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)

Verified step by step guidance

1. 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: $\mathrm{\backslash (\backslash sigma\backslash )\_1^2}=\mathrm{\backslash (\backslash sigma\backslash )\_2^2}$, and the alternative hypothesis (Ha) is that the variances are not equal: $\mathrm{\backslash (\backslash sigma\backslash )\_1^2}\backslash (\backslash neq\backslash )\mathrm{\backslash (\backslash sigma\backslash )\_2^2}$.
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 $s^2\; =\; \backslash (\backslash frac\{1\}\{n-1\}\backslash )\; \backslash (\backslash sum\backslash )\; (x\_i\; -\; \backslash (\backslash bar\{x\}\backslash ))^2$, where $n$ is the sample size, $x\_i$ are the data points, and $\backslash (\backslash bar\{x\}\backslash )$ is the sample mean.
3. Step 3: Determine the critical value and rejection region. Since this is a two-tailed F-test at significance level $\backslash (\backslash alpha\backslash )\; =\; 0.05$, 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.
4. Step 4: Calculate the test statistic F. The test statistic is the ratio of the larger sample variance to the smaller sample variance: $F\; =\; \backslash (\backslash frac\{s\_\{larger\}\backslash )^2\}\{s\_\{smaller\}^2\}$. This ensures the test statistic is always greater than or equal to 1.
5. 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.

Key Concepts

Two-Sample F-Test for Variances

The two-sample F-test compares the variances of two independent samples to determine if they come from populations with equal variances. The test statistic is the ratio of the larger sample variance to the smaller one, and it follows an F-distribution under the null hypothesis. This test is sensitive to the assumption of normality in the populations.

Hypothesis Testing Framework

Hypothesis testing involves stating a null hypothesis (H0) and an alternative hypothesis (Ha), selecting a significance level (α), and determining a rejection region based on critical values. The test statistic is calculated from sample data and compared to the critical value to decide whether to reject H0 or fail to reject it, guiding conclusions about the population.

Assumptions of the F-Test

The F-test assumes that the samples are independent, randomly selected, and drawn from normally distributed populations. Violations of these assumptions can affect the validity of the test results. Ensuring these conditions helps maintain the accuracy and reliability of conclusions about variance differences.