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Chapter 15—Inference about population variances
MULTIPLE CHOICE
1. Which of the following statements is false?
A.
The chi-squared distribution is positively skewed.
B.
The chi-squared distribution is symmetrical.
C.
All the values of the chi-squared distribution are positive.
D.
The shape of the chi-squared distribution depends on the number of degrees of freedom.
2. Which of the following statements is correct regarding the percentile points of the chi-squared
distribution?
A.
20.99,12 = 26.2170
B.
20.95,12 = 0.102587.
C.
20.95,12 = 28.2995.
D.
20.99,12 = 3.57056.
3. Two independent samples are drawn from two normal populations, where the population variances are
assumed to be equal. The sampling distribution of the ratio of the two sample variances is:
A.
a normal distribution.
B.
Student t-distribution.
C.
an F-distribution.
D.
a chi-squared distribution.
4. The ratio of two independent chi-squared variables, each divided by its number of degrees of freedom,
is:
A.
normally distributed.
B.
Student t distributed.
C.
chi-squared distributed.
D.
F distributed.
5. The F-distribution is the sampling distribution of the ratio of:
A.
two normal population variances.
B.
two normal population means.
C.
two sample variances, provided that the samples are independently drawn from two
normal populations.
D.
two sample variances, provided that the sample sizes are large.
6. Which of the following statements is not correct for an F-distribution?
A.
Variables that are F-distributed range from 0 to .
B.
The exact shape of the distribution is determined by two numbers of degrees of freedom.
C.
The number of degrees of freedom for the denominator is always smaller than the number
of degrees of freedom for the numerator.
D.
The number of degrees of freedom for the numerator can be larger than, smaller than, or
equal to the number of degrees of freedom for the denominator.
7. The sampling distribution of the ratio of two sample variances
1
s
/
2
s
is said to be F-distributed
provided that:
A.
the samples are independent.
B.
the populations are normal with equal variances.
C.
the samples are dependent and their sizes are large.
D.
the samples are independently drawn from two normal populations.
8. Which of the following statements is correct regarding the percentile points of the F-distribution?
A.
0.05,10,20 0.95,10,20
1/F F=
.
B.
0.05,10,20 0.05,20,10
1/F F=
.
C.
0.95,10,20 0.95,20,10
1/F F=
.
D.
0.95,10,20 0.05,20,10
1/F F=
.
9. In testing for the equality of two population variances, when the populations are normally distributed,
the 10% level of significance has been used. To determine the rejection region, it will be necessary to
refer to the F table corresponding to an upper-tail area of:
A.
0.90.
B.
0.05.
C.
0.20.
D.
0.10.
10. In constructing a 95% interval estimate for the ratio of two population variances,
2
1
/
2
2
, two
independent samples of sizes 41 and 61 are drawn from the populations. If the sample variances are
515 and 920, then the upper confidence limit is:
A.
0.321.
B.
1.009.
C.
0.311.
D.
0.974.
TRUE/FALSE
1. The value of with degrees of freedom, such that the area to its left under the chi-squared curve
is equal to A, is denoted by , while denotes the value such that the area to its right is
A.
2. We define as the value of the F with and degrees of freedom such that the area to
its right under the F curve is A, while is defined as the value such that the area to its
left is A.
3. The value in an F-distribution with and degrees of freedom such that the area to
its left is 0.95 is 4.74.
4. The value in an F-distribution with and degrees of freedom such that the area to its
left is 0.975 is 5.05.
5. The value in an F-distribution with and degrees of freedom such that the area to its
left is 0.99 is 0.036.
6. The value in an F-distribution with and degrees of freedom such that the area to its
right is 0.05 is 3.37.
7. The value in a chi-squared distribution with 5 degrees of freedom such that the area to its right is 0.10
is 1.61031.
8. The value in a chi-squared distribution with 6 degrees of freedom such that the area to its left is 0.05 is
12.5916.
9. The value in a chi-squared distribution with 8 degrees of freedom such that the area to its left is 0.95 is
15.5073.
10. The value in a chi-squared distribution with 4 degrees of freedom such that the area to its right is 0.99
is 0.29711.
11. To find the value in a chi-squared distribution with 10 degrees of freedom such that the area to its left
is 0.01, we find the point in the same distribution such that the area to its right is 0.99.
12. When comparing two population variances, we use the ratio
2 2
1 2
/
rather than the difference
2 2
1 2
−
.
13. The test statistic employed to test
2 2
0 1 2 1: /H
=
is
2 2
1 2
/F s s=
, which is F-distributed with
1 1 2 2
1 and 1n n
= − = −
degrees of freedom, provided that the two populations are F-distributed.
14. When the necessary conditions are met, a two-tail test is being conducted at
= 0.05 to
test
2 2
0 1 2 1: /H
=
. The two sample variances are
2 2
1 2
400 and 800ss= =
, and the sample sizes are
1 2
25 and 25nn= =
. The calculated value of the test statistic will be F = 2.
15. When the necessary conditions are met, a two-tail test is being conducted at
= 0.05 to
test
2 2
0 1 2 1: /H
=
. The two sample variances are
2 2
1 2
500 and 900ss= =
, and the sample sizes are
1 2
21 and 31nn= =
. The rejection region is F > 2.20 or F < 0.4255.
SHORT ANSWER
1. Random samples from two normal populations produced the following statistics:
=
1
n
10,
=
2
1
s
30.
=
2
n
15,
=
2
2
s
20.
Is there enough evidence at the 1% significance level to infer that the variance of population 1 is larger
than the variance of population 2?
Random samples from two normal populations produced the following statistics:
=
1
n
25,
=
2
1
s
75.
=
2
n
13,
=
2
2
s
130.
Use this information to answer the following question(s).
2. Is there enough evidence at the 5% significance level to infer that the two population variances differ?
3. Estimate with 95% confidence the ratio of the two population variances.
4. Briefly describe what the interval estimate in the previous question tells you.
5. Briefly explain how to use the interval estimate in the previous question to test the hypothesis of equal
population variances.
6. A statistician wants to test for the equality of means in two independent samples drawn from normal
populations. However, he will not perform the equal-variance t-test of the difference between the
population means if the condition necessary for its use is not satisfied. The data are as follows:
Sample 1:
7
9
6
15
7
10
8
12
Sample 2:
2
25
9
15
10
18
5
22
27
3
Given the data above, can the statistician conclude at the 5% significance level that the required
condition is not satisfied?
7. A statistician wants to test for the equality of means in two independent samples drawn from normal
populations. However, he will not perform the equal-variance t-test of the difference between the
population means if the condition necessary for its use is not satisfied. The data are as follows:
Sample 1:
7
9
6
15
7
10
8
12
Sample 2:
2
25
9
15
10
18
5
22
27
3
Estimate with 95% confidence the ratio of the two population variances.
8. A statistician wants to test for the equality of means in two independent samples drawn from normal
populations. However, he will not perform the equal-variance t-test of the difference between the
population means if the condition necessary for its use is not satisfied. The data are as follows:
Sample 1:
7
9
6
15
7
10
8
12
Sample 2:
2
25
9
15
10
18
5
22
27
3
Briefly describe what the interval estimate in the previous question tells you.
9. An investor is considering two types of investment. She is quite satisfied that the expected return on
investment 1 is higher than the expected return on investment 2. However, she is quite concerned that
the risk associated with investment 1 is higher than that of investment 2. To help make her decision,
she randomly selects seven monthly returns on investment 1 and 10 monthly returns on investment 2.
She finds that the sample variances of investments 1 and 2 are 225 and 118, respectively.
Can she infer at the 5% significance level that the population variance of investment 1 exceeds that of
investment 2?
10. An investor is considering two types of investment. She is quite satisfied that the expected return on
investment 1 is higher than the expected return on investment 2. However, she is quite concerned that
the risk associated with investment 1 is higher than that of investment 2. To help make her decision,
she randomly selects seven monthly returns on investment 1 and ten monthly returns on investment 2.
She finds that the sample variances of investments 1 and 2 are 225 and 118, respectively.
Estimate with 95% confidence the ratio of the two population variances, and briefly describe what the
interval estimate tells you.
11. In a random sample of 20 patients who visited the emergency room of hospital 1, a researcher found
that the variance of the waiting time (in minutes) was 128.0. In a random sample of 15 patients in the
emergency room of hospital 2, the researcher found the variance to be 178.8.
Can we infer at the 5% level of significance that the population variances differ?
12. In a random sample of 20 patients who visited the emergency room of hospital 1, a researcher found
that the variance of the waiting time (in minutes) was 128.0. In a random sample of 15 patients in the
emergency room of hospital 2, the researcher found the variance to be 178.8.
Estimate with 95% confidence the ratio of the two population variances.
13. In a random sample of 20 patients who visited the emergency room of hospital 1, a researcher found
that the variance of the waiting time (in minutes) was 128.0. In a random sample of 15 patients in the
emergency room of hospital 2, the researcher found the variance to be 178.8.
Briefly describe what the interval estimate in the previous question tells you.
