A random survey of enrollment at 35 community colleges across the United States yielded the
following figures (source: Microsoft Bookshelf ): 6414; 1550; 2109; 9350; 21828; 4300; 5944; 5722;
2825; 2044; 5481; 5200; 5853; 2750; 10012; 6357; 27000; 9414; 7681; 3200; 17500; 9200; 7380; 18314;
6557; 13713; 17768; 7493; 2771; 2861; 1263; 7285; 28165; 5080; 11622. Assume the underlying
population is normal.
a. i. x =
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ii. sx = ________
iii. n = ________
iv. n − 1 =________
b. Define the Random Variables X and X, in words.
c. Which distribution should you use for this problem? Explain your choice.
d. Construct a 95% confidence interval for the population mean enrollment at community colleges
in the United States.
i. State the confidence interval.
ii. Sketch the graph.
iii. Calculate the error bound.
e. What will happen to the error bound and confidence interval if 500 community colleges were
surveyed? Why?
Exercise 8.9.4
From a stack of IEEE Spectrum magazines, announcements for 84 upcoming engineering confer-
ences were randomly picked. The mean length of the conferences was 3.94 days, with a standard
deviation of 1.28 days. Assume the underlying population is normal.
a. Define the Random Variables X and X, in words.
b. Which distribution should you use for this problem? Explain your choice.
c. Construct a 95% confidence interval for the population mean length of engineering conferences.
i. State the confidence interval.
ii. Sketch the graph.
iii. Calculate the error bound.
Exercise 8.9.5
(Solution on p. 373.)
Suppose that a committee is studying whether or not there is waste of time in our judicial system.
It is interested in the mean amount of time individuals waste at the courthouse waiting to be called
for service. The committee randomly surveyed 81 people. The sample mean was 8 hours with a
sample standard deviation of 4 hours.
a. i. x =________
ii. sx = ________
iii. n =________
iv. n − 1 =________
b. Define the Random Variables X and X, in words.
c. Which distribution should you use for this problem? Explain your choice.
d. Construct a 95% confidence interval for the population mean time wasted.
a. State the confidence interval.
b. Sketch the graph.
c. Calculate the error bound.
e. Explain in a complete sentence what the confidence interval means.
Exercise 8.9.6
Suppose that an accounting firm does a study to determine the time needed to complete one per-
son’s tax forms. It randomly surveys 100 people. The sample mean is 23.6 hours. There is a known
standard deviation of 7.0 hours. The population distribution is assumed to be normal.
a. i. x = ________
ii. σ =________
iii. sx =________
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353
iv. n = ________
v. n − 1 =________
b. Define the Random Variables X and X, in words.
c. Which distribution should you use for this problem? Explain your choice.
d. Construct a 90% confidence interval for the population mean time to complete the tax forms.
i. State the confidence interval.
ii. Sketch the graph.
iii. Calculate the error bound.
e. If the firm wished to increase its level of confidence and keep the error bound the same by
taking another survey, what changes should it make?
f. If the firm did another survey, kept the error bound the same, and only surveyed 49 people,
what would happen to the level of confidence? Why?
g. Suppose that the firm decided that it needed to be at least 96% confident of the population
mean length of time to within 1 hour. How would the number of people the firm surveys
change? Why?
Exercise 8.9.7
(Solution on p. 373.)
A sample of 16 small bags of the same brand of candies was selected. Assume that the population
distribution of bag weights is normal. The weight of each bag was then recorded. The mean
weight was 2 ounces with a standard deviation of 0.12 ounces. The population standard deviation
is known to be 0.1 ounce.
a. i. x = ________
ii. σ = ________
iii. sx =________
iv. n = ________
v. n − 1 = ________
b. Define the Random Variable X, in words.
c. Define the Random Variable X, in words.
d. Which distribution should you use for this problem? Explain your choice.
e. Construct a 90% confidence interval for the population mean weight of the candies.
i. State the confidence interval.
ii. Sketch the graph.
iii. Calculate the error bound.
f. Construct a 98% confidence interval for the population mean weight of the candies.
i. State the confidence interval.
ii. Sketch the graph.
iii. Calculate the error bound.
g. In complete sentences, explain why the confidence interval in (f) is larger than the confidence
interval in (e).
h. In complete sentences, give an interpretation of what the interval in (f) means.
Exercise 8.9.8
A pharmaceutical company makes tranquilizers. It is assumed that the distribution for the length
of time they last is approximately normal. Researchers in a hospital used the drug on a random
sample of 9 patients. The effective period of the tranquilizer for each patient (in hours) was as
follows: 2.7; 2.8; 3.0; 2.3; 2.3; 2.2; 2.8; 2.1; and 2.4 .
a. i. x = ________
ii. sx = ________
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iii. n = ________
iv. n − 1 = ________
b. Define the Random Variable X, in words.
c. Define the Random Variable X, in words.
d. Which distribution should you use for this problem? Explain your choice.
e. Construct a 95% confidence interval for the population mean length of time.
i. State the confidence interval.
ii. Sketch the graph.
iii. Calculate the error bound.
f. What does it mean to be “95% confident” in this problem?
Exercise 8.9.9
(Solution on p. 373.)
Suppose that 14 children were surveyed to determine how long they had to use training wheels.
It was revealed that they used them an average of 6 months with a sample standard deviation of
3 months. Assume that the underlying population distribution is normal.
a. i. x = ________
ii. sx = ________
iii. n = ________
iv. n − 1 = ________
b. Define the Random Variable X, in words.
c. Define the Random Variable X, in words.
d. Which distribution should you use for this problem? Explain your choice.
e. Construct a 99% confidence interval for the population mean length of time using training
wheels.
i. State the confidence interval.
ii. Sketch the graph.
iii. Calculate the error bound.
f. Why would the error bound change if the confidence level was lowered to 90%?
Exercise 8.9.10
Insurance companies are interested in knowing the population percent of drivers who always
buckle up before riding in a car.
a. When designing a study to determine this population proportion, what is the minimum num-
ber you would need to survey to be 95% confident that the population proportion is esti-
mated to within 0.03?
b. If it was later determined that it was important to be more than 95% confident and a new survey
was commissioned, how would that affect the minimum number you would need to survey?
Why?
Exercise 8.9.11
(Solution on p. 373.)
Suppose that the insurance companies did do a survey. They randomly surveyed 400 drivers and
found that 320 claimed to always buckle up. We are interested in the population proportion of
drivers who claim to always buckle up.
a. i. x = ________
ii. n = ________
iii. p’ = ________
b. Define the Random Variables X and P’, in words.
c. Which distribution should you use for this problem? Explain your choice.
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d. Construct a 95% confidence interval for the population proportion that claim to always buckle
up.
i. State the confidence interval.
ii. Sketch the graph.
iii. Calculate the error bound.
e. If this survey were done by telephone, list 3 difficulties the companies might have in obtaining
random results.
Exercise 8.9.12
Unoccupied seats on flights cause airlines to lose revenue. Suppose a large airline wants to esti-
mate its mean number of unoccupied seats per flight over the past year. To accomplish this, the
records of 225 flights are randomly selected and the number of unoccupied seats is noted for each
of the sampled flights. The sample mean is 11.6 seats and the sample standard deviation is 4.1
seats.
a. i. x = ________
ii. sx = ________
iii. n = ________
iv. n − 1 = ________
b. Define the Random Variables X and X, in words.
c. Which distribution should you use for this problem? Explain your choice.
d. Construct a 92% confidence interval for the population mean number of unoccupied seats per
flight.
i. State the confidence interval.
ii. Sketch the graph.
iii. Calculate the error bound.
Exercise 8.9.13
(Solution on p. 373.)
According to a recent survey of 1200 people, 61% feel that the president is doing an acceptable
job. We are interested in the population proportion of people who feel the president is doing an
acceptable job.
a. Define the Random Variables X and P’, in words.
b. Which distribution should you use for this problem? Explain your choice.
c. Construct a 90% confidence interval for the population proportion of people who feel the pres-
ident is doing an acceptable job.
i. State the confidence interval.
ii. Sketch the graph.
iii. Calculate the error bound.
Exercise 8.9.14
A survey of the mean amount of cents off that coupons give was done by randomly surveying one
coupon per page from the coupon sections of a recent San Jose Mercury News. The following data
were collected: 20¢; 75¢; 50¢; 65¢; 30¢; 55¢; 40¢; 40¢; 30¢; 55¢; $1.50; 40¢; 65¢; 40¢. Assume the
underlying distribution is approximately normal.
a. i. x = ________
ii. sx = ________
iii. n = ________
iv. n − 1 = ________
b. Define the Random Variables X and X, in words.
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c. Which distribution should you use for this problem? Explain your choice.
d. Construct a 95% confidence interval for the population mean worth of coupons.
i. State the confidence interval.
ii. Sketch the graph.
iii. Calculate the error bound.
e. If many random samples were taken of size 14, what percent of the confident intervals con-
structed should contain the population mean worth of coupons? Explain why.
Exercise 8.9.15
(Solution on p. 374.)
An article regarding interracial dating and marriage recently appeared in the Washington Post. Of
the 1709 randomly selected adults, 315 identified themselves as Latinos, 323 identified themselves
as blacks, 254 identified themselves as Asians, and 779 identified themselves as whites. In this
survey, 86% of blacks said that their families would welcome a white person into their families.
Among Asians, 77% would welcome a white person into their families, 71% would welcome a
Latino, and 66% would welcome a black person.
a. We are interested in finding the 95% confidence interval for the percent of all black families that
would welcome a white person into their families. Define the Random Variables X and P’,
in words.
b. Which distribution should you use for this problem? Explain your choice.
c. Construct a 95% confidence interval
i. State the confidence interval.
ii. Sketch the graph.
iii. Calculate the error bound.
Exercise 8.9.16
Refer to the problem above.
a. Construct three 95% confidence intervals.
i: Percent of all Asians that would welcome a white person into their families.
ii: Percent of all Asians that would welcome a Latino into their families.
iii: Percent of all Asians that would welcome a black person into their families.
b. Even though the three point estimates are different, do any of the confidence intervals overlap?
Which?
c. For any intervals that do overlap, in words, what does this imply about the significance of the
differences in the true proportions?
d. For any intervals that do not overlap, in words, what does this imply about the significance of
the differences in the true proportions?
Exercise 8.9.17
(Solution on p. 374.)
A camp director is interested in the mean number of letters each child sends during his/her camp
session. The population standard deviation is known to be 2.5. A survey of 20 campers is taken.
The mean from the sample is 7.9 with a sample standard deviation of 2.8.
a. i. x = ________
ii. σ = ________
iii. sx = ________
iv. n = ________
v. n − 1 = ________
b. Define the Random Variables X and X, in words.
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c. Which distribution should you use for this problem? Explain your choice.
d. Construct a 90% confidence interval for the population mean number of letters campers send
home.
i. State the confidence interval.
ii. Sketch the graph.
iii. Calculate the error bound.
e. What will happen to the error bound and confidence interval if 500 campers are surveyed?
Why?
Exercise 8.9.18
Stanford University conducted a study of whether running is healthy for men and women over
age 50. During the first eight years of the study, 1.5% of the 451 members of the 50-Plus Fitness
Association died. We are interested in the proportion of people over 50 who ran and died in the
same eight–year period.
a. Define the Random Variables X and P’, in words.
b. Which distribution should you use for this problem? Explain your choice.
c. Construct a 97% confidence interval for the population proportion of people over 50 who ran
and died in the same eight–year period.
i. State the confidence interval.
ii. Sketch the graph.
iii. Calculate the error bound.
d. Explain what a “97% confidence interval” means for this study.
Exercise 8.9.19
(Solution on p. 374.)
In a recent sample of 84 used cars sales costs, the sample mean was $6425 with a standard deviation
of $3156. Assume the underlying distribution is approximately normal.
a. Which distribution should you use for this problem? Explain your choice.
b. Define the Random Variable X, in words.
c. Construct a 95% confidence interval for the population mean cost of a used car.
i. State the confidence interval.
ii. Sketch the graph.
iii. Calculate the error bound.
d. Explain what a “95% confidence interval” means for this study.
Exercise 8.9.20
A telephone poll of 1000 adult Americans was reported in an issue of Time Magazine. One of the
questions asked was “What is the main problem facing the country?” 20% answered “crime”. We
are interested in the population proportion of adult Americans who feel that crime is the main
problem.
a. Define the Random Variables X and P’, in words.
b. Which distribution should you use for this problem? Explain your choice.
c. Construct a 95% confidence interval for the population proportion of adult Americans who feel
that crime is the main problem.
i. State the confidence interval.
ii. Sketch the graph.
iii. Calculate the error bound.
d. Suppose we want to lower the sampling error. What is one way to accomplish that?
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e. The sampling error given by Yankelovich Partners, Inc. (which conducted the poll) is ± 3%. In
1-3 complete sentences, explain what the ± 3% represents.
Exercise 8.9.21
(Solution on p. 374.)
Refer to the above problem. Another question in the poll was “[How much are] you worried
about the quality of education in our schools?” 63% responded “a lot”. We are interested in the
population proportion of adult Americans who are worried a lot about the quality of education in
our schools.
1. Define the Random Variables X and P’, in words.
2. Which distribution should you use for this problem? Explain your choice.
3. Construct a 95% confidence interval for the population proportion of adult Americans wor-
ried a lot about the quality of education in our schools.
i. State the confidence interval.
ii. Sketch the graph.
iii. Calculate the error bound.
4. The sampling error given by Yankelovich Partners, Inc. (which conducted the poll) is ± 3%.
In 1-3 complete sentences, explain what the ± 3% represents.
Exercise 8.9.22
Six different national brands of chocolate chip cookies were randomly selected at the supermarket.
The grams of fat per serving are as follows: 8; 8; 10; 7; 9; 9. Assume the underlying distribution is
approximately normal.
a. Calculate a 90% confidence interval for the population mean grams of fat per serving of choco-
late chip cookies sold in supermarkets.
i. State the confidence interval.
ii. Sketch the graph.
iii. Calculate the error bound.
b. If you wanted a smaller error bound while keeping the same level of confidence, what should
have been changed in the study before it was done?
c. Go to the store and record the grams of fat per serving of six brands of chocolate chip cookies.
d. Calculate the mean.
e. Is the mean within the interval you calculated in part (a)? Did you expect it to be? Why or why
not?
Exercise 8.9.23
A confidence interval for a proportion is given to be (– 0.22, 0.34). Why doesn’t the lower limit of
the confidence interval make practical sense? How should it be changed? Why?
8.9.1 Try these multiple choice questions.
The next three problems refer to the following: According to a Field Poll, 79% of California adults
(actual results are 400 out of 506 surveyed) feel that “education and our schools” is one of the top is-
sues facing California. We wish to construct a 90% confidence interval for the true proportion of Cali-
fornia adults who feel that education and the schools is one of the top issues facing California. (Source:
http://field.com/fieldpollonline/subscribers/)
Exercise 8.9.24
(Solution on p. 374.)
A point estimate for the true population proportion is:
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359
A. 0.90
B. 1.27
C. 0.79
D. 400
Exercise 8.9.25
(Solution on p. 374.)
A 90% confidence interval for the population proportion is:
A. (0.761, 0.820)
B. (0.125, 0.188)
C. (0.755, 0.826)
D. (0.130, 0.183)
Exercise 8.9.26
(Solution on p. 374.)
The error bound is approximately
A. 1.581
B. 0.791
C. 0.059