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Chapter 10 - Comparisons Involving Means, Experimental Design, and Analysis of Variance
b.
4
c.
5
d.
18
94. Refer to Exhibit 10-15. The mean square between treatments (MSTR) is
a.
36
b.
16
c.
64
d.
15
95. Refer to Exhibit 10-15. If at a 5% significance level we want to determine whether or not the means of the populations
are equal, the critical value of F is
a.
5.80
b.
2.93
c.
3.16
d.
2.90
96. Refer to Exhibit 10-15. The conclusion of the test is that the means
a.
are equal
b.
may be equal
c.
are not equal
d.
None of these alternatives is correct.
Exhibit 10-16
The following is part of an ANOVA table, which was the results of three treatments and a total of 15 observations.
Source of Variation
Sum of
Squares
Degrees of
Freedom
Mean
Square
F
Between Treatments
64
Within Treatments (Error)
96
Total
97. Refer to Exhibit 10-16. The number of degrees of freedom corresponding to between treatments is
a.
12
b.
2
c.
3
d.
4
Chapter 10 - Comparisons Involving Means, Experimental Design, and Analysis of Variance
98. Refer to Exhibit 10-16. The number of degrees of freedom corresponding to within treatments is
a.
12
b.
2
c.
3
d.
15
99. Refer to Exhibit 10-16. The mean square between treatments (MSTR) is
a.
36
b.
16
c.
8
d.
32
100. Refer to Exhibit 10-16. If at a 5% level of significance, we want to determine whether or not the means of the
populations are equal, the critical value of F is
a.
4.75
b.
19.41
c.
3.16
d.
1.96
101. Refer to Exhibit 10-16. The computed test statistics is
a.
32
b.
8
c.
0.667
d.
4
102. Refer to Exhibit 10-16. The conclusion of the test is that the means
a.
are equal
b.
may be equal
c.
are not equal
d.
None of these alternatives is correct.
103. In a completely randomized design involving four treatments, the following information is provided.
Chapter 10 - Comparisons Involving Means, Experimental Design, and Analysis of Variance
Treatment 1
Treatment 2
Treatment 3
Treatment 4
Sample Size
50
18
15
17
Sample Mean
32
38
42
48
The overall mean (the grand mean) for all treatments is
a.
40.0
b.
37.3
c.
48.0
d.
37.0
104. An ANOVA procedure is used for data obtained from five populations. five samples, each comprised of 20
observations, were taken from the five populations. The numerator and denominator (respectively) degrees of freedom for
the critical value of F are
a.
5 and 20
b.
4 and 20
c.
4 and 99
d.
4 and 95
105. The critical F value with 8 numerator and 29 denominator degrees of freedom at α = 0.01 is
a.
2.28
b.
3.20
c.
3.33
d.
3.64
106. An ANOVA procedure is used for data obtained from four populations. Four samples, each comprised of 30
observations, were taken from the four populations. The numerator and denominator (respectively) degrees of freedom for
the critical value of F are
a.
3 and 30
b.
4 and 30
c.
3 and 119
d.
3 and 116
107. Which of the following is not a required assumption for the analysis of variance?
a.
The random variable of interest for each population has a normal probability distribution.
b.
The variance associated with the random variable must be the same for each population.
c.
At least 2 populations are under consideration.
d.
Populations have equal means.
108. In an analysis of variance, one estimate of 2 is based upon the differences between the treatment means and the
a.
means of each sample
b.
overall sample mean
c.
sum of observations
d.
populations have equal means
109. In ANOVA, treatments should be assigned to experimental units on a
a.
systematic basis
b.
subjective basis
c.
controlled basis
d.
random basis
110. In ANOVA, the dependent variable is also known as the
a.
response variable
b.
control variable
c.
blind variable
d.
experimental variable
Subjective Short Answer
111. 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.
112. The following sample information is given concerning the ACT scores of high school seniors form two local schools.
School A
School B
= 14
= 15
Chapter 10 - Comparisons Involving Means, Experimental Design, and Analysis of Variance
= 25
= 23
= 16
= 10
Develop a 95% confidence interval estimate for the difference between the two populations.
113. Independent random samples taken on two university campuses revealed the following information concerning the
average amount of money spent on textbooks during the fall semester.
University A
University B
Sample Size
50
40
Average Purchase
$260
$250
Standard Deviation (σ)
$ 20
$ 23
We want to determine if, on the average, students at University A spent more on textbooks then the students at University
B.
a.
Compute the test statistic.
b.
Compute the p-value.
c.
What is your conclusion? Let α = .05.
114. 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 (σ2)
64
70
a.
Develop a 95% confidence interval estimate for the difference between the average time of the
two processes.
b.
Is there conclusive evidence to prove that one process takes longer than the other? If yes, which
process? Explain.
115. 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)
.6
4.9
Standard Deviation (σ)
0.5
0.6
Chapter 10 - Comparisons Involving Means, Experimental Design, and Analysis of Variance
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 α = 0.05.
116. 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.
Student
Score
Before the Course
Score
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.
117. 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.
118. 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.
Chapter 10 - Comparisons Involving Means, Experimental Design, and Analysis of Variance
Health Clinic
Group Practice
Sample size
50 visits
45 visits
Sample mean
$21
$19
Standard deviation (σ)
$2.75
$3.00
Develop a 95% confidence interval estimate for the difference between the average fees of the two offices.
119. 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?
a.
Determine the degrees of freedom for this test.
b.
Compute the test statistic.
c.
Compute the p-value.
d.
What is your conclusion? Let α = .05.
120. 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.
121. 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.
122. 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.
123. 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.
124. Consider the following hypothesis test:
μ1 - μ2 ≤ 0
μ1 - μ2 > 0
The following results are for two independent samples taken from two populations.
Sample 1
Sample 2
Sample Size
35
37
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.
125. The following are the test scores of two samples of students from University A and University B on a national
Chapter 10 - Comparisons Involving Means, Experimental Design, and Analysis of Variance
statistics examination. Develop a 95% confidence interval estimate for the difference between the mean scores of the two
populations.
University A
Scores
University B
Scores
86
83
n
64
87
σ
68
126. In order to estimate the difference between the average mortgages in the South and the North of the United States,
the following information was gathered.
South
North
Sample Size
40
45
Sample Mean (in $1,000)
$70
$75
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 = 0.03.
127. 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.
Full-Time
Part-Time
27
24
s
1.2
2
n
50
60
128. 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
129. 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.
a.
Give the hypotheses for the above.
b.
Compute the test statistic.
c.
At 95% confidence, test the hypotheses.
130. 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.
131. 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.
a.
Give the hypotheses for this problem.
Chapter 10 - Comparisons Involving Means, Experimental Design, and Analysis of Variance
b.
Compute the test statistic.
c.
At 95% confidence, test the hypotheses. That is, did the training program actually increase the
production rates?
132. 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
Degree
Bachelors
Degree
Sample Size
33
30
Sample Mean (in $1,000)
38
34
Sample Standard Deviation (in $1,000)
2.4
2
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.
133. 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 difference "d" be d = after - before.)
Individual
After
Before
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.
134. 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.
Chapter 10 - Comparisons Involving Means, Experimental Design, and Analysis of Variance
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.
135. 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.
136. 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.
137. 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.)
Salesperson
Sales After
Sales Before
1
94
90
2
82
84
3
90
84
4
76
70
5
79
80
6
85
80
138. 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
3
n
40
37
Develop a 95% confidence interval estimate for the difference between the mean semester hours taken by the two groups
of students.
139. 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.
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?
140. 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.
141. A credit company has gathered information regarding the average amount owed by people under 30 years old and by
Chapter 10 - Comparisons Involving Means, Experimental Design, and Analysis of Variance
people 30 years and over. Independent random samples were taken from both age groups. You are given the following
information.
Under 30
30 and Over
600
550
n
200
300
σ2
361
400
Construct a 95% confidence interval for the difference between the average amounts owed by the two age groups.
142. 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.
143. 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.
a.
State the null and the alternative hypotheses.
b.
Compute the test statistic.
c.
Compute the p-value. What is your conclusion?
144. 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 the difference "d" be d = A -
B.)
Rate of
Rate of
Person
Machine A
Machine B
1
47
52
2
53
58
3
50
47
Chapter 10 - Comparisons Involving Means, Experimental Design, and Analysis of Variance
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.
145. 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.)
Salesperson
Sales Before
Sales After
1
48
44
2
48
40
3
38
36
4
44
50
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.
146. 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 α = 0.05.
147. A potential investor conducted a 49 day survey in two theaters in order to determine the difference between the
average daily attendance at North Mall and South Mall Theaters. The North Mall Theater averaged 720 patrons per day
with a variance of 100; while the South Mall Theater averaged 700 patrons per day with a variance of 96. Develop an
interval estimate for the difference between the average daily attendance at the two theaters. Use a confidence coefficient
of 0.95.
148. 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.
H0: μ1 - μ2 ≥ 0
Ha: μ1 - μ2 < 0
a.
Determine the degrees of freedom.
b.
Compute the test statistic.
c.
At 95% confidence, test the hypotheses. Assume the two populations are normally distributed
and have equal variances.
149. 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)
90
80
Sample Standard Deviation
(in $1,000s)
6
8
Develop an interval estimate for the difference between the average salaries of the two sectors. Let α = .05.
150. Independent random samples taken at two companies provided the following information regarding annual salaries
of the employees.
Whitney Co.
Max Co.
Sample Size
72
50
Sample Mean (in $1,000)
48
43
Sample Standard Deviation (in $1,000)
12
10
Chapter 10 - Comparisons Involving Means, Experimental Design, and Analysis of Variance
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.
151. In order to estimate the difference between the yearly incomes of marketing managers in the East and West of the
United States, the following information was gathered.
East
West
n1 = 40
n2 = 45
= 72 (in $1,000)
= 78 (in $1,000)
s1 = 6 (in $1,000)
s2 = 8 (in $1,000)
a.
Develop an interval estimate for the difference between the average yearly incomes of the
marketing managers in the East and West. Use α = 0.05.
b.
At 95% confidence, use the p-value approach and test to determine if the average yearly
income of marketing managers in the East is significantly different from the West.
152. In order to estimate the difference between the average daily sales of two branches of a department store, the
following data has been gathered.
Downtown Store
North Mall Store
Sample size
n1 = 23 days
n2 = 26 days
Sample mean (in $1,000)
= 37
= 34
Sample standard deviation (in $1,000)
S1 = 4
S2 = 5
a.
Determine the point estimate of the difference between the means.
b.
Determine the degrees of freedom for this interval estimation.
c.
Compute the margin of error.
d.
Develop a 95% confidence interval for the difference between the two population means.
153. A potential investor conducted a 144 day survey in each theater in order to determine the difference between the
average daily attendance at the North Mall and South Mall theaters. The North Mall Theater averaged 630 patrons per
day; while the South Mall Theater averaged 598 patrons per day. From past information, it is known that the variance for
North Mall is 1,000; while the variance for the South Mall is 1,304. Develop a 95% confidence interval for the difference
between the average daily attendance at the two theaters.
154. Information regarding the ACT scores of samples of students in three different majors are given below.
Student's 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 a 5% level of significance, test to determine whether there is a significant difference in the
means of the three populations.
155. Information regarding the ACT scores of samples of students in four different majors are given below.
Student's 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
Chapter 10 - Comparisons Involving Means, Experimental Design, and Analysis of Variance
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 a 5% level of significance, test to determine whether there is a significant difference in the
means of the four populations.
156. 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
80
86
85
76
81
88
89
80
At a 5% level of significance, test to see if there is a significant difference in the average sales of the three stores. (Please
note that the sample sizes are not equal.)
157. 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.
Source of Variation
Sum of
Squares
Degrees of
Freedom
Mean
Square
F
