes.
a. Develop a 95% confidence interval estimate for the proportion of the population of checking account customers at UCU
that routinely balance their accounts in a timely manner.
b. Suppose UCU wants a 95% confidence interval estimate of the population proportion with a margin of error of E =
.025. How large a sample size is needed?
117. National Discount has 260 retail outlets throughout the United States. National evaluates each potential location for
a new retail outlet in part on the mean annual income of the households in the marketing area of the new location.
National develops an interval estimate of the mean annual income in a potential marketing area after taking a random
sample of households.
Chapter 8 – Interval Estimation
For a marketing area being studied, a sample of 36 households was taken and the sample mean income was $21,100.39.
Based on past experience, National Discount assumes a known value of
= $4500 for the population income standard
deviation.
a. Develop a 95% confidence interval for the mean annual income of households in this marketing area.
b. Suppose that National’s management team wants a 95% confidence interval estimate of the population mean with a
margin of error of E = $500. How large a sample size is needed?
118. A reporter for a student newspaper is writing an article on the cost of off-campus housing. A sample was taken of 10
one-bedroom units within a half-mile of campus and the rents paid. The sample mean is $550 and the sample standard
deviation is $60.05. Provide a 95% confidence interval estimate of the mean rent per month for the population of one–
bedroom units within a half-mile of campus. We will assume this population to be normally distributed.
119. Political Science, Inc. (PSI) specializes in voter polls and surveys designed to keep political office seekers informed
of their position in a race. Using telephone surveys, interviewers ask registered voters who they would vote for if the
election were held that day. In a recent election campaign, PSI found that 220 registered voters, out of 500 contacted,
favored a particular candidate.
a. PSI wants to develop a 95% confidence interval estimate for the proportion of the population of registered voters that
favors the candidate.
b. Suppose that PSI would like 99% confidence that the sample proportion is within +/– .03 of the population proportion.
How large a sample size is needed to provide the desired margin of error?
120. An apartment complex developer is considering building apartments in College Town, but first wants to do a market
study. A sample of monthly rent values ($) for studio apartments in College Town was taken. The data collected from
the 70-apartment sample resulted in a sample mean of $490.80. (Based on past experience, the developer assumes a
known value of
= $55 for the population standard deviation.)
a. Develop a 98% confidence interval for the mean monthly rent for all studio apartments in this city.
b. Suppose the apartment developer wants a 98% confidence interval estimate of the population mean with a margin of
error of E = $10. How large a sample size is needed?