Question

Performing a Two-Sample F-Test In Exercises 19–26, (a) identify the claim and state H0 and Ha, (b) find the critical value and identify the rejection region, (c) find the test statistic F, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.

Heart Transplant Waiting Times The table at the left shows a sample of the waiting times (in days) for a heart transplant for two age groups. At α=0.05, can you conclude that the variances of the waiting times differ between the two age groups? (Adapted from Organ Procurement and Transplantation Network)

Accepted Answer

Step 1: Identify the claim and state the hypotheses. The claim is that the variances of the waiting times differ between the two age groups (18–34 and 35–49). Therefore, the null hypothesis (H0) is that the variances are equal: \$$sigma_1^2$$ = \$$sigma_2^2$$, and the alternative hypothesis (Ha) is that the variances are not equal: \$$sigma_1^2$$ 
eq \$$sigma_2^2. $$Step 2: Calculate the sample variances for each age group. For each group, find the mean of the waiting times, then compute the variance using the formula  $$s^2$$ = \frac{1}{n-1} \sum (x_i - \bar{x}$$)^2$$ , where n is the sample size, x_i are the data points, and \bar{x} is the sample mean. Step 3: Determine the critical value and rejection region. Since this is a two-tailed F-test at significance level \alpha = 0.05, find the critical values from the F-distribution table using degrees of freedom $$df_1$$ = $$n_1 - 1 $$and $$df_2$$ = $$n_2 - 1 $$for the two samples. The rejection region consists of values of the test statistic less than the lower critical value or greater than the upper critical value. Step 4: Calculate the test statistic F. The test statistic is the ratio of the larger sample variance to the smaller sample variance:  F = \frac{s_{larger}^2}{s_{smaller}^2} . This ensures the test statistic is always greater than or equal to 1. Step 5: Make a decision and interpret the result. Compare the calculated F statistic to the critical values. If the test statistic falls into the rejection region, reject the null hypothesis; otherwise, fail to reject it. Then, interpret this decision in the context of the original claim about whether the variances of waiting times differ between the two age groups.

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주요 개념

Two-Sample F-Test for Variances

Hypothesis Testing Framework

Assumptions of the F-Test