(a) identify the claim and state H0 and Ha (b) find the...

(a) identify the claim and state $H_{0}$ and $H_{a}$ (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. If convenient, use technology. The table at the left shows a sample of the waiting times (in days) for a heart transplant for two age groups. At $\alpha=0.05,$ can you conclude that the variances of the waiting times differ between the two age groups?

(a) identify the claim and state $H_{0}$ and $H_{a}$ (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. If convenient, use technology.
The table at the left shows a sample of the waiting times (in days) for a heart transplant for two age groups. At $\alpha=0.05,$ can you conclude that the variances of the waiting times differ between the two age groups?

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide between two mutually exclusive statements about a population. It involves formulating a null hypothesis (H0), which assumes no effect or no difference, and an alternative hypothesis (Ha) that contradicts the null. Data is then collected and analyzed to determine whether there is enough evidence to reject H0 in favor of Ha, based on a chosen level of significance.

Null and Alternative Hypotheses

The null hypothesis (H0) in this context typically states that there is no difference in the variances between two populations, while the alternative hypothesis (Ha) suggests that there is a difference. Formulating these hypotheses correctly is essential, as they form the basis for the subsequent analysis and interpretation of the test results.

F-test for Equality of Variances

An F-test is used to compare the variances of two populations and to determine if they differ significantly. The test statistic is computed as the ratio of two sample variances, and under the null hypothesis, it follows an F distribution. This method is particularly useful when the populations are assumed to be normally distributed.

F Distribution

The F distribution is a probability distribution that arises when comparing two scaled chi-squared distributions. It is used in the context of variance analysis, especially in the F-test, to determine the critical values needed to define the rejection region for the null hypothesis.

Significance Level and Critical Value

The significance level (alpha) is the probability of rejecting the null hypothesis when it is true. It is used to determine the critical value(s) from the F distribution, which in turn define the rejection region. If the calculated test statistic falls into this region, the null hypothesis is rejected.

Test Statistic

The test statistic in an F-test is the ratio of the sample variances. Its value is compared to the critical value derived from the F distribution. This value quantifies the difference between the variances and is used to assess whether the observed difference is statistically significant.

Decision and Interpretation

Based on the comparison of the test statistic with the critical value, the decision is made to either reject or fail to reject the null hypothesis. The final step is interpreting this decision in the context of the research question, which in this situation involves determining if there is evidence that the waiting time variances differ between the two age groups.

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