Provide an appropriate response.
43)
Suppose that you want to perform a hypothesis test based on independent simple random
samples to compare the means of two populations. Several procedures are available,
namely, the pooled t–test, the nonpooled t–test, and the Mann–Whitney test. Give
circumstances for which the nonpooled t–test would be the most appropriate procedure.
Be sure to comment on the sizes of the samples, the distributions of the variable under
consideration on the populations, and the relative standard deviations of the distributions.
43)
44)
A statistician performs a paired Wilcoxon signed–rank test to compare the means of two
populations using a paired sample. Suppose that the sum of the positive ranks is much
larger than the sum of the negative ranks. Explain in your own words why this suggests
that µ1>µ2.
44)
Preliminary data analyses indicates that use of a paired t–test is reasonable. Perform the hypothesis test by using either
the critical–value approach or the P–value approach as indicated. Assume that the null hypothesis is H0 : µ1=µ2.
45)
A test of abstract reasoning is given to a random sample of students before and after
completing a formal logic course. The results are shown below.
Before 74 83 75 88 84 63 93 84 91 77
After 73 77 70 77 74 67 95 83 84 75
At the 5% significance level, do the data provide sufficient evidence to conclude that that
the mean score after the course differs from the mean score before the course? Use the
critical–value approach.
45)
Preliminary data analyses indicate that you can reasonably use nonpooled t–procedures on the given data. Apply a
nonpooled t–test to perform the required hypothesis test, using either the critical–value approach or the P–value approach
as indicated.
46)
A paint manufacturer wishes to compare the drying times of two different types of paint.
Independent random samples of 11 cans of type A and 9 cans of type B were selected and
applied to similar surfaces. The drying times (in minutes) were recorded. The summary
statistics are as follows.
Type A Type B
x1=76.4 x2=66.3
s1= 4.5 s2= 5.1
n1= 11 n2= 9
Do the data provide sufficient evidence to conclude that the mean drying time for type A
differs from the mean drying time for type B? Perform a t–test at the 1% significance level.
Use the P–value approach.
46)
Preliminary data analyses indicate that you can reasonably consider the assumptions for using the Mann–Whitney test
satisfied. Assume that the null hypothesis is H0: µ1=µ2. Use the Mann–Whitney test to carry out the required
hypothesis test.
47)
A researcher was interested in comparing the weight losses over a two–month period for
people on two different diets. Independent simple random samples of 7 people on Diet A
and 8 people on Diet B yielded the following weight losses (in pounds).
Diet A Diet B
7 3
2 8
14 0
5 1
13 11
21 3
16 6
4
At the 10% significance level, do the data provide sufficient evidence to conclude that the
mean weight loss for people on diet A differs from the mean weight loss for people on diet
B?
47)
Provide an appropriate response.
48)
A researcher wants to compare the mean systolic blood pressures for adults ages 20–29 and
adults ages 30–39. Identify the variable under consideration and the two populations.
Suppose the researcher wants to perform a hypothesis test to decide whether the mean
systolic blood pressure for adults aged 20–29 is lower than the mean systolic blood
pressure for adults aged 30–39. State the null and alternative hypotheses for the hypothesis
test.
48)
Preliminary data analyses indicates that use of a paired Wilcoxon signed–rank test is reasonable. Perform the hypothesis
test by using a paired Wilcoxon signed–rank test. Assume that the null hypothesis is H0 : µ1=µ2.
49)
Students are given scores on both the mathematics section and the verbal section of a
standardized test. The data for a random sample of students is given below.
Mathematics 347 440 327 456 427 349 377 398 425
Verbal 285 378 243 371 340 271 294 322 385
At the 5% significance level, do the data provide sufficient evidence to conclude that there
is a difference between the mean math score and the mean verbal score?
49)
Preliminary data analyses indicate that you can reasonably consider the assumptions for using pooled t–procedures
satisfied. Perform the required hypothesis test by using either the critical–value approach or the P–value approach as
indicated.
50)
A researcher was interested in comparing the resting pulse rate of people who exercise
regularly and people who do not exercise regularly. Independent simple random samples
of 16 people ages 30–40 who do not exercise regularly and 12 people ages 30–40 who do
exercise regularly were selected and the resting pulse rate of each person was measured.
The summary statistics are as follows.
Do Not Exercise Do Exercise
x1=72.0 x2=68.3
s1=10.7 s2=8.3
n1= 16 n2= 12
At the 2.5% significance level, do the data provide sufficient evidence to conclude that the
mean resting pulse rate of people who do not exercise regularly is greater than the mean
resting pulse rate of people who exercise regularly? Use the critical–value approach.
50)
24
Perform the required hypothesis test by using a Mann–Whitney test. Use a normal approximation for M. Find the mean
and standard deviation of M, the value of M and of the test statistic, the critical value(s), and state your conclusion.
51)
Independent simple random samples of 11 female employees and 11 male employees are
selected from one company and the weekly salaries of the employees are recorded. The
salaries (in dollars) are shown below. Use a significance level of 0.10 to test whether the
mean salary for female employees differs from the mean salary for male employees.
Female Male
350 420 470
385 675 520
540 400 550
450 640
410 460 650
545 720 810
660 500 880
700 750
51)
Preliminary data analyses indicate that you can reasonably use nonpooled t–procedures on the given data. Apply a
nonpooled t–test to perform the required hypothesis test, using either the critical–value approach or the P–value approach
as indicated.
52)
A researcher was interested in comparing the amount of time spent watching television by
women and by men. Independent random samples of 14 women and 17 men were
selected and each person was asked how many hours he or she had watched television
during the previous week. The summary statistics are as follows:
Sample 1 (women) Sample 2 (men)
x1=12.5 x2=13.9
s1= 3.9 s2= 5.2
n1= 14 n2= 17
At the 5% significance level, do the data provide sufficient evidence to conclude that the
mean amount of time spent watching television by women is less than the mean amount
of time spent watching television by men? Use the P–value approach.
52)
25
Provide an appropriate response.
53)
Suppose that you want to perform a hypothesis test to compare the means of two
populations using paired samples. How would you decide between the paired t–test and
the paired Wilcoxon signed–rank test?
53)
Perform the required hypothesis test by using a Mann–Whitney test. Use a normal approximation for M. Find the mean
and standard deviation of M, the value of M and of the test statistic, the critical value(s), and state your conclusion.
54)
A teacher uses two different CAI programs to remediate students. Independent simple
random samples of 11 students from program I and 14 students from program II are
selected. Results for each group on a standardized test are listed in the table below. At the
0.05 level of significance, do the data provide sufficient evidence to conclude that the mean
score for students going through program I differs from the mean score for students going
through program II?
Program I Program II
60 75 61 63
86 69 64 70
72 82 59
66 89 68 77
84 80 81 87
78 73 91 93
94 95
54)
Preliminary data analyses indicate that you can reasonably consider the assumptions for using pooled t–procedures
satisfied. Perform the required hypothesis test by using either the critical–value approach or the P–value approach as
indicated.
55)
A researcher was interested in comparing the amount of time spent watching television by
women and by men. Independent simple random samples of 14 women and 17 men were
selected, and each person was asked how many hours he or she had watched television
during the previous week. The summary statistics are as follows.
Women Men
x1=11.1 x2=16.9
s1=4.5 s2=4.7
n1= 14 n2= 17
At the 5% significance level, do the data provide sufficient evidence to conclude that the
mean time spent watching television by women is less than the mean time spent watching
television by men? Use the critical–value approach.
55)
Provide an appropriate response.
56)
Suppose that you want to perform a hypothesis test based on a simple random paired
sample to compare the means of two populations. Two procedures are available, namely,
the paired t–test and the paired Wilcoxon signed–rank test. Give circumstances for which
the Wilcoxon signed–rank test would be the more appropriate procedure. Be sure to
comment on the size of the sample and the distribution of the paired–difference variable.
56)
Preliminary data analyses indicates that use of a paired Wilcoxon signed–rank test is reasonable. Perform the hypothesis
test by using a paired Wilcoxon signed–rank test. Assume that the null hypothesis is H0 : µ1=µ2.
57)
Students take a math test before and after tutoring. The data for a random sample of
students is given below.
Before 54 72 59 82 46 80 63 36 44
After 68 70 67 78 55 90 63 52 40
At the 2.5% level of significance, do the data provide sufficient evidence to conclude that
the mean score before the tutoring is lower than the mean score after the tutoring?
57)
Provide an appropriate response.
58)
A researcher wishes to compare the incomes of men and women. She selects a random
sample of 80 married women. She records the weekly income of each of these women.
Using the husbands of these women as the sample of men, she also records the weekly
income of each man. Incomes are known to be normally distributed for both populations
(women and men). Furthermore, the two population standard deviations are known to be
equal. Is it reasonable to use the Mann–Whitney test on these data? Explain your answer.
58)
Preliminary data analyses indicate that you can reasonably use nonpooled t–procedures on the given data. Apply a
nonpooled t–test to perform the required hypothesis test, using either the critical–value approach or the P–value approach
as indicated.
59)
A paint manufacturer wished to compare the drying times of two different types of paint.
Independent simple random samples of 11 cans of type A and 9 cans of type B were
selected and applied to similar surfaces. The drying times, in hours, were recorded. The
summary statistics are as follows.
Type A Type B
x1=74.1 x2=63.9
s1= 4.5 s2= 5.1
n1=11 n2= 9
At the 1% significance level, do the data provide sufficient evidence to conclude that the
mean drying time for type A differs from the mean drying time for type B? Use the
critical–value approach.
59)
Preliminary data analyses indicate that you can reasonably consider the assumptions for using pooled t–procedures
satisfied. Perform the required hypothesis test by using either the critical–value approach or the P–value approach as
indicated.
60)
A researcher wishes to determine whether the systolic blood pressure of people who follow
a vegetarian diet is, on average, lower than the systolic blood pressure of those who follow
a nonvegetarian diet.
Independent simple random samples of 85 vegetarians and 75 nonvegetarians yielded the
following sample statistics:
Vegetarians Nonvegetarians
n1= 85 n2= 75
x1= 124.1 x2= 138.7
s1= 38.7 s2= 39.2
Use the sample data to test the claim that the mean systolic blood pressure for vegetarians
is lower than the mean systolic blood pressure for nonvegetarians. Test the claim using a
significance level of 0.01. Use the P–value approach.
60)
Provide an appropriate response.
61)
Suppose a researcher wants to perform a hypothesis test to compare the mean systolic
blood pressure for men with the mean systolic blood pressure for women, based on
independent samples. She obtains a random sample of 30 married couples and determines
the mean systolic blood pressure for the men, x1, and the mean systolic blood pressure for
the women, x2. She formulates the hypotheses as follows
H0: µ1=µ2
Ha: µ1µ2
and will reject the null hypothesis if x1–x2, the observed difference between the sample
means is too large. She will use a two–means z–test. Identify the flaw in her methodology.
61)
Preliminary data analyses indicate that you can reasonably use nonpooled t–procedures on the given data. Apply a
nonpooled t–test to perform the required hypothesis test, using either the critical–value approach or the P–value approach
as indicated.
62)
A researcher was interested in comparing the resting pulse rates of people who exercise
regularly and people who do not exercise regularly. Independent simple random samples
of 16 people ages 30–40 who do not exercise regularly and 12 people ages 30–40 who
exercise regularly were selected, and the resting pulse rate (in beats per minute) of each
person was measured. The summary statistics are as follows.
Do Not Exercise Do Exercise
x1=73.9 x2=69.1
s1= 10.9 s2= 8.2
n1= 16 n2= 12
At the 2.5% significance level, do the data provide sufficient evidence to conclude that the
mean resting pulse rate of people who do not exercise regularly is greater than the mean
resting pulse rate of people who exercise regularly? Use the critical–value approach.
62)
Preliminary data analyses indicate that you can reasonably consider the assumptions for using pooled t–procedures
satisfied. Perform the required hypothesis test by using either the critical–value approach or the P–value approach as
indicated.
63)
A researcher was interested in comparing the salaries of female and male employees at a
particular company. Independent simple random samples of 8 female employees and 15
male employees yielded the following weekly salaries (in dollars).
Female Male
495 722 518
760 562 904
556 880 1150
904 520 805
520 500 480
1005 1250 970
743 750 605
660 1640
At the 5% significance level, do the data provide sufficient evidence to conclude that at this
particular company the mean salary of female employees is less than the mean salary of
male employees? Use the critical–value approach. (Note: x1= 705.375, x2= 817.067, s1=
183.855, s2= 330.146.)
63)
Preliminary data analyses indicate that you can reasonably consider the assumptions for using the Mann–Whitney test
satisfied. Assume that the null hypothesis is H0: µ1=µ2. Use the Mann–Whitney test to carry out the required
hypothesis test.
64)
A researcher was interested in comparing the GPAs of students at two different colleges.
Independent simple random samples of 8 students from college A and 9 students from
college B yielded the following GPAs.
College A College B
2.8 3.6
3.2 3.9
3.7 3.3
2.5 4.0
2.6 3.8
3.0 2.9
4.0 3.6
2.7 4.0
3.5
At the 10% significance level, do the data provide sufficient evidence to conclude that the
mean GPA of students at college A differs from the mean GPA of students at college B?
64)
Provide an appropriate response.
65)
The pooled t–test requires the assumption that the two population standard deviations be
equal. What methods are available for checking this assumption? Which of these methods
is not recommended and why is it not recommended?
65)
Preliminary data analyses indicate that you can reasonably use nonpooled t–procedures on the given data. Apply a
nonpooled t–test to perform the required hypothesis test, using either the critical–value approach or the P–value approach
as indicated.
66)
A researcher was interested in comparing the GPAs of students at two different colleges.
Independent random samples of 8 students from college A and 13 students from college B
yielded the following GPAs:
College A College B
3.7 3.8 2.8
3.2 3.2 4.0
3.0 3.0 3.6
2.5 3.9 2.6
2.7 3.8 4.0
3.6 2.5 3.6
2.8 3.9
3.4
At the 10% significance level, do the data provide sufficient evidence to conclude that the
mean GPA of students at college A differs from the mean GPA of students at college B? Use
the P–value approach.
(Note: x1= 3.1125, x2= 3.4385, s1= 0.4357, s2= 0.5485.)
66)
Preliminary data analyses indicate that you can reasonably consider the assumptions for using pooled t–procedures
satisfied. Perform the required hypothesis test by using either the critical–value approach or the P–value approach as
indicated.
67)
A researcher was interested in comparing the response times of two different cab
companies. Companies A and B were each called at 50 randomly selected times. The calls
to company A were made independently of the calls to company B. The response times for
each call were recorded. The summary statistics were as follows:
Company A Company B
Mean response time 7.6 minutes 6.9 minutes
Standard deviation 1.4 minutes 1.7 minutes
At the 0.02 level of significance, do the data provide sufficient evidence to conclude that
the mean response time for company A differs from the mean response time for company
B? Use the P–value approach.
67)
Preliminary data analyses indicate that you can reasonably consider the assumptions for using the Mann–Whitney test
satisfied. Assume that the null hypothesis is H0: µ1=µ2. Use the Mann–Whitney test to carry out the required
hypothesis test.
68)
A researcher was interested in comparing the salaries of female and male employees at a
particular company. Independent simple random samples of 8 female employees and 9
male employees yielded the following weekly salaries (in dollars).
Female Male
495 480
660 605
914 1250
743 722
520 1150
760 518
1005 1540
556 750
530
At the 5% significance level, do the data provide sufficient evidence to conclude that at this
company the mean salary of female employees is lower than the mean salary of male
employees?
68)
Preliminary data analyses indicate that you can reasonably use nonpooled t–procedures on the given data. Apply a
nonpooled t–test to perform the required hypothesis test, using either the critical–value approach or the P–value approach
as indicated.
69)
A researcher was interested in comparing the salaries of female and male employees at a
particular company. Independent simple random samples of 8 female employees and 15
male employees yielded the following weekly salaries (in dollars).
Female Male
495 722 518
760 562 904
556 880 1150
904 520 805
520 500 480
1005 1250 970
743 750 605
660 1640
At the 5% significance level, do the data provide sufficient evidence to conclude that at this
company the mean salary of female employees is less than the mean salary of male
employees? Use the critical–value approach.
(Note: x1= 705.375, x2= 817.067, s1= 183.855, s2= 330.146.)
69)
Perform the required hypothesis test by using a Mann–Whitney test. Use a normal approximation for M. Find the mean
and standard deviation of M, the value of M and of the test statistic, the critical value(s), and state your conclusion.
70)
A person who commutes to work is choosing between two different routes. He tries the
first route 11 times and the second route 12 times and records the time of each trip. The
results (in minutes) are shown below. Use a significance level of 0.01 to test whether the
mean time for route 1 differs from the mean time for route 2.
Route 1 Route 2
35 42 41
33 48 40
39 50 46
36 40
41 46 38
49 53 36
51 57 53
45 50 55
70)
Preliminary data analyses indicate that you can reasonably use nonpooled t–procedures on the given data. Apply a
nonpooled t–test to perform the required hypothesis test, using either the critical–value approach or the P–value approach
as indicated.
71)
A researcher was interested in comparing the response times of two different cab
companies. Companies A and B were each called at 50 randomly selected times. The calls
to company A were made independently of the calls to company B. The response times for
each call were recorded. The summary statistics were as follows:
Company A Company B
Mean response time 7.6 minutes 6.9 minutes
Standard deviation 1.4 minutes 1.7 minutes
At the 0.02 level of significance, do the data provide sufficient evidence to conclude that
the mean response time for company A differs from the mean response time for company
B? Use the P–value approach.
71)
Perform the required hypothesis test by using a Mann–Whitney test. Use a normal approximation for M. Find the mean
and standard deviation of M, the value of M and of the test statistic, the critical value(s), and state your conclusion.
72)
Test whether the mean grade for students from college A differs from the mean grade for
students from College B. Independent simple random samples of 11 students from College
A and 11 students from College B are selected. Grades of the students are shown in the
table. Use a 0.05 level of significance.
College A 3.2 4.0 2.4 2.6 2.0 1.8 1.3 0.0 0.5 1.4 2.9
College B 2.4 1.9 0.3 0.8 2.8 3.0 3.1 3.1 3.1 3.5 3.5
72)
Preliminary data analyses indicates that use of a paired Wilcoxon signed–rank test is reasonable. Perform the hypothesis
test by using a paired Wilcoxon signed–rank test. Assume that the null hypothesis is H0 : µ1=µ2.
73)
Eleven runners are timed in the 100–meter dash and are timed again one month later after
following a new training program. The times (in seconds) are shown in the following table.
Before 12.1 12.4 11.7 11.5 11.0 11.8 12.3 10.8 12.6 12.7 10.7
After 11.9 12.4 11.8 11.4 11.2 11.5 12.0 10.9 12.0 12.2 11.1
At the 10% significance level, do the data provide sufficient evidence to conclude that the
training is effective in reducing times in the 100–meter dash?
73)
Provide an appropriate response.
74)
Suppose that you want to perform a hypothesis test to compare the means of two
populations using independent simple random samples. The two distributions of the
variable under consideration have the same shape. How would you decide between the
pooled t–test and the Mann–Whitney test?
74)
75)
Suppose that you want to perform a hypothesis test to compare the means of two
populations using a paired sample. Preliminary data analyses of the sample of paired
differences suggest that the distribution of the paired–difference variable is skewed to the
right . The sample of 20 paired differences contains an outlier. If it is not legitimate to
remove the outlier, is it reasonable to use a paired t–test? Is it reasonable to use a paired
Wilcoxon signed–rank test? Explain your thinking
75)
Preliminary data analyses indicates that use of a paired t–test is reasonable. Perform the hypothesis test by using either
the critical–value approach or the P–value approach as indicated. Assume that the null hypothesis is H0 : µ1=µ2.
76)
A coach uses a new technique in training middle distance runners. The times, in seconds,
for 8 different athletes to run 800 meters before and after this training are shown below.
Athlete A B C D E F G H
Before 119 111.5 108 108.1 108.7 108 117.2 116.4
After 119.6 110.2 105.6 108.9 106.9 108.1 113.6 112.5
At the 5% significance level, do the data provide sufficient evidence that the training helps
to improve times for the 800 meters? Use the critical–value approach.
76)