CHAPTER 10—INFERENCE ABOUT MEANS AND PROPORTIONS WITH TWO
POPULATIONS
MULTIPLE CHOICE
1. If we are interested in testing whether the proportion of items in population 1 is larger than the
proportion of items in population 2, the
a.
null hypothesis should state P1 – P2 < 0
b.
null hypothesis should state P1 – P2 0
c.
alternative hypothesis should state P1 – P2 > 0
d.
alternative hypothesis should state P1 – P2 < 0
2. To compute an interval estimate for the difference between the means of two populations, the t
distribution
a.
is restricted to small sample situations
b.
is not restricted to small sample situations
c.
can be applied when the populations have equal means
d.
None of these alternatives is correct.
3. When developing an interval estimate for the difference between two sample means, with sample sizes
of n1 and n2,
a.
n1 must be equal to n2
b.
n1 must be smaller than n2
c.
n1 must be larger than n2
d.
n1 and n2 can be of different sizes,
4. To construct an interval estimate for the difference between the means of two populations when the
standard deviations of the two populations are unknown, we must use a t distribution with (let n1 be the
size of sample 1 and n2 the size of sample 2)
a.
(n1 + n2) degrees of freedom
b.
(n1 + n2 – 1) degrees of freedom
c.
(n1 + n2 – 2) degrees of freedom
d.
n1 – n2 + 2
5. When each data value in one sample is matched with a corresponding data value in another sample, the
samples are known as
a.
corresponding samples
b.
matched samples
c.
independent samples
d.
None of these alternatives is correct.
6. Independent simple random samples are taken to test the difference between the means of two
populations whose variances are not known. The sample sizes are n1 = 32 and n2 = 40. The correct
distribution to use is the
a.
binomial distribution
b.
t distribution with 72 degrees of freedom
c.
t distribution with 71 degrees of freedom
d.
t distribution with 70 degrees of freedom
7. Independent simple random samples are taken to test the difference between the means of two
populations whose standard deviations are not known. The sample sizes are n1 = 25 and n2 = 35. The
correct distribution to use is the
a.
Poisson distribution
b.
t distribution with 60 degrees of freedom
c.
t distribution with 59 degrees of freedom
d.
t distribution with 58 degrees of freedom
8. If two independent large samples are taken from two populations, the sampling distribution of the
difference between the two sample means
a.
can be approximated by a Poisson distribution
b.
will have a variance of one
c.
can be approximated by a normal distribution
d.
will have a mean of one
9. The standard error of is the
a.
variance of
b.
variance of the sampling distribution of
c.
standard deviation of the sampling distribution of
d.
difference between the two means
10. The sampling distribution of is approximated by a
a.
normal distribution
b.
t-distribution with n1 + n2 degrees of freedom
c.
t-distribution with n1 + n2 – 1 degrees of freedom
d.
t-distribution with n1 + n2 + 2 degrees of freedom
Exhibit 10-1
Salary information regarding male and female employees of a large company is shown below.
Male
Female
Sample Size
64
36
Sample Mean Salary (in $1,000)
44
41
Population Variance
128
72
11. Refer to Exhibit 10-1. The point estimate of the difference between the means of the two populations
is
a.
-28
b.
3
c.
4
d.
-4
12. Refer to Exhibit 10-1. The standard error for the difference between the two means is
a.
4
b.
7.46
c.
4.24
d.
2.0
13. Refer to Exhibit 10-1. At 95% confidence, the margin of error is
a.
1.96
b.
1.645
c.
3.920
d.
2.000
14. Refer to Exhibit 10-1. The 95% confidence interval for the difference between the means of the two
populations is
a.
0 to 6.92
b.
-2 to 2
c.
-1.96 to 1.96
d.
-0.92 to 6.92
15. Refer to Exhibit 10-1. If you are interested in testing whether or not the average salary of males is
significantly greater than that of females, the test statistic is
a.
2.0
b.
1.5
c.
1.96
d.
1.645
16. Refer to Exhibit 10-1. The p-value is
a.
0.0668
b.
0.0334
c.
1.336
d.
1.96
17. Refer to Exhibit 10-1. At 95% confidence, the conclusion is the
a.
average salary of males is significantly greater than females
b.
average salary of males is significantly lower than females
c.
salaries of males and females are equal
d.
None of these alternatives is correct.
Exhibit 10-2
The following information was obtained from matched samples.
The daily production rates for a sample of workers before and after a training program are shown
below.
Worker
Before
After
1
20
22
2
25
23
3
27
27
4
23
20
5
22
25
6
20
19
7
17
18
18. Refer to Exhibit 10-2. The point estimate for the difference between the means of the two populations
is
a.
-1
b.
-2
c.
0
d.
1
19. Refer to Exhibit 10-2. The null hypothesis to be tested is H0: d = 0. The test statistic is
a.
-1.96
b.
1.96
c.
0
d.
1.645
20. Refer to Exhibit 10-2. Based on the results of question 18, the
a.
null hypothesis should be rejected
b.
null hypothesis should not be rejected
c.
alternative hypothesis should be accepted
d.
None of these alternatives is correct.
Exhibit 10-3
A statistics teacher wants to see if there is any difference in the abilities of students enrolled in
statistics today and those enrolled five years ago. A sample of final examination scores from students
enrolled today and from students enrolled five years ago was taken. You are given the following
information.
Today
Five Years Ago
82
88
2
112.5
54
n
45
36
21. Refer to Exhibit 10-3. The point estimate for the difference between the means of the two populations
is
a.
58.5
b.
9
c.
-9
d.
-6
22. Refer to Exhibit 10-3. The standard error of is
a.
12.9
b.
9.3
c.
4
d.
2
23. Refer to Exhibit 10-3. The 95% confidence interval for the difference between the two population
means is
a.
-9.92 to -2.08
b.
-3.92 to 3.92
c.
-13.84 to 1.84
d.
-24.228 to 12.23
24. Refer to Exhibit 10-3. The test statistic for the difference between the two population means is
a.
-.47
b.
-.65
c.
-1.5
d.
-3
25. Refer to Exhibit 10-3. The p-value for the difference between the two population means is
a.
.0014
b.
.0028
c.
.4986
d.
.9972
26. Refer to Exhibit 10-3. What is the conclusion that can be reached about the difference in the average
final examination scores between the two classes? (Use a .05 level of significance.)
a.
There is a statistically significant difference in the average final examination scores
between the two classes.
b.
There is no statistically significant difference in the average final examination scores
between the two classes.
c.
It is impossible to make a decision on the basis of the information given.
d.
There is a difference, but it is not significant.
Exhibit 10-4
The following information was obtained from independent random samples.
Assume normally distributed populations with equal variances.
Sample 1
Sample 2
27. Refer to Exhibit 10-4. The point estimate for the difference between the means of the two populations
is
a.
0
b.
2
c.
3
d.
15
28. Refer to Exhibit 10-4. The standard error of is
a.
3.0
b.
4.0
c.
8.372
d.
19.48
29. Refer to Exhibit 10-4. The degrees of freedom for the t-distribution are
a.
22
b.
21
c.
20
d.
19
30. Refer to Exhibit 10-4. The 95% confidence interval for the difference between the two population
means is
a.
-5.372 to 11.372
b.
-5 to 3
c.
-4.86 to 10.86
d.
-2.65 to 8.65
Exhibit 10-5
The following information was obtained from matched samples.
Individual
Method 1
Method 2
1
7
5
2
5
9
3
6
8
4
7
7
5
5
6
31. Refer to Exhibit 10-5. The point estimate for the difference between the means of the two populations
is
a.
-1
b.
0
c.
1
d.
2
32. Refer to Exhibit 10-5. The 95% confidence interval for the difference between the two population
means is
a.
-3.776 to 1.776
b.
-2.776 to 2.776
c.
-1.776 to 2.776
d.
0 to 3.776
33. Refer to Exhibit 10-5. The null hypothesis tested is H0: d = 0. The test statistic for the difference
between the two population means is
a.
2
b.
0
c.
-1
d.
-2
34. Refer to Exhibit 10-5. If the null hypothesis is tested at the 5% level, the null hypothesis
a.
should be rejected
b.
should not be rejected
c.
should be revised
d.
None of these alternatives is correct.
Exhibit 10-6
The management of a department store is interested in estimating the difference between the mean
credit purchases of customers using the store’s credit card versus those customers using a national
major credit card. You are given the following information.
Store’s Card
Major Credit Card
Sample size
64
49
Sample mean
$140
$125
Population standard deviation
$10
$8
35. Refer to Exhibit 10-6. A point estimate for the difference between the mean purchases of the users of
the two credit cards is
a.
2
b.
18
c.
265
d.
15
36. Refer to Exhibit 10-6. At 95% confidence, the margin of error is
a.
1.694
b.
3.32
c.
1.96
d.
15
37. Refer to Exhibit 10-6. A 95% confidence interval estimate for the difference between the average
purchases of the customers using the two different credit cards is
a.
49 to 64
b.
11.68 to 18.32
c.
125 to 140
d.
8 to 10
Exhibit 10-7
In order to estimate the difference between the average hourly wages of employees of two branches of
a department store, the following data have been gathered.
Downtown Store
North Mall Store
Sample size
25
20
Sample mean
$9
$8
Sample standard deviation
$2
$1
38. Refer to Exhibit 10-7. A point estimate for the difference between the two sample means is
a.
1
b.
2
c.
3
d.
4
39. Refer to Exhibit 10-7. A 95% interval estimate for the difference between the two population means is
a.
0.078 to 1.922
b.
1.922 to 2.078
c.
1.09 to 4.078
d.
1.078 to 2.922
Exhibit 10-8
In order to determine whether or not there is a significant difference between the hourly wages of two
companies, the following data have been accumulated.
Company A
Company B
Sample size
80
60
Sample mean
$6.75
$6.25
Population standard deviation
$1.00
$0.95
40. Refer to Exhibit 10-8. A point estimate for the difference between the two sample means is
a.
20
b.
0.50
c.
0.25
d.
1.00
41. Refer to Exhibit 10-8. The test statistic is
a.
0.098
b.
1.645
c.
2.75
d.
3.01
42. Refer to Exhibit 10-8. The p-value is
a.
0.0013
b.
0.0026
c.
0.0042
d.
0.0084
43. Refer to Exhibit 10-8. The null hypothesis
a.
should be rejected
b.
should not be rejected
c.
should be revised
d.
None of these alternatives is correct.
Exhibit 10-9
Two major automobile manufacturers have produced compact cars with the same size engines. We are
interested in determining whether or not there is a significant difference in the MPG (miles per gallon)
of the two brands of automobiles. A random sample of eight cars from each manufacturer is selected,
and eight drivers are selected to drive each automobile for a specified distance. The following data
show the results of the test.
Driver
Manufacturer A
Manufacturer B
1
32
28
2
27
22
3
26
27
4
26
24
5
25
24
6
29
25
7
31
28
8
25
27
44. Refer to Exhibit 10-9. The mean for the differences is
a.
0.50
b.
1.5
c.
2.0
d.
2.5
45. Refer to Exhibit 10-9. The test statistic is
a.
1.645
b.
1.96
c.
2.096
d.
2.256
46. Refer to Exhibit 10-9. At 90% confidence the null hypothesis
a.
should not be rejected
b.
should be rejected
c.
should be revised
d.
None of these alternatives is correct.
Exhibit 10-10
The results of a recent poll on the preference of shoppers regarding two products are shown below.
Product
Shoppers Surveyed
Shoppers Favoring
This Product
A
800
560
B
900
612
47. Refer to Exhibit 10-10. The point estimate for the difference between the two population proportions
in favor of this product is
a.
52
b.
100
c.
0.44
d.
0.02
48. Refer to Exhibit 10-10. The standard error of is
a.
52
b.
0.044
c.
0.0225
d.
100
49. Refer to Exhibit 10-10. At 95% confidence, the margin of error is
a.
0.064
b.
0.044
c.
0.0225
d.
52
50. Refer to Exhibit 10-10. The 95% confidence interval estimate for the difference between the
populations favoring the products is
a.
-0.024 to 0.064
b.
0.6 to 0.7
c.
0.024 to 0.7
d.
0.02 to 0.3
Exhibit 10-11
An insurance company selected samples of clients under 18 years of age and over 18 and recorded the
number of accidents they had in the previous year. The results are shown below.
Under Age of 18
Over Age of 18
n1 = 500
n2 = 600
Number of accidents = 180
Number of accidents = 150
We are interested in determining if the accident proportions differ between the two age groups.
51. Refer to Exhibit 10-11 and let pu represent the proportion under and po the proportion over the age of
18. The null hypothesis is
a.
pu – po 0
b.
pu – po 0
c.
pu – po 0
d.
pu – po = 0
52. Refer to Exhibit 10-11. The pooled proportion is
a.
0.305
b.
0.300
c.
0.027
d.
0.450
53. Refer to Exhibit 10-11. The test statistic is
a.
0.96
b.
1.96
c.
2.96
d.
3.96
54. Refer to Exhibit 10-11. The p-value is
a.
less than 0.001
b.
more than 0.10
c.
0.0228
d.
0.3
Exhibit 10-12
The results of a recent poll on the preference of teenagers regarding the types of music they listen to
are shown below.
Music Type
Teenagers Surveyed
Teenagers Favoring
This Type
Pop
800
384
Rap
900
450
55. Refer to Exhibit 10-12. The point estimate for the difference between the proportions is
a.
-0.02
b.
0.048
c.
100
d.
66
56. Refer to Exhibit 10-12. The standard error of is
a.
0.48
b.
0.50
c.
0.03
d.
0.0243
57. Refer to Exhibit 10-12. The 95% confidence interval for the difference between the two proportions is
a.
384 to 450
b.
0.48 to 0.5
c.
0.028 to 0.068
d.
-0.068 to 0.028
Exhibit 10-13
In order to determine whether or not there is a significant difference between the hourly wages of two
companies, the following data have been accumulated.
Company 1
Company 2
n1 = 80
n2 = 60
= $10.80
= = $10.00
= $2.00
= $1.50
58. Refer to Exhibit 10-13. The null hypothesis for this test is
a.
1 – 2 0
b.
1 – 2 > 0
c.
1 – 2 < 0
d.
1 – 2 = 0
59. Refer to Exhibit 10-13. The point estimate of the difference between the means is
a.
b.
c.
d.
−
60. Refer to Exhibit 10-13. The test statistic has a value of
a.
b.
c.
d.
61. Refer to Exhibit 10-13. The p-value is
a.
b.
c.
d.
62. In testing the null hypothesis H0:
1 –
2 = 0, the computed test statistic is z = -1.66. The
corresponding p-value is
a.
.0485
b.
.0970
c.
.9515
d.
.9030
63. A company wants to identify which of two production methods has the smaller completion time. One
sample of workers is selected and each worker first uses one method and then uses the other method.
The sampling procedure being used to collect completion time data is based on
a.
cross samples
b.
pooled samples
c.
independent samples
d.
matched samples
PROBLEM
1. 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
b.
Since the range of the interval is from positive to positive, it indicates that there is conclusive
evidence (at 95%) that Model A has a larger mean.
2. The following sample information is given concerning the ACT scores of high school seniors form two
local schools.
School A
School B
= 14
= 15
= 25
= 23
= 16
= 10
Develop a 95% confidence interval estimate for the difference between the two populations.
3. 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 ()
$ 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 = .05.
a.
Z = 2.17
b.
c.
students at University B
4. 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 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.
a.
-7.92 to -0.08
b.
Yes, since the range of the interval is from a negative to a negative value, it indicates that
process B takes longer.
5. 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 ()
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.
6. 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.
a.
Test statistic t = 1.391
b.
7. Of 200 UTC seniors surveyed, 60 were planning on attending Graduate School. At UTK, 400 seniors
were surveyed and 100 indicated that they were planning to attend Graduate School.
b.
-2.77
a.
Determine a 95% confidence interval estimate for the difference between the proportion of
seniors at the two universities that were planning to attend Graduate School.
b.
Is there conclusive evidence to prove that the proportion of students from UTC who plan to go
to Graduate School is significantly more than those from UTK? Explain.
8. Of 300 female registered voters surveyed, 120 indicated they were planning to vote for the incumbent
president; while of 400 male registered voters, 140 indicated they were planning to vote for the
incumbent president.
a.
Compute the test statistic.
b.
At alpha = .05, test to see if there is a significant difference between the proportions of females
and males who plan to vote for the incumbent president. (Use the p-value approach.)
a.
Z = 1.35
b.
9. Of 150 Chattanooga residents surveyed, 60 indicated that they participated in a recycling program. In
Knoxville, 120 residents were surveyed and 36 claimed to recycle.
a.
Determine a 95% confidence interval estimate for the difference between the proportion of
residents recycling in the two cities.
b.
From your answer in Part a, is there sufficient evidence to conclude that there is a significant
difference in the proportion of residents participating in a recycling program?
a.
-0.0134 to 0.2134
b.
No, because the interval for the proportions ranges from a negative to a positive value.
10. 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.
ANS:
a.
-0.026 to 0.126
b.
No, the range of the interval is from a negative to a positive value.
11. 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
Standard deviation ()
$2.75
$3.00
Develop a 95% confidence interval estimate for the difference between the average fees of the two
offices.
12. 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 = .05.
128
b.
Test statistic t = -3.438
d.
13. Among a sample of 50 M.D.’s (medical doctors) in the city of Memphis, Tennessee, 10 indicated they
make house calls; while among a sample of 100 M.D.’s in Atlanta, Georgia, 18 said they make house
calls. Determine a 95% interval estimate for the difference between the proportion of doctors who
make house calls in the two cities.
14. Consider the following results for two samples randomly taken from two populations.
a.
-0.70 to 8.70
b.
No, because the range of the interval if from negative to positive.
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.
15. 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
2.974 to 9.026
16. 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
b.
3.31
c.
2.69 to 9.31
a.
60
b.
0.277 to 7.723