A sample of 25 concession stand purchases at the October 22 matinee of Bride of Chucky showed
a mean purchase of $5.29 with a standard deviation of $3.02. For the October 26 evening showing
of the same movie, for a sample of 25 purchases the mean was $5.12 with a standard deviation of
$2.14. The means appear to be very close, but not the variances. At á = .05, is there a difference
in variances? Show all steps clearly, including an illustration of the decision rule. (Data are from
a project by statistics students Kim Dyer, Amy Pease, and Lyndsey Smith.)
To test if there is a difference in variances between the two samples, we can use the F-test for variances. The F-test compares the variances of the two samples to determine if they are statistically different.
Here are the steps to perform the F-test for variances:
Step 1: State the null and alternative hypotheses.
- Null hypothesis (H0): The variances of the two samples are equal.
- Alternative hypothesis (Ha): The variances of the two samples are not equal.
Step 2: Set the significance level (α).
- In this case, α is given as 0.05.
Step 3: Calculate the F-statistic.
- The F-statistic is calculated as the ratio of the larger sample variance to the smaller sample variance.
- F = S1^2 / S2^2, where S1^2 is the larger sample variance and S2^2 is the smaller sample variance.
Step 4: Determine the critical value.
- The critical value for the F-statistic is obtained from the F-distribution table.
- The degrees of freedom for the two samples are n1-1 and n2-1, where n1 and n2 are the sample sizes.
- In this case, both samples have the same size of 25, so the degrees of freedom are 24 for each sample.
- Based on the degrees of freedom and the significance level, find the critical value.
Step 5: Compare the calculated F-statistic with the critical value.
- If the calculated F-statistic is greater than the critical value, reject the null hypothesis and conclude that there is a difference in variances.
- If the calculated F-statistic is less than or equal to the critical value, do not reject the null hypothesis and conclude that there is no significant difference in variances.
Illustration of the decision rule:
- Draw an F-distribution curve with the critical values marked for the given significance level (α = 0.05) and the degrees of freedom for both samples.
- Shade the rejection region in the F-distribution curve based on the calculated F-statistic and the critical value.
Performing the calculations and comparing the F-statistic with the critical value will determine whether there is a difference in variances between the two samples.