The image presents a statistical problem that involves testing for differences in variance between two classes (Class A and Class B). The questions are as follows:

Problem 1

A researcher wants to determine if there is a difference in the study time variance between two classes. Given are:

• Class A times (hours): 5, 7, 9, 10, 6
• Class B times (hours): 8, 9, 7, 5, 8, 10

The question requires performing an F-test at a significance level of $\alpha = 5\%$ to determine if there is a significant difference in variance.

Problem 2

A researcher wants to know if there is a difference in variance of test scores between two classes, given the following scores:

• Class A scores: 70, 75, 80, 65, 85, 72, 77, 68, 84, 79, 73, 76, 82, 69, 78, 81, 71, 83
• Class B scores: 88, 85, 80, 90, 86, 84, 82, 87, 89, 85, 83, 88, 81, 90, 86, 85, 84, 89

With $\alpha = 5\%$, the researcher needs to determine if there is a difference in the variance between the two classes.

Solution Approach

For both questions, we can use the F-test for variance comparison. Here's how:

1. Calculate the variance of each dataset (Class A and Class B).
2. Determine the F-ratio by dividing the larger variance by the smaller variance.
3. Compare the F-ratio to the critical F-value at the given significance level (5% or 0.05) based on the degrees of freedom for each class.
4. Interpret the results: If the calculated F-ratio is greater than the critical F-value, there is a significant difference in variance.

Would you like me to go through the calculations for each problem, including finding variances, F-ratios, and determining if there is a significant difference?

Further Questions

1. How is the F-test critical value determined, and how does it depend on sample sizes?
2. How does increasing or decreasing the sample size impact the F-test results?
3. What are the assumptions for using the F-test in variance comparison?
4. How would we proceed if the data did not meet the normality assumption required for the F-test?
5. Can an F-test determine differences in both variance and mean simultaneously?

Tip

When comparing variances using the F-test, always arrange the variances so that the larger is in the numerator to simplify interpretation.