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8 Sums, Differences, and Combinations of Random Variables

# 8 Sums, Differences, and Combinations of Random Variables

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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.

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

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

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

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.

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

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

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

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

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

Dataset Exercises

Datasets are required to solve these

exercises and can be found at http://1pass.thomson.com

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.

Preparing for an exam? Assess your

progress by taking the post-test at http://1pass.thomson.com.

Do you need a live tutor for homework problems? Access

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

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:

of the material

• Allow you to take an examprep quiz

• Provide a Personalized Learning

Plan targeting resources that

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

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