Exam 15: Inference About Population Variances

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.

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

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?

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.

Random samples from two normal populations produced the following statistics: $n _ { 1 } =$ 25, $s _ { 1 } ^ { 2 } =$ 75. $n _ { 2 } =$ 13, $S _ { 2 } ^ { 2 } =$ 130. Estimate with 95% confidence the ratio of the two population variances.

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?

Which of the following is the test statistic for σ²?

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.

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

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.

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

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:

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.

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.

Random samples from two normal populations produced the following statistics: $n _ { 1 } =$ 25, $s _ { 1 } ^ { 2 } =$ 75. $n _ { 2 } =$ 13, $S _ { 2 } ^ { 2 } =$ 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 = ( $s _ { 1 } ^ { 2 } /$ $S _ { 2 } ^ { 2 }$ ) / F_0.025,24,12 = 0.191. UCL = ( $s _ { 1 } ^ { 2 } /$ $S _ { 2 } ^ { 2 }$ )F_0.025,12,24 = 1.465.

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

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?

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?

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.

Random samples from two normal populations produced the following statistics: $n _ { 1 } =$ 10, $s _ { 1 } ^ { 2 } =$ 40. $n _ { 2 } =$ 15, $S _ { 2 } ^ { 2 } =$ 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?