17. During the primary elections of 2004, candidate A showed the following pre-election voter support in
Tennessee and Mississippi.
Voters Surveyed
Voters Favoring
Candidate A
Tennessee
500
295
Mississippi
700
357
a.
Develop a 95% confidence interval estimate for the difference between the proportion of voters
favoring candidate A in the two states.
b.
Is there conclusive evidence that one of the two states had a larger proportion of voters’
support? If yes, which state? Explain.
18. 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.
a.
49
b.
-1.753 to 5.753
c.
No, the interval ranges from negative to positive.
19. The results of a recent poll on the preference of voters regarding the presidential candidates are shown
below.
Voters Surveyed
Voters Favoring
This Candidate
Candidate A
200
150
Candidate B
300
195
a.
Develop a 90% confidence interval estimate for the difference between the proportion of voters
favoring each candidate.
b.
Does your confidence interval provide conclusive evidence that one of the candidates is
favored more? Explain.
a.
0.023 to 0.137
support.
20. Consider the following hypothesis test:
1 – 2 0
1 – 2 > 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.
65
b.
Test statistic t = 2.0
21. 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
Scores
University B
Scores
86
83
n
64
87
68
22. 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
40
45
0.032 to 0.168
b.
Yes, the range of interval is from positive to positive, indicating Candidate A had the larger
$70
$75
$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.
23. 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
24. 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
df = 56; test statistic t = 2.021; p-value (2 tailed) is between .02 and .05; reject H0
25. 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
79
b.
-7.605 to -2.395 (in thousands)
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.
26. 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
c.
test statistic t = 3.221
d.
27. 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.
b.
Compute the test statistic.
b.
t = -2.31
c.
At 95% confidence, test the hypotheses. That is, did the training program actually increase the
production rates?
28. In a sample of 40 Democrats, 6 opposed the President’s foreign policy, while of 50 Republicans, 8
were opposed to his policy. Determine a 90% confidence interval estimate for the difference between
the proportions of the opinions of the individuals in the two parties.
29. In a sample of 100 Republicans, 60 favored the President’s anti-drug program. While in a sample of
150 Democrats, 84 favored his program. At 95% confidence, test to see if there is a significant
difference in the proportions of the Democrats and the Republicans who favored the President’s
anti-drug program.
30. In a random sample of 200 Republicans, 160 opposed the new tax laws. While in a sample of 120
Democrats, 84 opposed the new tax laws. Determine a 95% confidence interval estimate for the
difference between the proportions of Republicans and Democrats opposed to this new law.
31. Two independent random samples of annual starting salaries for individuals with master’s and
bachelor’s degrees in business were taken and the results are shown below
Master’s
Degree
Bachelor’s
Degree
33
30
38
34
2.4
2
a.
What are the degrees of freedom for the t distribution?
b.
test statistic t = 5.8
c.
b.
Provide a 95% confidence interval estimate for the difference between the salaries of the two
groups.
32. During the recent primary elections, the democratic presidential candidate showed the following
pre-election voter support in Alabama and Mississippi.
State
Voters Surveyed
Voters Favoring the
Democratic Candidate
Alabama
800
440
Mississippi
600
360
a.
We want to determine whether or not the proportions of voters favoring the Democratic
candidate were the same in both states. Provide the hypotheses.
b.
Compute the test statistic.
c.
Determine the p-value; and at 95% confidence, test the above hypotheses.
b.
Z = -1.87
c.
33. 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.
a.
60
b.
2.89 to 5.11 (thousands)
34. 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.
b.
t = 1.632
35. 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.
36. 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
b.
t = 1.89
Provide a 98% confidence interval estimate for the difference between the test scores of the two
universities.
37. 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
38. The office of records at a university has stated that the proportion of incoming female students who
major in business has increased. A sample of female students taken several years ago is compared with
a sample of female students this year. Results are summarized below. Has the proportion increased
significantly? Test at alpha = .10.
Sample Size
No. Majoring in Business
Previous Sample
250
50
Present Sample
300
69
39. 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.
40. 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?
41. 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
42. 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
361
400
Construct a 95% confidence interval for the difference between the average amounts owed by the two
age groups.
43. 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.
44. 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?
b.
Test statistic t = 13.1
45. 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.)
Person
Machine A Productivity
Machine B Productivity
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
c.
t = -2.169
d.
46. The reliability of two types of machines used in the same manufacturing process is to be tested. The
first machine failed to operate correctly in 90 out of 300 trials while the second type failed to operate
correctly in 50 out of 250 trials.
a.
Give a point estimate for the difference between the population proportions of these machines.
b.
Calculate the pooled estimate of the population proportion.
c.
Carry out a hypothesis test to check whether there is a statistically significant difference in the
reliability for the two types of machines using a .10 level of significance.
b.
0.2545
c.
test statistic Z = 2.68, p-value = 0.0074, reject H0
47. 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.)
c.
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.
48. 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.
49. The results of a recent poll on the preference of voters regarding presidential candidates are shown
below.
Candidate
Voters
Surveyed
Voters Favoring
This Candidate
A
400
192
B
450
225
b.
t = 0.68
c.
At 95% confidence, test to determine whether or not there is a significant difference between the
preferences for the two candidates.
50. 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.
51. From production line A, a sample of 500 items is selected at random, and it is determined that 30 items
are defective. In a sample of 300 items from production process B (which produces identical items to
line A), there are 12 defective items. Determine a 95% confidence interval estimate for the difference
between the proportions of defectives in the two lines.
52. 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: 1 – 2 0
Ha: 1 – 2 < 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.
a.
81
b.
t = -2.887
H0: P1 – P2 = 0
Ha: P1 – P2 0
53. 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 =
.05.
54. Independent random samples taken at two companies provided the following information regarding
annual salaries of the employees.
Whitney Co.
Max Co.
72
50
48
43
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
b.
significant difference in the average salaries
55. 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 = 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.
ANS:
56. 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.
3
b.
46 (rounded down from 46.56)
2.591
3 2.591 (In dollars it will be from $409 to $5,591)
57. Babies weighing less than 5.5 pounds at birth are considered “low-birth-weight babies.” In the United
States, 7.6% of newborns are low-birth-weight babies. The following information was accumulated
from samples of new births taken from two counties.
Hamilton
Shelby
Sample size
150
200
Number of “low-birth-weight babies
18
22
a.
Develop a 95% confidence interval estimate for the difference between the proportions of
low-birth-weight babies in the two counties.
b.
Is there conclusive evidence that one of the proportions is significantly more than the other?
If yes, which county? Explain, using the results of part (a). Do not perform any test.
-.0577 to .0777
b.
Since the range of the interval is from negative to positive, there is no indication that one
proportion is significantly different (at 95% confidence) from the other.
58. A poll was taken this year asking college students if they considered themselves overweight. A similar
poll was taken five years ago. Results are summarized below. Has the proportion increased
significantly? Let = 0.05.
a.
-9.084 to –2.916
Test statistic t = -3.937 (df = 80), p-value .005, reject Ho and conclude that there is a
Sample Size
Number Considered
Themselves Overweight
Present Sample
300
150
Previous sample
275
121
59. 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.
60. A comparative study of organic and conventionally grown produce was checked for the presence of E.
coli. Results are summarized below. Is there a significant difference in the proportion of E. Coli in
organic vs. conventionally grown produce? Test at = 0.10.
Sample Size
E. Coli Prevalence
Organic
200
3
Conventional
500
20
61. Starting annual salaries for business school graduates majoring in finance and management infor-
mation systems (MIS) were collected in two independent random samples summarized below. Based
on previous studies, the population standard deviations for Finance and MIS salaries are estimated to
be $2,100 and $2,600, respectively.
Finance MIS
n1 = 60 n2 = 50
1
x
= $43,200
2
x
= $46,500
a. Develop a 95% confidence interval estimate of the difference between the starting salaries for the
two majors.
b. Using
= .10, test to determine if the average starting salary for an MIS graduate is $4,000 more
than the starting salary for a finance graduate. Use both the critical value and p-value approach-
es to hypothesis testing. (Hint: the null hypothesis is H0:
1 –
2 = $4,000, where
1 is the aver-
age starting salary of MIS graduates.)
62. A manager is thinking of providing, on a regular basis, in-house training for employees preparing for
an inventory management certification exam. In the past, some employees received the in-house
training before taking the exam, while others did not. Independent random samples taken from the
company’s records provided the following exam scores for 10 workers who did not receive in-house
training and 8 workers who did receive training. (The manager is confident that the distributions of
both populations’ exam scores are approximately normal.)
No Training Training
76 80
80 66
60 71
91 79
73 94
77 74
82 83
68 78
75
86
a. Develop a 95% confidence interval estimate for the difference between the average test scores for
the two populations of employees.
b. Using
= .05, test for any difference between the average test scores for the two populations of
employees.
63. A survey was recently conducted to determine if consumers spend more on computer-related
purchases via the Internet or store visits. Assume a sample of 8 respondents provided the following
data on their computer-related purchases during a 30-day period. Using a .05 level of significance,
can we conclude that consumers spend more on computer-related purchases by way of the Internet
than by visiting stores?
Expenditures (dollars)
Respondent
In-Store
Internet
1
132
225
2
90
24
3
119
95
4
16
55
5
85
13
6
248
105
7
64
57
8
49
0
64. A movie based on a best-selling novel was recently released. Six hundred viewers of the movie, 235
of whom had previously read the novel, were asked to rate the quality of the movie. The survey
showed that 141 of the novel readers gave the movie a rating of excellent, while 248 of the non-readers
gave the movie an excellent rating.
a. Develop an interval estimate of the difference between the proportions of the two populations, us-
ing a .05 level of significance, as the basis for your decision.
b. Can we conclude, on the basis of a hypothesis test about p1 – p2, that the proportion of the
non-readers of the novel who thought the movie was excellent is greater than the proportion of
readers of the novel who thought the movie was excellent? Use a .05 level of significance.
(Hint: this is a one-tailed test.)