106. Refer to Exhibit 10-16. The null hypothesis is to be tested at the 5% level of significance. The p-value
is
a.
less than .01
b.
between .01 and .025
c.
between .025 and .05
d.
between .05 and .10
107. Refer to Exhibit 10-16. The null hypothesis
a.
should be rejected
b.
should not be rejected
c.
was designed incorrectly
d.
None of these alternatives is correct.
PROBLEM
1. The following sample information is given concerning the ACT scores of high school seniors form two
local schools.
School A
School B
n1 = 14
n2 = 15
= 25
= 23
= 16
= 10
Develop a 95% confidence interval estimate for the difference between the two populations.
2. Maxforce, Inc. manufactures racquetball racquets by two different manufacturing processes (A and B).
Because the management of this company is interested in estimating the difference between the
average time it takes each process to produce a racquet, they select independent samples from each
process. The results of the samples are shown below.
Process A
Process B
Sample Size
32
35
Sample Mean (in minutes)
43
47
Population Variance (2)
64
70
a.
Develop a 95% confidence interval estimate for the difference between the average times of
the two processes.
b.
Is there conclusive evidence to prove that one process takes longer than the other? If yes,
which process? Explain.
-7.92 to -0.08
3. Consider the following results for two samples randomly taken from two normal populations with
equal variances.
Sample I
Sample II
Sample Size
28
35
Sample Mean
48
44
Population Standard Deviation
9
10
a.
Develop a 95% confidence interval for the difference between the two population means.
b.
Is there conclusive evidence that one population has a larger mean? Explain.
-0.70 to 8.70
No, because the range of the interval if from negative to positive.
4. The business manager of a local health clinic is interested in estimating the difference between the fees
for extended office visits in her center and the fees of a newly opened group practice. She gathered the
following information regarding the two offices.
Health Clinic
Group Practice
Sample Size
50 visits
45 visits
Sample Mean
$21
$19
Population Standard Deviation
$2.75
$3.00
Develop a 95% confidence interval estimate for the difference between the average fees of the two
offices.
5. Independent random samples taken on two university campuses revealed the following information
concerning the average amount of money spent on non-textbook purchases at the university’s
bookstore during the fall semester.
University A
University B
Sample Size
50
40
Average Purchase
$260
$250
Population Standard Deviation()
$20
$23
We want to determine if, on the average, students at University A spent more on non-textbook
purchases at the university’s bookstore than the students at University B.
Compute the test statistic.
Compute the p-value.
What is your conclusion? Let = .05.
students at University B
6. The management of Recover Fast Hospital (RFH) claims that the average length of stay in their
hospital after a major surgery is less than the average length of stay at General Hospital (GH). The
following data have been accumulated to test their claim.
RFH
GH
Sample size
45
58
Mean (in days)
4.6
4.9
Population Standard Deviation ()
0.5
0.6
a.
Formulate the hypotheses.
b.
Compute the test statistic.
c.
Using the p-value approach, test to see if the average length of stay in RFH is significantly
less than the average length of stay in GH. Let = 0.05.
a.
b.
-2.77
7. In order to estimate the difference between the average Miles per Gallon of two different models of
automobiles, samples are taken and the following information is collected.
Model A
Model B
Sample Size
60
55
Sample Mean
28
25
Sample Variance
16
9
a.
At 95% confidence develop an interval estimate for the difference between the average
Miles per Gallon for the two models.
b.
Is there conclusive evidence to indicate that one model gets a higher MPG than the other?
Why or why not? Explain.
1.714 to 4.286
8. Consider the following results for two samples randomly taken from two populations.
Sample A
Sample B
Sample Size
31
35
Sample Mean
106
102
Sample Standard Deviation
8
7
a.
Determine the degrees of freedom for the t distribution.
b.
Develop a 95% confidence interval for the difference between the two population means.
60
0.277 to 7.723
9. Consider the following results for two samples randomly taken from two populations.
Sample A
Sample B
Sample Size
25
38
Sample Mean
66
60
Sample Standard Deviation
5
7
a.
What are the degrees of freedom for the t distribution?
b.
At 95% confidence, compute the margin of error.
c.
Develop a 95% confidence interval for the difference between the two population means.
60
3.026
2.974 to 9.026
10. Consider the following results for two samples randomly taken from two populations.
Sample A
Sample B
Sample Size
20
25
Sample Mean
28
22
Sample Standard Deviation
5
6
a.
Determine the degrees of freedom for the t distribution.
b.
At 95% confidence, what is the margin of error?
c.
Develop a 95% confidence interval for the difference between the two population means.
42
3.31
2.69 to 9.31
11. Consider the following results for two samples randomly taken from two populations.
Sample A
Sample B
Sample Size
28
30
Sample Mean
24
22
Sample Standard Deviation
8
6
a.
Determine the degrees of freedom for the t distribution.
b.
Develop a 95% confidence interval for the difference between the two population means.
c.
Is there conclusive evidence that one population has a larger mean? Explain.
49
-1.753 to 5.753
No, the interval ranges from negative to positive.
12. The following are the test scores of two samples of students from University A and University B on a
national statistics examination. Develop a 95% confidence interval estimate for the difference between
the mean scores of the two populations.
University A
University B
Scores
Scores
86
83
n
64
87
6
8
13. In order to estimate the difference between the average mortgages in the southern states and the
northern states, the following information was gathered.
Southern
Northern
Sample Size
40
45
Sample Mean (in $1,000)
$170
$175
Sample Standard Deviation (in $1,000)
$5
$7
a.
Compute the degrees of freedom for the t distribution.
b.
Develop an interval estimate for the difference between the average of the mortgages in the
South and North. Let alpha = 0.05.
79
-7.605 to -2.395 (in thousands)
14. A credit company has gathered information regarding the average amount owed by people under 30
years old and by people over 30 years. Independent random samples were taken from both age groups.
You are given the following information.
Under 30
Over 30
600
550
n
200
300
2
361
400
Construct a 95% confidence interval for the difference between the average amounts owed by the two
age groups.
15. Two independent random samples of annual starting salaries for individuals with masters and
bachelors degrees in business were taken and the results are shown below.
Masters
Bachelors
Degree
Degree
Sample Size
33
30
Sample Mean (in $1,000)
58
54
Sample Standard Deviation (in $1,000)
2.4
2.0
a.
What are the degrees of freedom for the t distribution?
b.
Provide a 95% confidence interval estimate for the difference between the salaries of the two
groups.
60
2.89 to 5.11 (thousands)
16. Samples of employees of Companies A and B provided the following information regarding the ages
of employees.
Company A
Company B
Sample Size
32
36
Average Age
42
47
Variance
16
36
Develop a 97% confidence interval for the difference between the average ages of the employees of
the two companies.
17. Test scores on a standardized test from samples of students from two universities are given below.
UA
UB
Sample Size
28
41
Average Test Score
84
82
Variance
64
100
Provide a 98% confidence interval estimate for the difference between the test scores of the two
universities.
18. In order to estimate the difference between the average age of male and female employees at the
Young Corporation, the following information was gathered.
Male
Female
Sample Size
32
36
Sample Mean
25
23
Sample Standard Deviation
4
6
Develop a 95% confidence interval estimate for the difference between the average age of male and
female employees at the Young Corporation.
19. In order to estimate the difference between the average yearly salaries of top managers in private and
governmental organizations, the following information was gathered.
Private
Governmental
Sample Size
50
60
Sample Mean (in $1,000s)
190
180
Sample Standard Deviation (in $1,000s)
6
8
Develop an interval estimate for the difference between the average salaries of the two sectors. Let =
.05.
20. The following information regarding the ages of full-time and part-time students are given. Using the
following data, develop an interval estimate for the difference between the mean ages of the two
populations. Use a 5% level of significance. The degrees of freedom for the t distribution are 106.
Full-Time
Part-Time
27
24
s
1.5
2
n
50
60
PTS: 1
21. The following information regarding the number of semester hours taken from random samples of day
and evening students is provided.
Day
Evening
16
12
s
4
2
n
140
160
Develop a 95% confidence interval estimate for the difference between the mean semester hours taken
by the two groups of students.
22. A random sample of 89 tourists in the Grand Bahamas showed that they spent an average of $2,860 (in
a week) with a standard deviation of $126; and a sample of 64 tourists in New Province showed that
they spent an average of $2,935 (in a week) with a standard deviation of $138. We are interested in
determining if there is any significant difference between the average expenditures of those who
visited the two islands?
Determine the degrees of freedom for this test.
Compute the test statistic.
Compute the p-value.
What is your conclusion? Let = .05.
128
Test statistic t = -3.438
23. Consider the following hypothesis test:
H0:
1 −
2 0
Ha:
1 −
2 0
The following results are for two independent samples taken from two populations.
Sample 1
Sample 2
Sample Size
35
34
Sample Mean
43
37
Sample Variance
140
170
a.
Determine the degrees of freedom for the t distribution.
b.
Compute the test statistic.
c.
Determine the p-value and test the above hypotheses.
65
Test statistic t = 2.0
24. Recently, a local newspaper reported that part time students are older than full time students. In order
to test the validity of its statement, two independent samples of students were selected.
Full Time
Part Time
26
24
s
2
3
n
42
31
a.
Give the hypotheses for the above.
b.
Determine the degrees of freedom.
c.
Compute the test statistic.
d.
At 95% confidence, test to determine whether or not the average age of part time students is
significantly more than full time students.
b.
49
test statistic t = 3.221
d.
p-value (0.0011) is less than .005, reject H0
25. Independent random samples of managers’ yearly salaries (in $1000) taken from governmental and
private organizations provided the following information. At 95% confidence, test to determine if there
is a significant difference between the average salaries of the managers in the two sectors.
Government
Private
80
75
s
9
10
n
28
31
26. Independent random samples taken at two local malls provided the following information regarding
purchases by patrons of the two malls.
Hamilton Place
Eastgate
Sample Size
85
93
Average Purchase
$143
$150
Standard Deviation
$22
$18
We want to determine whether or not there is a significant difference between the average purchases
by the patrons of the two malls.
Give the hypotheses for the above.
Compute the test statistic.
At 95% confidence, test the hypotheses.
c.
27. The Dean of Students at UTC has said that the average grade of UTC students is higher than that of the
students at GSU. Random samples of grades from the two schools are selected, and the results are
shown below.
UTC
GSU
Sample Size
14
12
Sample Mean
2.85
2.61
Sample Standard Deviation
0.40
0.35
Sample Mode
2.5
3.0
a.
Give the hypotheses.
b.
Compute the test statistic.
c.
At a 0.1 level of significance, test the Dean of Students’ statement.
c.
28. Zip, Inc. manufactures Zip drives on two different manufacturing processes. Because the management
of this company is interested in determining if process 1 takes less manufacturing time, they selected
independent samples from each process. The results of the samples are shown below.
Process 1
Process 2
Sample Size
27
22
Sample Mean (in minutes)
10
14
Sample Variance
16
25
a.
State the null and alternative hypotheses.
b.
Determine the degrees of freedom for the t test.
c.
Compute the test statistic
d.
At 95% confidence, test to determine if there is sufficient evidence to indicate that process 1
takes a significantly shorter time to manufacture the Zip drives.
b.
39
c.
-3.042
d.
29. A recent Time magazine reported the following information about a sample of workers in Germany
and the United States.
United States
Germany
Average length of workweek (hours)
42
38
Sample Standard Deviation
5
6
Sample Size
600
700
We want to determine whether or not there is a significant difference between the average workweek
in the United States and the average workweek in Germany.
State the null and the alternative hypotheses.
Compute the test statistic.
Compute the p-value. What is your conclusion?
Test statistic t = 13.1
c.
30. Independent random samples taken at two companies provided the following information regarding
annual salaries of the employees.
Marissa, Inc
Jason, Inc.
Sample Size
72
50
Sample Mean (in $1,000)
48
43
Sample Standard Deviation (in $1,000)
12
10
a.
We want to determine whether or not there is a significant difference between the average
salaries of the employees at the two companies. Compute the test statistic.
b.
Compute the p-value; and at 95% confidence, test the hypotheses.
test statistics t = 2.5
significant difference in the average salaries.
31. Two independent samples are drawn from two populations, and the following information is provided.
Population 1
Population 2
n
34
52
55
65
s
14
18
We want to test the following hypotheses.
Ho:
1 −
2 0
Ha:
1 −
2 0
Determine the degrees of freedom.
Compute the test statistic.
At 95% confidence, test the hypotheses. Assume the two populations are normally distributed
and have equal variances.
81
32. In order to determine whether or not a driver’s education course improves the scores on a driving
exam, a sample of 6 students were given the exam before and after taking the course. The results are
shown below.
Let d = Score After – Score Before.
Score
Score
Student
Before the Course
After the Course
1
83
87
2
89
88
3
93
91
4
77
77
5
86
93
6
79
83
a.
Compute the test statistic.
b.
At 95% confidence using the p-value approach, test to see if taking the course actually
increased scores on the driving exam.
Test statistic t = 1.391
evidence to show that the course increased the scores.
33. The daily production rates for a sample of factory workers before and after a training program are
shown below. Let d = After – Before.
Worker
Before
After
1
6
9
2
10
12
3
9
10
4
8
11
5
7
9
We want to determine if the training program was effective.
Give the hypotheses for this problem.
Compute the test statistic.
At 95% confidence, test the hypotheses. That is, did the training program actually increase the
production rates?
a.
b.
test statistic t = 5.88
c.
8
5
34. The following data present the number of computer units sold per day by a sample of 6 salespersons
before and after a bonus plan was implemented. Let d = before – after.
Salesperson
Before
After
1
3
6
2
7
5
3
6
6
4
8
7
5
7
8
6
9
8
At 95% confidence, test to see if the bonus plan was effective. That is, did the bonus plan actually
increase sales?
35. A test on world history was given to a group of individuals before and also after a film on the history
of the world was presented. The results are given below. We want to determine if the film significantly
increased the test scores. (For the following matched samples, let the d = after − before.)
Individual
After
Before
0
4
1
92
86
2
86
88
3
89
84
4
90
90
5
93
85
6
88
90
7
97
91
a.
Give the hypotheses for this problem.
b.
Compute the test statistic.
c.
At 95% confidence, test the hypotheses.
c.
36. The following shows the monthly sales in units of six salespersons before and after a bonus plan was
introduced. At 95% confidence, determine whether the bonus plan has increased sales significantly.
(For the following matched samples, let the difference “d” be: d = after − before.)
Monthly Sales
Salesperson
After
Before
1
94
90
2
82
84
3
90
84
4
76
70
5
79
80
6
85
80
37. Allied Corporation is trying to determine whether to purchase Machine A or B. It has leased the two
machines for a month. A random sample of 5 employees has been taken. These employees have gone
through a training session on both machines. Below you are given information on their productivity
rate on both machines. (Let d = Machine A − Machine B.)
Productivity Rate
Person
Machine A
Machine B
1
47
52
2
53
58
3
50
47
4
55
60
5
45
53
a.
State the null and alternative hypotheses for a two-tailed test.
b.
Find the mean and standard deviation for the difference.
c.
Compute the test statistic.
d.
Test the null hypothesis stated in Part a at the 10% level.
b.
-4 and 4.123
d.
38. A company attempts to evaluate the potential for a new bonus plan by selecting a sample of 4
salespersons to use the bonus plan for a trial period. The weekly sales volume before and after
implementing the bonus plan is shown below. (For the following matched samples, let the difference
“d” be d = after − before.)
Weekly Sales
Salesperson
Before
After
1
44
48
2
40
48
3
36
38
4
50
44
a.
State the hypotheses.
b.
Compute the test statistic.
c.
Use Alpha = .05 and test to see if the bonus plan will result in an increase in the mean
weekly sales.
39. The following information was obtained from matched samples regarding the productivity of four
individuals using two different methods of production.
Individual
Method 1
Method 2
1
6
8
2
9
5
3
7
6
4
7
5
5
8
6
6
9
5
7
6
3
Let d = Method 1 − Method 2. Is there a significant difference between the productivity of the two
methods? Let = 0.05.
40. Information regarding the ACT scores of samples of students in three different majors is given below.
Major
Management
Finance
Accounting
28
22
29
26
23
27
25
24
26
27
22
28
21
24
25
19
26
26
27
27
28
17
29
20
17
28
20
23
24
28
28
29
Sums
230
225
338
Means
23
25
26
Variances
18
6.75
9.33
a.
Set up the ANOVA table for this problem.
b.
At 95% confidence test to determine whether there is a significant difference in the means of
the three populations.
a.
Between Groups
Within Groups
Total
value is greater than 0.1.
Actual p-value = 0.1248
41. Information regarding the ACT scores of samples of students in four different majors is given below.
Major
Management
Marketing
Finance
Accounting
29
22
29
28
27
22
27
26
21
25
27
25
28
26
28
20
22
27
24
21
28
20
20
19
28
23
20
27
23
25
30
24
28
27
29
21
24
28
23
29
27
31
27
24
Sum
318
245
234
312
Mean
26.50
24.50
26.00
24.00
Variance
10.09
6.94
14.50
9.00
a.
Set up the ANOVA table for this problem.
b.
At 95% confidence, test to determine whether there is a significant difference in the means
of the three populations.
Between Groups
Within Groups
Total
Using the F table (3 numerator and 40 denominator degrees of freedom) for F = 1.666, the p–
value is greater than 0.1.
Actual p-value = 0.1897
Since the p-value is = 0.05, do not reject Ho. (Also the test statistic F = 1.666 2.84.)
42. Guitars R. US has three stores located in three different areas. Random samples of the sales of the
three stores (in $1000) are shown below.
Store 1
Store 2
Store 3
80
85
79
75
86
85
76
81
88
89
80
80
a.
Compute the overall sample mean .
b.
At 95% confidence, test to see if there is a significant difference in the average sales of the
three stores. Set up the complete ANOVA table. Please note that the sample sizes are not
equal.
a.
82
b.
Source of Variation
Between Groups
Within Groups
Total
value is greater than 0.1.
Actual p-value = 0.4580
do not reject Ho.)
43. In a completely randomized experimental design, 18 experimental units were used for the first
treatment, 10 experimental units for the second treatment, and 15 experimental units for the third
treatment. Part of the ANOVA table for this experiment is shown below.
ANOVA
Source of Variation
SS
df
MS
F
Between Groups
?
?
?
3
Within Groups
?
?
6
Total
?
?
a.
Fill in all the blanks in the above ANOVA table.
b.
At 95% confidence, test to see if there is a significant difference among the means.
Source of Variation
Between Groups
Within Groups
Total
is between 0.05 and 0.1
Actual p-value = 0.0611
44. Random samples were selected from three populations. The data obtained are shown below.
Treatment 1
Treatment 2
Treatment 3
37
43
28
33
39
32
36
35
33
38
38
40
a.
Compute the overall sample mean .
b.
At 95% confidence, test to see if there is a significant difference in the means of the three
populations. Show the complete ANOVA table. Please note that the sample sizes are not
equal.
36
b.
Source of Variation
Between Groups
Within Groups
Total
Using the F table (2 numerator and 9 denominator degrees of freedom) for F = 8.71, the p–
value is less than 0.01.
Actual p-value = 0.0079
45. In a completely randomized experimental design, 7 experimental units were used for the first
treatment, 9 experimental units for the second treatment, and 14 experimental units for the third
treatment. Part of the ANOVA table for this experiment is shown below.
ANOVA
Source of Variation
SS
df
MS
F
Between Groups
?
?
?
4.5
Within Groups
?
?
4
Total
?
?
a.
Fill in all the blanks in the above ANOVA table.
b.
At 95% confidence, test to see if there is a significant difference among the means.
Source of Variation
Between Groups
36
18