16. 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
17. 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.
18. 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.
19. 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.
20. 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.
21. 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.
22. 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
c.
As the level of confidence increases, the confidence interval gets wider.
23. 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
.
b.
Construct a 99% confidence interval for
.
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.
24. 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.
25. 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
?
b.
Construct a 95% confidence interval for
.
c.
Construct a 90% confidence interval for
.
d.
Discuss why the 90% and 95% confidence intervals are different.
40
b.
17.613 to 62.387
23.445 to 56.555
d.
As the level of confidence increases, the confidence interval gets wider.
26. The monthly incomes from a random sample of faculty at a university are shown below.
Monthly Income ($1000s)
3.0
4.0
6.0
a.
77.756 to 91.444
b.
73.412 to 95.788
c.
As the level of confidence increases, the confidence interval gets wider.
d.
Decrease in width since the margin of error decreased.
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 normally distributed. Give your answer in dollars.
27. 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?
28. 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.
29. 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.
30. 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?
31. 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?
32. 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?
33. 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?
34. 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?
35. 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?
36. 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?
37. 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.
38. 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.
39. 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.
40. 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.
41. 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.
42. 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
43. 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
44. 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
no
no
yes
yes
yes
no
yes
yes
no
no
no
yes
yes
no
yes
yes
no
no
45. 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
46. 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.
47. 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?
48. 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%?
49. 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.
50. 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?
51. 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 in-
come in a potential marketing area after taking a random sample of households.
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 popu-
lation mean with a margin of error of E = $500. How large a sample size is needed?
52. 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 $60.05. Provide a 95% confidence interval estimate of
the mean rent per month for the population of one-bedroom units within a half-mile of campus. We
will assume this population to be normally distributed.
53. Political Science, Inc. (PSI) specializes in voter polls and surveys designed to keep political office
seekers informed of their position in a race. Using telephone surveys, interviewers ask registered vot-
ers who they would vote for if the election were held that day.
In a recent election campaign, PSI found that 220 registered voters, out of 500 contacted, favored a
particular candidate.
a. PSI wants to develop a 95% confidence interval estimate for the proportion of the population of
registered voters that favors the candidate.
b. Suppose that PSI would like 99% confidence that the sample proportion is within +/- .03 of the
population proportion. How large a sample size is needed to provide the desired margin of error?
54. An apartment complex developer is considering building apartments in College Town, but first wants
to do a market study. A sample of monthly rent values ($) for studio apartments in College Town was
taken. The data collected from the 70-apartment sample is presented below. (Based on past expe-
rience, 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?
55. 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.
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. We will assume this population to be normally distributed.
91
78
93
57
75
52
99
80
105
62
104
74
62
68
97
73
77
65
80
109
56. 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 systematic 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 ac-
count 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 mar-
gin of error of E = .025. How large a sample size is needed?