Deck 15: Inference About Population Variances

Question 1

Which of the following best describes the sampling distribution of s₁²/s₂² , if we have independently sampled from two normal populations?

Answer 1

D

Question 2

In constructing a 95% interval estimate for the ratio of two population variances, / , 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:

Answer 2

B

Question 3

Which of the following best describes the Chi-square distribution?

Answer 3

D

Question 4

The sampling distribution of the ratio of two sample variances / is said to be F-distributed provided that:

Question 5

Which of the following statements is correct regarding the percentile points of the chi-squared distribution?

Question 6

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:

Question 7

Which of the following statements is false? $\begin{array}{|l|l|}\hline A.&\text {The chi-squared distribution is positively skewed. }\\\hline B.&\text { The chi-squared distribution is symmetrical.}\\\hline C.&\text {All the values of the chi-squared distribution are positive. }\\\hline D.&\text {The shape of the chi-squared distribution depends on the number of degrees of freedom. }\\\hline \end{array}$

Question 8

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:

Question 9

Which of the following statements is correct regarding the percentile points of the F-distribution?

Question 10

The ratio of two independent chi-squared variables, each divided by its number of degrees of freedom, is:

Question 11

The value in an F-distribution with $v _ { 1 } = 5$ and $v _ { 2 } = 10$ degrees of freedom such that the area to its left is 0.95 is 4.74.

Question 12

Which of the following statements is not correct for an F-distribution?

Question 13

Which of the following is the most common null hypothesis used when testing for equality of two population variances?

Question 14

A study wants to investigate whether the population variance is greater than 8, if a random sample of size 40, yielded a variance of 10. Which of the following are the correct null hypothesis and alternative hypotheses?

Question 15

Random samples from two normal populations produced the following statistics: s₁ = 3 n₁=30
S₂ = 4 n₂=30
What is the value of the test statistic if we wanted to test the hypothesis that the two populations differ?

Question 16

Which of the following is the test statistic for σ²?

Question 17

A study wants to investigate whether the population variance is greater than 8, if a random sample of size 40, yielded a variance of 10. Which of the following is the correct value of the test statistic?

Question 18

How many degrees of freedom are used for an F statistic?

Question 19

The F-distribution is the sampling distribution of the ratio of:

Question 20

Which of the following best describes the number of degrees of freedom used in a Chi-square test for a value of the population variance?

Question 21

Random samples from two normal populations produced the following statistics: 10, 40. 15, 20.
Is there enough evidence at the 5% significance level to infer that the variance of population 1 is larger than the variance of population 2?

Question 22

The value in an F-distribution with $v _ { 1 } = 4$ and $v _ { 2 } = 8$ degrees of freedom such that the area to its left is 0.975 is 5.05.

Question 23

The test statistic employed to test $H _ { 0 } : \sigma _ { 1 } ^ { 2 } / \sigma _ { 2 } ^ { 2 } = 1$ is $F = s _ { 1 } ^ { 2 } / s _ { 2 } ^ { 2 }$ , which is F-distributed with $v _ { 1 } = n _ { 1 } - 1 \text { and } v _ { 2 } = n _ { 2 } - 1$ degrees of freedom, provided that the two populations are F-distributed.

Question 24

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.

Question 25

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.

Question 26

When the necessary conditions are met, a two-tail test is being conducted at $\alpha$ = 0.05 to test $H _ { 0 } : \sigma _ { 1 } ^ { 2 } / \sigma _ { 2 } ^ { 2 } = 1$ . The two sample variances are $s _ { 1 } ^ { 2 } = 400 \text { and } s _ { 2 } ^ { 2 } = 800$ , and the sample sizes are $n _ { 1 } = 25 \text { and } n _ { 2 } = 25$ . The calculated value of the test statistic will be F = 2.

Question 27

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: Estimate with 95% confidence the ratio of the two population variances.

Question 28

When comparing two population variances, we use the difference $\sigma _ { 1 } ^ { 2 } - \sigma _ { 2 } ^ { 2 }$ rather than the ratio $\sigma _ { 1 } ^ { 2 } / \sigma _ { 2 } ^ { 2 }$ .

Question 29

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.

Question 30

The value in an F-distribution with $v _ { 1 } = 3$ and $v _ { 2 } = 7$ degrees of freedom such that the area to its left is 0.99 is 0.036.

Question 31

When the necessary conditions are met, a two-tail test is being conducted at $\alpha$ = 0.05 to test $H _ { 0 } : \sigma _ { 1 } ^ { 2 } / \sigma _ { 2 } ^ { 2 } = 1$ . The two sample variances are $s _ { 1 } ^ { 2 } = 500 \text { and } s _ { 2 } ^ { 2 } = 900$ , and the sample sizes are $n _ { 1 } = 21 \text { and } n _ { 2 } = 31$ . The rejection region is F > 2.20 or F < 0.4255.

Question 32

The value in an F-distribution with $v _ { 1 } = 6$ and $v _ { 2 } = 9$ degrees of freedom such that the area to its right is 0.05 is 3.37.

Question 33

Random samples from two normal populations produced the following statistics: 25, 75. 13, 130.
Briefly explain how to use the 95% confidence the ratio of the two population variances to test the hypothesis of equal population variances.
LCL = ( ) / F_0.025,24,12 = 0.191.
UCL = ( )F_0.025,12,24 = 1.465.

Question 34

Random samples from two normal populations produced the following statistics: 25, 75. 13, 130.
Estimate with 95% confidence the ratio of the two population variances.

Question 35

Random samples from two normal populations produced the following statistics: 25, 75. 13, 130.
Briefly describe the 95% confidence the ratio of the two population variances:
LCL = ( ) / F_0.025,24,12 = 0.191.
UCL = ( )F_0.025,12,24 = 1.465.

Question 36

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: Briefly describe what the interval estimate in the previous question tells you.

Question 37

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 left is 0.99.

Question 38

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.

Question 39

A university lecturer wants to investigate if the variance of final marks of students in two of her courses differs. She takes a random sample of 25 students from the mathematics course she lectures and finds the student's final marks had a variance of 5. She takes a random sample of 13 students from the statistics course she lectures and finds the variance of 10. Assuming that the final grades of students in her mathematics and in her statistics course are normally distributed, is there enough evidence at the 5% significance level for this lecturer to infer that the two population variances differ?

Question 40

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: Given the data above, can the statistician conclude at the 5% significance level that the required condition is not satisfied?

Question 41

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.

Question 42

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?

Question 43

What are the rejection regions for each of the following sets of hypotheses?

Question 44

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?

Question 45

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.

Question 46

For each of the following hypothesis tests, state for what values of χ² we would reject Ho.
a. Ho: σ₁² = 10
HA: σ₁² ≠ 10
α = 0.10 and n = 50
b. Ho: σ₁² = 4
HA: σ₁² > 4
α = 0.05 and n = 30
c. Ho: σ₁² = 0.50
HA: σ₁² < 0.50
α = 0.01 and n = 45

Question 47

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.