8 Sums, Differences, and Combinations of Random Variables
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Testing Hypotheses About Means
13.11
13.12
13.13
13.14
tient revenue. A legislator is concerned that there
is inadequate enforcement of this regulation. He
plans to audit a random sample of 30 nonproﬁt
hospitals and assess the percentage of net patient
revenue they spend on charity care. He will then
test whether the population mean is at least 4%.
● Give the value of the test statistic t in each of the
following situations.
a. H0: m ϭ 50, x ϭ 60 , s ϭ 90, n ϭ 100.
b. Null value ϭ 100, sample mean ϭ 98, s ϭ 15,
sample size ϭ 40.
c. H0: m ϭ 250, x ϭ 270 , standard error ϭ 5, n ϭ 100.
● Find the p-value and draw a sketch showing the
p-value area for each of the following situations in
which the value of t is the test statistic for the hypotheses given:
a. H0: m ϭ m0, Ha: m Ͼ m0, n ϭ 28, t ϭ 2.00.
b. H0: m ϭ m0, Ha: m Ͼ m0, n ϭ 28, t ϭ Ϫ2.00.
c. H0: m ϭ m0, Ha: m m0, n ϭ 81, t ϭ 2.00.
d. H0: m ϭ m0, Ha: m m0, n ϭ 81, t ϭ Ϫ2.00.
● Use Table A.2 to ﬁnd the critical value and rejection region in each of the following situations. Then
determine whether the null hypothesis would be rejected. In each case the null hypothesis is H0: m ϭ 100.
a. Ha: m Ͼ 100, n ϭ 21, a ϭ .05, test statistic t ϭ 2.30.
b. Ha: m Ͼ 100, n ϭ 21, a ϭ .01, test statistic t ϭ 2.30.
c. Ha: m 100, n ϭ 21, a ϭ .05, test statistic t ϭ 2.30.
d. Ha: m 100, n ϭ 21, a ϭ .01, test statistic t ϭ 2.30.
e. Ha: m Ͼ 100, n ϭ 10, a ϭ .05, test statistic t ϭ 1.95.
f. Ha: m Ͻ 100, n ϭ 10, a ϭ .05, test statistic t ϭ Ϫ1.95.
g. Ha: m Ͻ 100, n ϭ 10, a ϭ .05, test statistic t ϭ 1.95.
h. Ha: m 100, n ϭ 10, a ϭ .05, test statistic t ϭ 1.95.
◆ The dataset cholest on the CD for this book includes cholesterol levels for heart attack patients and
for a group of control patients. It is recommended
that people try to keep their cholesterol level below
200. The following Minitab output is for the control
patients:
Test of mu = 200.00 vs mu < 200.00
Variable
control
N
30
Mean
193.13
StDev
22.30
SE Mean
4.07
T
–1.69
P
0.051
a. What are the null and alternative hypotheses being
tested? Write them in symbols.
b. What is the mean cholesterol level for the sample
of control patients?
c. How many patients were in the sample?
d. Use the formula for the standard error of the mean
to show how to compute the value of 4.07 reported
by Minitab.
e. What values does Minitab report for the test statistic and the p-value?
f. Identify the numbers that were used to compute
the t-statistic, and verify that the reported value is
correct.
g. What conclusion would be made in this situation,
using a .05 level of signiﬁcance?
● Basic skills
◆ Dataset available but not required
589
13.15 Suppose a study is done to test the null hypothesis
H0: m ϭ 100. A random sample of n ϭ 50 observations
results in x ϭ 102 and s ϭ 15.
a. What is the null standard error in this case?
b. Plug numbers into the formula
Sample statistic Ϫ Null value
Null standard error
c. On the basis of the information given, can the
p-value for this test be found? If so, ﬁnd it. If not,
explain what additional information would be
needed.
13.16 It has been hypothesized that the mean pulse rate
for college students is about 72 beats per minute. A
sample of Penn State students recorded their sexes
and pulse rates. Assume that the samples are representative of all Penn State men and women for pulse
rate measurements. The summary statistics were as
follows:
Sex
n
Mean
StDev
Female
Male
35
57
76.90
70.42
11.60
09.95
a. Test whether the pulse rates of all Penn State men
have a mean of 72.
b. Test whether the pulse rates of all Penn State
women have a mean of 72.
c. Write a sentence or two summarizing the results
of parts (a) and (b) in words that would be understood by someone with no training in statistics.
13.17 Refer to Exercise 13.10(a), which posed the following
research question: “Many cars have a recommended
tire pressure of 32 psi (pounds per square inch). At a
roadside vehicle safety checkpoint, ofﬁcials plan to
randomly select 50 cars for which this is the recommended tire pressure and measure the actual tire
pressure in the front left tire. They want to know
whether drivers on average have too little pressure in
their tires.” Suppose the experiment is conducted,
and the mean and standard deviation for the 50 cars
tested are 30.1 psi and 3 psi, respectively. Carry out
the ﬁve steps to test the appropriate hypotheses.
13.18 A cell phone company knows that the mean length of
calls for all of its customers in a certain city is 9.2 minutes. The company is thinking about offering a senior
discount to attract new customers but ﬁrst wants to
know whether the mean length of calls for current
customers who are seniors (65 and over) is the same
as it is for the general customer pool. The only way to
identify seniors is to conduct a survey and ask people
whether they are over age 65. Using this method, the
company contacts a random sample of 200 seniors
and records the length of their last call. The sample
mean and standard deviation for the 200 calls are
8 minutes and 10 minutes, respectively.
a. Do you think the data collected on the 200 seniors
are approximately bell-shaped? Explain.
Bold-numbered exercises answered in the back
590
Chapter 13
b. Is it valid to conduct a one-sample t-test in this situation? Explain.
c. In spite of how you may have answered part (b),
carry out the ﬁve steps to test the hypotheses of interest in this situation.
13.19 A university is concerned that it is taking students too
long to complete their requirements and graduate;
the average time for all students is 4.7 years. The dean
of the campus honors program claims that students
who participated in that program in their ﬁrst year
have had a lower mean time to graduation. Unfortunately, there is no automatic way to pull the records
of all of the thousands of students who have participated in the program; they must be pulled individually and checked. A random sample of 30 students
who had participated is taken, and the mean and
standard deviation for the time to completion for
those students are 4.5 years and 0.5 year, respectively.
Carry out the ﬁve steps to test the hypotheses of interest in this situation.
13.20 ◆ The survey in the UCDavis2 data set on the CD accompanying this book asked students if they preferred to sit in the front, middle, or back of the class
and also asked them their heights. The following data
are the heights for 15 female students who said they
prefer to sit in the back of the class.
68, 62, 65, 69, 68, 69, 64, 66, 69, 68, 62, 64, 67, 68, 65
The mean height for the population of college females is 65 inches. Carry out the ﬁve steps to test the
claim that the mean height for females who prefer
to sit in the back of the room is higher than it is for
the general population. In other words, test whether
females who prefer to sit in the back of the room are
taller than average.
13.21 ◆ Refer to Exercise 13.20. The following data are the
heights for the 38 females who said they prefer to sit
in the front of the classroom
66, 63, 63, 66, 65.5, 63, 60, 64, 63, 68, 68, 66, 62.5, 65,
64, 63, 66, 63, 63, 67, 66, 66, 62, 65, 63.5, 60, 61, 62,
63, 60, 65, 62, 63, 63, 62, 65, 63, 66
The mean height for the population of college females is 65 inches. Carry out the ﬁve steps to test the
claim that the mean height for females who prefer to
sit in the front of the room is lower than it is for the
general population. In other words, test whether females who prefer to sit in the front of the room are
shorter than average.
Section 13.3
13.22 ● Explain how a paired t-test and a one-sample t-test
are different and how they are the same.
13.23 ● If you were given a data set consisting of pairs of
observations for which the question of interest was if
the population mean of the differences was 0, explain
the steps you would take to determine whether it is
valid to use a paired t-test.
13.24 ● Give the value of the test statistic t in each of the
following situations, then ﬁnd the p-value or p-value
range for a two-tailed test.
● Basic skills
◆ Dataset available but not required
a. H0: md ϭ 0, d ϭ 4, sd ϭ 15, n ϭ 50 .
b. H0: md ϭ 0, d ϭ Ϫ4, sd ϭ 15, n ϭ 50 .
c. H0: md ϭ 0, d ϭ 0, sd ϭ 15, n ϭ 50 .
13.25 Most people complain that they gain weight during
the December holidays, and Yanovski et al. (2000)
wanted to determine whether that was the case. They
sampled the weights of 195 adults in mid-November
and again in early to mid-January. The mean weight
change for the sample was a gain of 0.37 kg, with a
standard deviation of 1.52 kg. State and test the appropriate hypotheses. Be sure to carefully deﬁne the
population parameter(s) you are testing.
13.26 ◆ In Exercise 11.68 a study was reported in which
students were asked to place as many dried beans
into a cup as possible in 15 seconds with their dominant hand, and again with their nondominant hand
(in randomized order). The differences in number
of beans (dominant hand–nondominant hand) for
15 students were as follows:
4, 4, 5, 1, Ϫ2, 0, 2, 4, Ϫ3, 0, 0, 0, Ϫ2, 2, 1
The data also are given in the dataset beans on the
CD for this book.
a. The research question was whether students have
better manual dexterity with their dominant hand
than with their nondominant hand. Write the null
and alternative hypotheses.
b. Check the necessary conditions for doing a one
sample t-test.
c. Carry out the test using a ϭ .05.
d. Carry out the test using a ϭ .10.
e. Write a conclusion about this situation that would
be understood by other students of statistics.
13.27 ◆ Data from the dataset UCDavisl on the CD for this
book included information on height (height) and
mother’s height (momheight) for 93 female students.
Here is the output from the Minitab paired t procedure comparing these heights:
Paired T for height – momheight
height
momheight
Difference
N
93
93
93
Mean
64.4495
63.1645
1.28495
StDev
2.5226
2.6284
2.64719
SE Mean
0.2616
0.2726
0.27450
95% lower bound for mean difference: 0.82884
T-Test of mean difference = 0 (vs > 0): T-Value = 4.68
P-Value = 0.000
a. It has been hypothesized that college students are
taller than they were a generation ago and therefore that college women should be signiﬁcantly
taller than their mothers. State the null and alternative hypotheses to test this claim. Be sure to
deﬁne any parameters you use.
b. Using the information in the Minitab output, the
test statistic is t ϭ 4.68. Identify the numbers that
were used to compute the t-statistic, and verify
that the stated value is correct.
c. What are the degrees of freedom for the test
statistic?
Bold-numbered exercises answered in the back
Testing Hypotheses About Means
d. Carry out the remaining steps of the hypothesis test.
e. Draw a sketch that illustrates the connection
between the t-statistic and the p-value in this
problem.
13.28 In Case Study 5.1 (p. 179), results were presented for
a sample of 63 men who were asked to report their
actual weight and their ideal weight. The mean difference between actual and ideal weight was 2.48
pounds, and the standard deviation of the differences
was 13.77 pounds. Is there sufﬁcient evidence to conclude that for the population of men represented by
this sample the actual and ideal weights differ, on
average? Justify your answer by showing all steps of a
hypothesis test.
13.29 Although we have not emphasized it, the paired t-test
can be used to test hypotheses in which the null value
is something other than 0. For example, suppose that
the proponents of a diet plan claim that the mean
amount of weight lost in the ﬁrst three weeks of following the plan is 10 pounds. A consumer advocacy
group is skeptical and measures the beginning and
ending weights for a random sample of 20 people
who follow the plan for three weeks. The mean and
standard deviation for the difference in weight at the
two times are 8 pounds and 4 pounds, respectively.
a. What is the parameter of interest? Be sure to specify the appropriate population.
b. What are the null and alternative hypotheses?
c. What is the value of the test statistic?
d. What is the p-value for the test?
e. What conclusion can the consumer advocacy
group make?
13.30 A company manufactures a homeopathic drug that it
claims can reduce the time it takes to overcome jet
lag after long-distance ﬂights. A researcher would like
to test that claim. She recruits nine people who take
frequent trips from San Francisco to London and assigns them to take a placebo for one of their trips and
the drug for the other trip, in random order. She then
asks them how many days it took to recover from jet
lag under each condition. The results are as follows:
Person
Placebo
Drug
1
2
3
4
5
6
7
8
9
7
4
8
4
5
4
6
6
5
6
3
2
7
8
8
6
4
2
Carry out the ﬁve steps to test the appropriate
hypotheses.
13.31 Many people have high anxiety about visiting the
dentist. Researchers want to know if this affects blood
pressure in such a way that the mean blood pressure
while waiting to see the dentist is higher than it is
an hour after the visit. Ten individuals have their systolic blood pressures measured while they are in the
dentist’s waiting room and again an hour after the
conclusion of the visit to the dentist. The data are as
follows:
● Basic skills
◆ Dataset available but not required
591
Person
B.P. Before
B.P. After
1
2
3
4
5
6
7
8
9
10
132
118
135
137
149
140
133
139
119
107
121
116
128
122
132
124
119
115
110
103
a. Write the parameter of interest in this situation.
b. Write the null and alternative hypotheses of
interest.
c. Carry out the remaining steps to test the hypotheses you speciﬁed in part (b).
Section 13.4
Exercises 13.32 to 13.43 correspond to the two Lessons in Section 13.4. Lesson 1 exercises are 13.32 to 13.37; Lesson 2 exercises are 13.38 to 13.43.
13.32 ● In each of the following situations, determine
whether the alternative hypothesis was Ha: m1 Ϫ m2
Ͼ 0, Ha: m1 Ϫ m2 Ͻ 0, or Ha: m1 Ϫ m2 0.
a. H0: m1 Ϫ m2 ϭ 0, t ϭ 2.33, df ϭ 8, p-value ϭ 0.048.
b. H0: m1 Ϫ m2 ϭ 0, t ϭ Ϫ2.33, df ϭ 8, p-value ϭ 0.024.
c. H0: m1 Ϫ m2 ϭ 0, t ϭ 2.33, df ϭ 8, p-value ϭ 0.976.
d. H0: m1 Ϫ m2 ϭ 0, t ϭ Ϫ2.33, df ϭ 8, p-value ϭ 0.976.
13.33 ● For each of the following situations, identify
whether a paired t-test or a two-sample t-test is
appropriate:
a. The weights of a sample of 15 marathon runners
were taken before and after a training run to test
whether marathon runners lose dangerous levels
of ﬂuids during a run.
b. Random samples of 200 new freshmen and 200
new transfer students at a university were given a
50-question test on current events to test whether
the level of knowledge of current events differs for
new freshmen and transfer students.
c. Sixty students were matched by initial pulse rate,
with the two with the highest pulse forming a
pair, and so on. Within each pair, one student was
randomly chosen to drink a caffeinated beverage,
while the other one drank an equivalent amount of
water. Their pulse rates were measured 10 minutes
later, to test whether caffeine consumption elevates pulse rates.
13.34 ● Calculate the value of the unpooled test statistic t
in each of the following situations. In each case, assume the null hypothesis is H0: m1 Ϫ m2 ϭ 0.
a. x1 ϭ 35 , s1 ϭ 10, n1 ϭ 100; x2 ϭ 33 , s2 ϭ 9, n2 ϭ 81.
b. The difference in sample means is 48,
s.e. 1x1 Ϫ x2 2 ϭ 22.
c. Minitab output:
Sample 1
Sample 2
N
68
68
Mean
80.58
78.55
StDev
4.22
3.31
SE Mean
0.51
0.40
13.35 ● Do hardcover and softcover books likely to be
found on a professor’s shelf have the same average
number of pages? Data on the number of pages for
Bold-numbered exercises answered in the back
592
Chapter 13
eight hardcover and seven softcover books from
a professor’s shelf were presented in Example 4.2
(p. 121) and are in the ﬁle ProfBooks on the CD for
this book. The Minitab output from the “2-sample t”
procedure follows. Assume that the books are equivalent to a random sample.
Cover
Hard
Soft
N
8
7
Mean
307
429.4
StDev
134
80.9
Lib. arts
Non-Lib. arts
95% CI for mu (Hard) – mu (Soft): (–246, 2)
T-Test mu (Hard) = mu (Soft)(vs not =): T = –2.16
P = 0.054
DF = 11
Cambridge: 188.5, 183.0, 194.5, 185.0, 214.0,
203.5, 186.0, 178.5
Oxford: 186.0, 184.5, 204.0, 184.5, 195.5, 202.5,
174.0, 183.0
13.37 Case Study 1.1 presented data given in response to
the question “What is the fastest you have ever driven
a car? ____ mph.” The summary statistics are:
Females: n ϭ 102, mean ϭ 88.4, standard deviation ϭ14.4
Males: n ϭ 87, mean ϭ 107.4, standard deviation ϭ 17.4
Assuming that these students represent a random
sample of college students, test whether the mean
fastest speed driven by college men and college
women is equal versus the alternative that it is higher
for men.
13.38 Example 11.7 (p. 457) presented results for the number of hours slept the previous night from a survey
given in two statistics classes. One class was a liberal
arts class; the other class was a general introductory
class. The survey was given following a Sunday night
after classes had started. For simplicity, let’s assume
that these classes represent a random sample of sleep
hours for college students in liberal arts and nonliberal arts majors. The data are as follows:
● Basic skills
◆ Dataset available but not required
Mean
St. Dev.
25
148
7.66
6.81
1.34
1.73
a. Test the hypothesis that the mean number of hours
of sleep for the two populations of students are
equal versus the alternative that they are not equal.
Use the unpooled t-test. (Note: The approximate
df ϭ 38 for the unpooled test.)
b. The ﬁgure below displays a dotplot of the data.
Brieﬂy explain what is indicated about the necessary conditions for doing a two-sample t-test.
c. Repeat the hypothesis test using the pooled procedure. Compare the results to those in part (a), and
discuss which procedure you think is more appropriate in this situation.
SE Mean
47
31
a. Give the null and alternative hypotheses using
symbols.
b. What is the value of the test statistic t?
c. Identify the numbers that were used to compute
the t-statistic, and verify that the reported value is
correct.
d. What conclusion would be made using a .05 level
of signiﬁcance? Write the conclusion in statistical
terms and in the context of the problem.
13.36 Example 2.14 (p. 43) gave the weights of eight rowers
on each of the Cambridge and Oxford crew teams.
The weights are shown again here. Assuming that
these men represent appropriate random samples,
test the hypothesis that the mean weight of rowers on
the Cambridge and Oxford crew teams are equal versus the alternative that they are not equal.
n
Non–liberal arts
Liberal arts
2
7
Hours of sleep
12
13.39 Example 11.3 (p. 449) presented data from a study in
which sedentary men were randomly assigned to be
placed on a diet or exercise for a year to lose weight.
Forty-two men were placed on a diet, while the remaining 47 were put on an exercise routine. The
group on a diet lost an average of 7.2 kg, with a standard deviation of 3.7 kg. The men who exercised lost
an average of 4.0 kg, with a standard deviation of
3.9 kg.
a. State and test appropriate null and alternative hypotheses to determine whether the mean weight
loss would be different under the two routines for
the population of men similar to those in this study.
b. Explain how you decided whether to do a pooled
or unpooled test in part (a).
13.40 Do students sleep more in Pennsylvania or in California? Data from surveys in elementary statistics
classes at Penn State University and the University of
California at Davis resulted in the following summary
statistics for the number of hours students sleep:
UC Davis
Penn State
n
Mean
St. Dev.
S.E. Mean
173
190
6.93
7.11
1.71
1.95
0.13
0.14
Assume that these students are representative of all
students at those two schools. Is there sufﬁcient evi-
Bold-numbered exercises answered in the back
Testing Hypotheses About Means
dence to conclude that the mean hours of sleep are
different at the two schools? Carry out all steps of the
hypothesis test, and deﬁne all parameters.
13.41 Students in a statistics class at Penn State were asked,
“About how many minutes do you typically exercise
in a week?” Responses from the women in the class
were as follows:
60, 240, 0, 360, 450, 200, 100, 70, 240, 0, 60, 360, 180,
300, 0, 270
Responses from the men in the class were as follows:
180, 300, 60, 480, 0, 90, 300, 14, 600, 360, 120, 0, 240
a. Draw appropriate graphs to check whether the
conditions for conducting a two-sample t-test are
met. Discuss the results of your graphs.
b. What additional assumption or condition is required if conclusions are to be made about
amount of exercise for the population of all Penn
State students on the basis of these sample results?
c. Assume that the conditions are met, and conduct a
test to determine whether the mean amount of exercise differs for men and women.
13.42 Researchers speculate that drivers who do not wear a
seatbelt are more likely to speed than drivers who do
wear one. The following data were collected on a random sample of 20 drivers who were clocked to see
how fast they were driving (miles per hour), and then
were stopped to see whether they were wearing a seat
belt (Y ϭ yes, N ϭ no).
Driver
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Speed
62 60 72 85 68 64 72 72 75 63 62 84 76 60 66 63 64 80 52 64
Seatbelt
Y Y N N Y Y Y N Y Y N N N Y N N Y Y Y Y
Do these results support the claim that the mean
speed is higher for the population of drivers who do
not wear seatbelts than for the population of drivers
who do?
a. Carry out the ﬁve steps of hypotheses testing using
the unpooled procedure.
b. Repeat part (a) using the pooled procedure.
c. Compare the unpooled and pooled results and
discuss which procedure is more appropriate.
d. Carry out the unpooled test using the rejection
region approach with a ϭ .05.
13.43 In Example 11.12 (p. 471) a study by Slutske, Piasecki,
and Hunt-Carter (2003; and on the CD for this book)
was presented, in which the mean number of hangover symptoms was compared for students whose
parents have alcohol problems and students whose
parents do not. Researchers are interested in knowing if the mean number of hangover symptoms is
higher for the population of students whose parents
have alcohol problems than for the population whose
parents do not. The sample statistics are as follows:
● Basic skills
◆ Dataset available but not required
Group
Parental alcohol problems (n1 ؍282)
No parental alcohol problems (n2 ؍945)
593
Mean
Standard
deviation
x1 ϭ 5.9
x2 ϭ 4.9
s 1 ϭ 3.6
s 2 ϭ 3.4
a. Carry out the ﬁve steps of hypothesis testing using
the unpooled procedure.
b. Repeat part (a) using the pooled procedure.
c. Compare the unpooled and pooled results and
discuss which procedure is more appropriate.
Section 13.5
13.44 ● In each of the following cases, explain whether
the null hypothesis H0: m ϭ 25 can be rejected. Use
a ϭ .05.
a. 95% conﬁdence interval for m is (10 to 30),
Ha: m 25.
b. 95% conﬁdence interval for m is (26 to 50),
Ha: m 25.
c. 90% conﬁdence interval for m is (10 to 30),
Ha: m Ͼ 25.
d. 90% conﬁdence interval for m is (10 to 30),
Ha: m Ͻ 25.
e. 90% conﬁdence interval for m is (26 to 50),
Ha: m Ͼ 25.
f. 90% conﬁdence interval for m is (26 to 50),
Ha: m Ͻ 25.
13.45 ● Refer to the rules for the relationship between conﬁdence intervals and two-sided alternatives, given in
the two bullets on page 574.
a. Rewrite the rules speciﬁcally for a ϭ .05.
b. Rewrite the rules speciﬁcally for a ϭ .01.
13.46 ● Refer to the rules for the relationship between
conﬁdence intervals and one-sided tests given in
the three bullets and sentence preceding them on
page 575.
a. Rewrite the rules speciﬁcally for a ϭ .05.
b. Rewrite the rules speciﬁcally for a ϭ .01.
13.47 Each of the following presents a two-sided 95% conﬁdence interval and the alternative hypothesis of a
corresponding hypothesis test. In each case, state a
conclusion for the test, including the level of signiﬁcance you are using.
a. C.I. for m is (101 to 105), Ha: m 100.
b. C.I. for p is (.12 to .28), Ha: p Ͻ .10.
c. C.I. for m1 Ϫ m2 is (3 to 15), Ha: m1 Ϫ m2 Ͼ 0.
d. C.I. for p1 Ϫ p2 is (Ϫ.15 to .07), Ha: p1 Ϫ p2 0.
13.48 As was stated in Section 13.5, “a conﬁdence interval
can be used as an alternative way to conduct a twosided signiﬁcance test.” If a test were conducted by
using this method, would the p-value for the test be
available? Explain.
13.49 For each of the following situations, can you conclude whether a 90% conﬁdence interval for m would
include the value 10? If so, make the conclusion. If
not, explain why you can’t tell.
a. H0: m ϭ 10, Ha: m Ͻ 10, do not reject the null hypothesis for a ϭ .05.
Bold-numbered exercises answered in the back
594
Chapter 13
b. H0: m ϭ 10, Ha: m Ͻ 10, reject the null hypothesis for
a ϭ .05.
c. H0: m ϭ 10, Ha: m 10, do not reject the null hypothesis for a ϭ .10.
d. H0: m ϭ 10, Ha: m 10, reject the null hypothesis for
a ϭ .10.
Section 13.6
13.50 ● Refer to the ﬁve parameters given in Table 13.3
(p. 578). For each of the following situations, identify
which of the ﬁve is or are appropriate. Give the parameter(s) in symbols, and deﬁne what the symbols
represent in the context of the situation.
a. Researchers want to compare the mean running
times for the 50-yard dash for ﬁrst-grade boys and
girls. They select a random sample of 35 boys and
32 girls in ﬁrst grade and time them running the
50-yard dash.
b. Researchers want to know what proportion of a
certain type of tree growing in a national forest suffers from a disease. They test a representative
sample of 200 of the trees from around the forest
and ﬁnd that 15 of them have the disease.
c. Researchers want to know what percentage of
adults have a fear of going to the dentist. They also
want to know the average number of visits made
to a dentist in the past 10 years for adults who have
that fear. They ask a random sample of adults
whether or not they fear going to the dentist and
also how many times they have gone in the past
10 years.
d. Refer to part (c). Researchers also want to compare
the average number of visits made by adults who
fear going to the dentist with the average number
of visits for those who don’t have the fear.
13.51 ● Refer to Exercise 13.50. In each case, explain
whether a conﬁdence interval, a hypothesis test, or
both would be more appropriate.
13.52 Give an example of a situation for which the appropriate inference procedure would be each of the
following:
a. A hypothesis test for one proportion.
b. A hypothesis test and conﬁdence interval for a
paired difference mean.
c. A conﬁdence interval for one mean.
d. A conﬁdence interval for the difference in two
means for independent samples.
13.53 Refer to each of the following scenarios from exercises in various chapters of this book. In each case,
determine the most appropriate inference procedure(s), including the appropriate parameter. If you
think inference about more than one parameter may
be of interest, answer the question for all parameters
of interest. Explain your choices.
a. A sample of college students was asked whether
they would return the money if they found a wallet
on the street. Of the 93 women, 84 said “yes,” and
of the 75 men, 53 said “yes.” Assume these students represent all college students.
● Basic skills
◆ Dataset available but not required
b. A study was conducted on pregnant women and
subsequent development of their children (Olds
et al., 1994). One of the questions of interest was
whether the IQ of children would differ for mothers who smoked at least 10 cigarettes a day during
pregnancy and those who did not smoke at all.
c. Max likes to keep track of birthdays of people he
meets. He has 170 birthdays on his birthday calendar. One cold January night, he comes up with
the theory that people are more likely to be born
in October than they would be if all 365 days
were equally likely. He consults his birthday calendar and ﬁnds that 22 of the 170 birthdays are in
October.
d. The dataset cholest reports cholesterol levels of
heart attack patients 2, 4, and 14 days after the
heart attack. Data for days 2 and 4 were available
for 28 patients. The mean difference (2 day Ϫ
4 day) was 23.29, and the standard deviation of the
differences was 38.28.
Section 13.7
13.54 ● Compute the effect size for each of the following
situations, and state whether it would be considered
closer to a small, medium, or large effect:
a. In a one-sample test with n ϭ 100, the test statistic
is t ϭ 2.24.
b. In a one-sample test with n ϭ 50, the test statistic
is t ϭ Ϫ2.83.
c. In a paired-difference test with n ϭ 30 pairs, the
test statistic is t ϭ 1.48.
d. In a test for the difference in two means with independent samples, with n1 ϭ 40 and n2 ϭ 50, the test
statistic is t ϭ Ϫ2.33.
13.55 ● Refer to Table 13.4 (p. 585), which presents power
for a one-sided, one-sample t-test. In a test of H0:
md ϭ 0 versus Ha: md Ͼ 0, suppose the truth is that the
population of differences is a normal distribution
with mean md ϭ 2 and standard deviation s ϭ 4.
a. Recalling the Empirical Rule from Chapter 2, draw
a picture of this distribution, showing the ranges
into which 68% and 95% of the differences fall.
b. On your picture, indicate where the null value of
0 falls.
c. If a sample of size 20 is taken, what is the power for
the test, assuming that a .05 level of signiﬁcance
will be used?
d. Explain in words what probability the power of the
test represents.
13.56 ● In Table 13.4 (p. 585), it is shown that the power is
.40 for a one-sided, one-sample t-test with .05 level of
signiﬁcance, n ϭ 50, and true effect size of 0.2. Would
the power be higher or lower for each of the following
changes?
a. The true effect size is 0.4.
b. The sample size used is n ϭ 75.
c. The level of signiﬁcance used is .01.
13.57 (Computer software required.) Find the power for the
following one-sample t-test situations. In each case,
assume a .05 level of signiﬁcance will be used.
Bold-numbered exercises answered in the back
Testing Hypotheses About Means
13.58
13.59
13.60
13.61
a. Effect size ϭ 0.3, sample size ϭ 45, Ha: m Ͼ m0.
10, true mean ϭ 13,
b. Sample size ϭ 30, Ha: m
s ϭ 4.
c. Effect size ϭ Ϫ1.0, sample size ϭ 15, Ha: m Ͻ m0.
(Computer software required.) In parts (a) to (d),
ﬁnd the sample size necessary to achieve power of .80
for a one-sample t-test with Ha: m Ͼ m0 and level of
signiﬁcance of .05 for each of the following effect
sizes:
a. 0.2
b. 0.4
c. 0.6
d. 0.8
e. Make a scatterplot of the sample size (vertical axis)
versus the effect size for the effect sizes in parts (a)
to (d). Does the relationship between the effect
size and the sample size required to achieve 80%
power appear to be linear? If not, what is the nature of the relationship?
For a z-test for one proportion, a possible effect size
measure is 1p1 Ϫ p0 2 > 1p0 11 Ϫ p0 2 where p0 is the
null value and p1 is the true population proportion.
a. What is the relationship between this effect size
and the z-test statistic for this situation?
b. Does the effect size ﬁt the relationship “Test statistic ϭ Size of effect ϫ Size of study”? If not, explain
why not. If so, show how it ﬁts.
c. What would be a reasonable way to estimate this
effect size?
Refer to the effect-size measure in Exercise 13.59. For
parts (a) to (c), compute the effect size.
a. p1 ϭ .35, p0 ϭ .25.
b. p1 ϭ .15, p0 ϭ .05.
c. p1 ϭ .95, p0 ϭ .85.
d. On the basis of the results in parts (a) to (c), does
this effect size stay the same when p1 Ϫ p0 stays the
same? Explain.
e. In statistical software for computing power for
a test for a single proportion, unlike for a single
mean, both p1 and p0 must be speciﬁed, rather than
just the difference between them. Explain why it is
not enough to specify the difference, using the results of the previous parts of this exercise.
Explain why it is more useful to compare effect sizes
than p-values in trying to determine whether many
studies about the same topic have found similar
results.
Section 13.8
13.62 ● Refer to the list of items in Section 13.8. Explain
which ones should be of concern if the sample size(s)
for a test are large.
13.63 ● Refer to the list of items in Section 13.8. Explain
which ones should be of concern if the sample size(s)
for a test are small.
13.64 Refer to item 6 in the list of concerns in Section 13.8.
Is that statement the same thing as saying that the
null hypothesis is likely to be true in about 1 out of
20 tests that have achieved statistical signiﬁcance?
Explain.
● Basic skills
◆ Dataset available but not required
595
13.65 Refer to the list of items in Section 13.8.
a. For which of the concerns would the p-value for a
test be useful to have? Explain why in each case.
b. For which of the concerns would a conﬁdence interval estimate for the parameter be useful to have?
Explain why in each case.
c. For which of the concerns would the sample size(s)
be useful to know? Explain why in each case.
Chapter Exercises
13.66 For each of the following research questions, specify
the parameter and the value that constitute the null
hypothesis of “parameter ϭ null value.” In other
words, deﬁne the population parameter of interest
and specify the null value that is being tested.
a. Do a majority of Americans between the ages of 18
and 30 think that the use of marijuana should be
legalized?
b. Is the mean of the Math SAT scores in California in
a given year different from the target mean of 500
set by the test developers?
c. Is the mean age of death for left-handed people
lower than that for right-handed people?
d. Is there a difference in the proportions of male and
female college students who smoke cigarettes?
13.67 Refer to Exercise 13.66. In each case, specify whether
the alternative hypothesis would be one-sided or
two-sided.
13.68 Suppose a one-sample t-test of H0: m ϭ 0 versus
0 results in a test statistic of t ϭ 0.65 with
Ha: m
df ϭ 14. Suppose a new study is done with n ϭ 150,
and the sample mean and standard deviation turn
out to be exactly the same as in the ﬁrst study.
a. What conclusion would you reach in the original
study, using a ϭ .05?
b. What conclusion would you reach in the new
study, using a ϭ .05?
c. Compare your results in parts (a) and (b) and
comment.
13.69 A study is conducted to ﬁnd out whether results of an
IQ test are signiﬁcantly higher after listening to
Mozart than after sitting in silence. Explain what has
happened in each of the following scenarios:
a. A type 1 error was committed.
b. A type 2 error was committed.
c. The power of the test was too low to detect the difference that actually exists.
d. The power of the test was so high that a very
small difference resulted in a statistically signiﬁcant ﬁnding.
13.70 In a random sample of 170 married British couples,
the difference between the husband’s and wife’s ages
had a mean of 2.24 years and a standard deviation of
4.1 years.
a. Test the hypothesis that British men are signiﬁcantly older than their wives, on average.
b. Explain what is meant by the use of the term
signiﬁcant in part (a), and discuss how it compares
with the everyday use of the word.
Bold-numbered exercises answered in the back
596
Chapter 13
13.71 Suppose a highway safety researcher makes modiﬁcations to the design of a highway sign. The researcher believes that the modiﬁcations will make
the mean maximum distance at which drivers are
able to read the sign greater than 450 feet. The maximum distances (in feet) at which n ϭ 16 drivers can
read the sign are as follows:
440
490
600
540
540
600
240
440
360
600
490
400
490
540
440
490
Use this sample to determine whether the mean
maximum sign-reading distance at which drivers can
read the sign is greater than 450. Show all ﬁve steps
of a hypothesis test, and be sure to state a conclusion. If conditions are not met, make appropriate adjustments. Also, describe any assumptions that you
make.
13.72 In Exercise 11.70, a study is described in which the
mean IQs at age 4 for children of smokers and nonsmokers were compared. The mean for the children
of the 66 nonsmokers was 113.28 points, while for
the children of the 47 smokers it was 103.12 points.
Assume that the pooled standard deviation for the
two samples is 13.5 points and that a pooled t-test is
appropriate. Test the hypothesis that the population
mean IQ at age 4 is the same for children of smokers
and of nonsmokers versus the alternative that it is
higher for children of nonsmokers.
13.73 Refer to Exercise 13.72. One of the statements made
in the research article and reported in Exercise 11.70
was “After control for confounding background variables . . . the average difference observed at 36 and
48 months was reduced to 4.35 points (95% CI: 0.02,
8.68).” Use this statement to test the null hypothesis that the difference in the mean IQ scores for the
two populations is 0 versus the alternative that it is
greater than 0. (Notice that the results have now been
adjusted for confounding variables such as parents’
IQ, whereas the data given in Exercise 13.72 had not.)
13.74 It is believed that regular physical exercise leads to a
lower resting pulse rate. Following are data for n ϭ 20
individuals on resting pulse rate and whether the individual regularly exercises or not. Assuming that this
is a random sample from a larger population, use this
sample to determine whether the mean pulse is lower
for those who exercise. Clearly show all ﬁve steps of
the hypothesis test.
Person
Pulse
Regularly
Exercises
Person
Pulse
Regularly
Exercises
1
2
3
4
5
6
7
8
9
10
72
62
72
84
60
63
66
72
75
64
No
Yes
Yes
No
Yes
Yes
No
No
Yes
Yes
11
12
13
14
15
16
17
18
19
20
62
84
76
60
52
60
64
80
68
64
No
No
No
Yes
Yes
No
Yes
Yes
Yes
Yes
● Basic skills
◆ Dataset available but not required
13.75 In an experiment conducted by one of the authors,
ten students in a graduate-level statistics course were
given this question about the population of Canada: “The population of the U.S. is about 270 million.
To the nearest million, what do you think is the population of Canada?” (The population of Canada at the
time was slightly over 30 million.) The responses were
as follows:
20, 90, 1.5, 100, 132, 150, 130, 40, 200, 20
Eleven other students in the same class were given
the same question with different introductory information: “The population of Australia is about
18 million. To the nearest million, what do you think
is the population of Canada?” The responses were
as follows:
12, 20, 10, 81, 15, 20, 30, 20, 9, 10, 20
The experiment was done to demonstrate the anchoring effect, which is that responses to a survey
question may be “anchored” to information provided to introduce the question. In this experiment,
the research hypothesis was that the individuals who
saw the U.S. population ﬁgure would generally give
higher estimates of Canada’s population than would
the individuals who saw the Australia population
ﬁgure.
a. Write null and alternative hypotheses for this experiment. Use proper notation.
b. Test the hypotheses stated in part (a). Be sure to
state a conclusion in the context of the experiment.
c. As a step in part (b), you should have created a
graphical summary to verify necessary conditions.
Do you think that any possible violations of the
necessary conditions have affected the results of
part (b) in a way that produced a misleading conclusion? Explain why or why not.
13.76 ◆ The dataset cholest on the CD for this book reports
cholesterol levels of heart attack patients 2, 4, and
14 days after the heart attack. Data for days 2 and
4 were available for 28 patients. The mean difference
(2 day Ϫ 4 day) was 23.29, and the standard deviation
of the differences was 38.28. A histogram of the differences was approximately bell-shaped. Is there sufﬁcient evidence to indicate that the mean cholesterol
level of heart attack patients decreases from the second to fourth days after the heart attack?
13.77 Refer to Exercise 13.76. Suppose physicians will use
the answer to that question to decide whether to
retest patients’ cholesterol levels on day 4. If there is
no conclusive evidence that the cholesterol level goes
down, they will use the day 2 level to decide whether
to prescribe drugs for high cholesterol. If there is evidence that cholesterol goes down between days 2 and
4, patients will all be retested on day 4, and the prescription drug decision will be made then.
a. What are the consequences of type 1 and type 2 errors for this setting?
b. Which type of error do you think is more serious?
Bold-numbered exercises answered in the back
Testing Hypotheses About Means
Dataset Exercises
Datasets are required to solve these
exercises and can be found at http://1pass.thomson.com
or on your CD.
13.78 Refer to Exercises 13.76 and 13.77. Using the dataset
cholest, determine whether there is sufﬁcient evidence to conclude that the cholesterol level drops, on
average, from day 2 to day 14 after a heart attack.
13.79 The dataset UCDavis2 includes information on Sex,
Height, and Dadheight. Use the data to test the hypothesis that college men are taller, on average, than
their fathers. Assume that the male students in the
survey represent a random sample of college men.
13.80 The dataset UCDavis2 includes grade point average
GPA and answers to the question: “Where do you typically sit in a classroom (circle one): Front, Middle,
Back.” The answer to this question is coded as F, M, B
for the variable Seat. Assuming these students represent all college students, test whether there is a difference in mean GPA for students who sit in the front
versus the back of the classroom.
● Basic skills
◆ Dataset available but not required
597
13.81 The dataset deprived includes information on selfreported amount of sleep per night and whether a
person feels sleep-deprived for n ϭ 86 college students (Source: Laura Simon, Pennsylvania State University). Assume the students represent a random
sample of college students. Is the mean number of
hours of sleep per night lower for the population
of students who feel sleep-deprived than it is for
the population of students who do not? Use an unpooled test.
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vMentor at http://1pass.thomson.com for one-on-one tutoring from
a statistics expert.
Bold-numbered exercises answered in the back
Royalty-free/Corbis
14
How is her handspan related to her height?
S e e F i g u r e 1 4 . 1 (p. 600)
598
Inference About
Simple Regression
In Chapter 5, we used regression to describe a relationship in a sample. Now we make
inferences about the population represented by the sample. What is the relationship between handspan and height in the population? What is the mean handspan for people
who are 65 inches tall? What interval covers the handspans of most individuals of that
height?
W
e learned in Chapter 5 that a straight line often describes the
pattern of a relationship between two quantitative variables.
For instance, in Example 5.1, we explored the relationship
between the handspans (cm) and heights (in.) of 167 college students
and found that the pattern of the relationship in this sample could be
described by the equation
Average handspan ϭ Ϫ3 ϩ 0.35 1Height2
An equation like the one relating handspan to height is called a regression
equation, and the term simple regression is sometimes used to describe
the analysis of a straight-line relationship (linear relationship) between a
response variable ( y variable) and an explanatory variable (x variable).
In Chapter 5, we used regression methods only to describe a sample
and did not make statistical inferences about the larger population. Now
we consider how to make inferences about a relationship in the population represented by the sample. Some questions involving the population
that we might ask when analyzing a relationship are the following:
Throughout the chapter, this icon
introduces a list of resources on the
StatisticsNow website at http://
1pass.thomson.com that will:
• Help you evaluate your knowledge
of the material
• Allow you to take an examprep quiz
• Provide a Personalized Learning
Plan targeting resources that
address areas you should study
1. Does the observed relationship also occur in the population? For
example, is the observed relationship between handspan and height
strong enough to conclude that a similar relationship also holds in
the population?
2. For a linear relationship, what is the slope of the regression line in
the population? For example, in the larger population, what is the
slope of the regression line that connects handspans to heights?
3. What is the mean value of the response variable ( y) for individuals
with a speciﬁc value of the explanatory variable (x)? For example,
what is the mean handspan in a population of people who are
65 inches tall?
599