Chapter 14—Hypothesis testing: Comparing two populations
MULTIPLE CHOICE
1. In testing the difference between two population means, using two independent samples, we use the
pooled variance in estimating the standard error of the sampling distribution of the sample mean
difference if the:
A.
sample sizes are both large.
B.
populations are normally distributed.
C.
populations have equal variances.
D.
populations are normal with equal variances.
2. In testing whether the means of two normal populations are equal, summary statistics computed for
two independent samples are as follows:
n1 = 25, = 7.30, s1 = 1.05.
n2 = 30, = 6.80, s2 = 1.20.
Assume that the population variances are equal. Then the standard error of the sampling distribution of
the sample mean difference is equal to:
A.
0.3072.
B.
0.0917.
C.
0.3028.
D.
0.0944.
3. In testing the difference between two population means, using two independent samples, the sampling
distribution of the sample mean difference is normal if the:
A.
sample sizes are both greater than 30.
B.
populations are normal.
C.
populations are non-normal and the sample sizes are large.
D.
All of the above are required conditions.
4. A political analyst in Perth surveys a random sample of Labor Party members and compares the results
with those obtained from a random sample of Liberal Party members. This would be an example of:
A.
independent samples.
B.
dependent samples.
C.
independent samples only if the sample sizes are equal.
D.
dependent samples only if the sample sizes are equal.
5. In testing the difference between the means of two normal populations with known population
standard deviations the test statistic calculated from two independent random samples equals 2.56. If
the test is two-tailed and the 1% level of significance has been specified, the conclusion should be:
A.
to reject the null hypothesis.
B.
not to reject the null hypothesis.
C.
the test is inconclusive.
D.
None of the above answers is correct.
6. In testing the difference between two population means, for which the population variances are
unknown and are not assumed to be equal, two independent samples of large sizes are drawn from the
populations. Which of the following tests is appropriate?
A.
Z-test.
B.
Pooled-variances t-test.
C.
Unequal variances t–test.
D.
Matched pairs t–test.
7. Which of the following is a required condition for using the normal approximation to the binomial
distribution in testing the difference between two population proportions?
A.
n1p1 > 5 and n2p2 > 5.
B.
n2(1 – p2) > 5.
C.
n1p1 > 5, n1(1 – p1) > 5, n2p2 > 5 and n2(1 – p2) > 5.
D.
n1(1 – p1) > 5 and n2(1 – p2) > 5.
8. A test is being conducted to test the difference between two population means, using data that are
gathered from a matched pairs experiment. If the paired differences are normal, then the distribution
used for testing is the:
A.
normal.
B.
binomial.
C.
Student t.
D.
F.
9. In testing the difference between the means of two normal populations, using two independent
samples, when the population variances are unknown and unequal, the sampling distribution of the
resulting statistic is:
A.
normal.
B.
Student t.
C.
approximately normal.
D.
approximately Student t.
10. When the necessary conditions are met, a two-tail test is being conducted to test the difference
between two population means, but your statistical software provides only a one-tail area of 0.028 as
part of its output. The p–value for this test will be:
A.
0.972.
B.
0.05.
C.
0.056.
D.
0.014.
11. When the necessary conditions are met, a two-tail test is being conducted to test the difference
between two population proportions. The two sample proportions are p1 = 0.25 and p2 = 0.20 and the
standard error of the sampling distribution of is 0.04. The calculated value of the test
statistic will be:
A.
z = 0.25.
B.
z = 1.25.
C.
t = 0.25.
D.
t = 0.80.
12. We can design a matched pairs experiment when the data collected are:
A.
observational.
B.
experimental.
C.
controlled.
D.
All the above are correct.
13. A sample of size 100 selected from one population has 53 successes, and a sample of size 150 selected
from a second population has 95 successes. The test statistic for testing the equality of the population
proportions is equal to:
A.
–11.1051.
B.
–17.1107.
C.
–0.0944.
D.
–1.0944.
14. For testing the difference between two population proportions, the pooled proportion estimate should
be used to compute the value of the test statistic when the:
A.
populations are normally distributed.
B.
sample sizes are small.
C.
samples are independently drawn from the populations.
D.
null hypothesis states that the two population proportions are equal.
15. In testing the null hypothesis H0 = = 0, if H0 is false, the test could lead to:
A.
a Type I error.
B.
a Type II error.
C.
either a Type I or a Type II error.
D.
none of the above.
16. A sample of size 150 from population 1 has 40 successes. A sample of size 250 from population 2 has
30 successes. The value of the test statistic for testing the null hypothesis that the proportion of
successes in population 1 exceeds the proportion of successes in population 2 by 0.05 is:
A.
1.645.
B.
2.327.
C.
1.960.
D.
1.977.
17. Two samples of sizes 25 and 35 are independently drawn from two normal populations, where the
unknown population variances are assumed to be equal. The number of degrees of freedom of the
equal-variances t-test statistic are:
A.
60.
B.
59.
C.
58.
D.
35.
18. If some natural relationship exists between each pair of observations that provides a logical reason to
compare the first observation of sample 1 with the first observation of sample 2, the second
observation of sample 1 with the second observation of sample 2, and so on, the samples are referred
to as:
A.
matched samples.
B.
independent samples.
C.
weighted samples.
D.
random samples.
19. In testing the difference between the means of two normally distributed populations, the number of
degrees of freedom associated with the unequal-variances t-test statistic usually results in a non-integer
number. It is recommended that you:
A.
round up to the nearest integer.
B.
round down to the nearest integer.
C.
change the sample sizes until the number of degrees of freedom becomes an integer.
D.
assume that the population variances are equal, and then use df = n1 + n2 – 2.
20. The symbol refers to:
A.
the difference in the means of two dependent populations.
B.
the difference in the means of two independent populations.
C.
the matched pairs differences.
D.
the mean difference in the pairs of observations taken from two dependent samples.
21. The number of degrees of freedom associated with the t-test, when the data are gathered from a
matched pairs experiment with 30 pairs, is:
A.
58.
B.
29.
C.
14.
D.
13.
22. Two independent samples of sizes 20 and 30 are randomly selected from two normally distributed
populations. Assume that the population variances are unknown but equal. In order to test the
difference between the population means, 1 – 2, the sampling distribution of the sample mean
difference, , is:
A.
normally distributed.
B.
t-distributed with 50 degrees of freedom.
C.
t-distributed with 48 degrees of freedom.
D.
F-distributed with 19 and 29 degrees of freedom.
23. Two independent samples of sizes 40 and 50 are randomly selected from two populations to test the
difference between the population means, 2 – 1. The sampling distribution of the sample mean
difference is:
A.
normally distributed.
B.
approximately normal.
C.
t-distributed with 88 degrees of freedom.
D.
chi-squared distributed with 90 degrees of freedom.
24. Two independent samples of sizes 25 and 35 are randomly selected from two normal populations with
equal variances. In order to test the difference between the population means, the test statistic is:
A.
a standard normal random variable.
B.
an approximately standard normal random variable.
C.
Student t-distributed with 58 degrees of freedom.
D.
Student t-distributed with 33 degrees of freedom.
25. When the necessary conditions are met, a two-tail test is being conducted to test the difference
between two population proportions. If the value of the test statistic z is 2.05, then the p-value is:
A.
0.4798.
B.
0.0404.
C.
0.2399.
D.
0.0202.
26. A sample of size 100 selected from one population has 60 successes, and a sample of size 150 selected
from a second population has 95 successes. The test statistic for testing the equality of the population
proportions is equal to:
A.
–0.5319.
B.
0.7293.
C.
–0.419.
D.
0.2702.
27. For testing the difference between two population proportions, the pooled proportion estimate should
be used to compute the value of the test statistic when the:
A.
populations are normally distributed.
B.
sample sizes are small.
C.
samples are independently drawn from the populations.
D.
null hypothesis states that the two population proportions are equal.
SHORT ANSWER
1. In random samples of 25 and 22 from each of two normal populations, we find the following statistics:
x1-bar = 56, s1 = 8.
x2-bar = 62, s2 = 8.5.
Assume that the population variances are equal.
Test with = 0.05 to determine whether we can infer that the population means differ.
2. In random samples of 25 and 22 from each of two normal populations, we find the following statistics:
x1-bar = 56, s1 = 8.
x2-bar = 62, s2 = 8.5.
Assume that the population variances are equal.
Estimate with 95% confidence the difference between the two population means.
3. In testing the hypotheses:
H0 : 1 – 2 = 13.
H1 : 1 – 2 < 13.
two random samples from two normal populations produced the following results:
n1 = 40, x1-bar = 21, s1 = 5.
n2 = 32, x2-bar = 18, s2 = 6.
Assume that the population variances are equal.
What conclusion can we draw at the 1% significance level?
4. In testing the hypotheses
H0: p1 – p2 = 0
H1: p1 – p2 < 0,
we find the following statistics:
n1 = 400, x1 = 105.
n2 = 500, x2 = 140.
What conclusion can we draw at the 10% significance level?
5. In testing the hypotheses
H0: p1 – p2 = 0
H1: p1 – p2 < 0,
we find the following statistics:
n1 = 400, x1 = 105.
n2 = 500, x2 = 140.
Estimate with 90% confidence the difference between the two population proportions.
6. In testing the hypotheses:
H0: D = 5
H1: D 5,
two random samples from two normal populations produced the following statistics:
nD = 36, xD = 7.8, sD = 7.5.
What conclusion can we draw at the 5% significance level?
7. Test the following hypotheses at the 5% level of significance:
H0: 1 – 2 = 0
H1: 1 – 2 < 0,
given the following statistics:
n1 = 10, x1 = 58.6, s1 = 13.45.
n2 = 10, x2 = 64.6, s2 = 11.15.
Estimate with 95% confidence the difference between the two population means.
8. In testing the hypotheses:
H0: p1 – p2 = 0.10
H1: p1 – p2 > 0.10,
we found the following statistics:
n1 = 350, x1 = 178.
n2 = 250, x2 = 112.
What conclusion can we draw at the 10% significance level?
9. In testing the hypotheses:
H0: p1 – p2 = 0.10
H1: p1 – p2 > 0.10,
we found the following statistics:
n1 = 350, x1 = 178.
n2 = 250, x2 = 112.
a. What is the p-value of the test?
b. Estimate with 90% confidence the difference between the two population proportions.
10. In testing the hypotheses:
H0: 1 – 2 = 0
H1: 1 – 2 0,
two random samples from two normal populations produced the following statistics:
n1 = 51, x1-bar = 35, s1 = 28.
n2 = 40, x2-bar = 28, s2 = 10.
Assume that the two population variances are different.
What conclusion can we draw at the 5% significance level?
11. In testing the hypotheses:
H0: 1 – 2 = 0
H1: 1 – 2 0,
two random samples from two normal populations produced the following statistics:
n1 = 51, x1-bar = 35, s1 = 28.
n2 = 40, x2-bar = 28, s2 = 10.
Assume that the two population variances are different.
a. Estimate with 95% confidence the difference between the two population means.
b. Explain how to use this confidence interval for testing the hypotheses.
12. The following data were generated from a matched pairs experiment:
Pair:
1
2
3
4
5
6
7
Sample 1:
8
15
7
9
10
13
11
Sample 2:
12
18
8
9
12
11
10
Determine whether these data are sufficient to infer at the 10% significance level that the two
population means differ.
13. The following data were generated from a matched pairs experiment:
Pair:
1
2
3
4
5
6
7
Sample 1:
8
15
7
9
10
13
11
Sample 2:
12
18
8
9
12
11
10
a. Estimate with 90% confidence the mean difference.
b. Briefly describe what the interval estimate in part a. tells you, and explain how to use it to test the
hypotheses.
14. In testing the hypotheses:
H0: p1 – p2 = 0.10
H1 : p1 – p2 0.10,
we find the following statistics:
n1 = 150, x1 = 72.
n2 = 175, x2 = 70.
What conclusion can we draw at the 5% significance level?
15. In testing the hypotheses:
H0: p1 – p2 = 0.10
H1 : p1 – p2 0.10,
we find the following statistics:
n1 = 150, x1 = 72.
n2 = 175, x2 = 70.
a. What is the p-value of the test?
b. Briefly explain how to use the p-value to test the hypotheses.
16. In testing the hypotheses:
H0: p1 – p2 = 0.10
H1 : p1 – p2 0.10,
we find the following statistics:
n1 = 150, x1 = 72.
n2 = 175, x2 = 70.
a. Estimate with 95% confidence the difference between the two population proportions.
b. Explain how to use the confidence interval in part a. to test the hypotheses.
17. In testing the hypotheses:
H0: p1 – p2 = 0
H1: p1 – p2 > 0,
we find the following statistics:
n1 = 200, x1 = 80.
n2 = 200, x2 = 140.
What conclusion can we draw at the 5% significance level?
18. In testing the hypotheses:
H0: p1 – p2 = 0
H1: p1 – p2 > 0,
we find the following statistics:
n1 = 200, x1 = 80.
n2 = 200, x2 = 140.
a. What is the p-value of the test?
b. Estimate with 95% confidence the difference between the two population proportions.
19. In order to test the hypotheses:
H0: 1 – 2 = 0
H1: 1 – 2 0,
we independently draw a random sample of 18 observations from a normal population with standard
deviation of 15, and another random sample of 12 from a second normal population with standard
deviation of 25.
If we set the level of significance at 10%, determine the power of the test when 1 – 2 = 5.
20. In order to test the hypotheses:
H0: 1 – 2 = 0
H1: 1 – 2 0,
we independently draw a random sample of 18 observations from a normal population with standard
deviation of 15, and another random sample of 12 from a second normal population with standard
deviation of 25.
a. If we set the level of significance at 5%, determine the power of the test when 1 – 2 = 5.
b. Describe the effect of reducing the level of significance on the power of the test.
21. A simple random sample of ten firms was asked how much money (in thousands of dollars) they spent
on employee training programs this year and how much they plan to spend on these programs next
year. The data are shown below.
1
2
3
4
5
6
7
8
9
10
25
31
12
15
21
36
18
5
9
17
21
30
18
20
22
36
20
10
8
15
Assume that the populations of amount spent on employee training programs are normally distributed.
Can we infer at the 5% significance level that more money will be spent on employee training
programs next year than this year?
22. A simple random sample of ten firms was asked how much money (in thousands of dollars) they spent
on employee training programs this year and how much they plan to spend on these programs next
year. The data are shown below.
1
2
3
4
5
6
7
8
9
10
25
31
12
15
21
36
18
5
9
17
21
30
18
20
22
36
20
10
8
15
Assume that the populations of amount spent on employee training programs are normally distributed.
a. Estimate with 95% confidence the mean difference.
b. Briefly explain what the interval estimate in part a. tells you.
ANS: