Test the given claim. Data Set 7 "IQ and Lead" in Appendix B...

Test the given claim. Data Set 7 "IQ and Lead" in Appendix B lists full IQ scores for a random sample of subjects with low lead levels in their blood and another random sample of subjects with high lead levels in their blood. The statistics are summarized on the top of the next page. Use a 0.05 significance level to test the claim that IQ scores of people with low lead levels vary more than IQ scores of people with high lead levels. $$\begin{aligned} &\text { Low Lead Level: } n=78, \bar{x}=92.88462, s=15.34451\\ &\text { High Lead Level: } n=21, \bar{x}=86.90476, s=8.988352 \end{aligned}$$

Test the given claim.
Data Set 7 "IQ and Lead" in Appendix B lists full IQ scores for a random sample of subjects with low lead levels in their blood and another random sample of subjects with high lead levels in their blood. The statistics are summarized on the top of the next page. Use a 0.05 significance level to test the claim that IQ scores of people with low lead levels vary more than IQ scores of people with high lead levels.
$$\begin{aligned}
&\text { Low Lead Level: } n=78, \bar{x}=92.88462, s=15.34451\\
&\text { High Lead Level: } n=21, \bar{x}=86.90476, s=8.988352
\end{aligned}$$

Key Concepts

P-Value

The p-value is the probability of observing the test statistic, or one more extreme, assuming that the null hypothesis is true. It is compared to the significance level to determine whether to reject the null hypothesis. If the p-value is less than the significance level, the null hypothesis is rejected, supporting the alternative hypothesis.

Significance Level

The significance level, often denoted as alpha (?), represents the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true. In this problem, a significance level of 0.05 indicates that there is a 5% risk of concluding that the IQ scores of people with low lead levels vary more when they might not.

Degrees of Freedom

Degrees of freedom in the context of an F-test refer to the number of independent values in each sample that are free to vary. For the F-test, two degrees of freedom are needed: one for the numerator (based on the sample with the larger variance) and one for the denominator. Accurate determination of these values is crucial for finding the correct critical value from the F-distribution.

Null and Alternative Hypotheses

When comparing variances, the null hypothesis typically states that the variances are equal, while the alternative hypothesis indicates that one variance is greater than the other. In this context, the null hypothesis would claim that the variability in both groups is the same, whereas the alternative hypothesis asserts that the IQ scores of people with low lead levels have greater variability than those with high lead levels.

Hypothesis Testing

Hypothesis testing is a statistical procedure used to determine whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. This involves formulating null and alternative hypotheses, selecting an appropriate test statistic, and comparing the computed statistic to a critical value or using a p-value to decide whether to reject the null hypothesis.

F-Test for Variance Comparison

The F-test is used to compare the variances of two independent samples. The test statistic is calculated as the ratio of the two sample variances. In this case, it is often set up so that the larger sample variance is in the numerator, which aligns with the claim being tested. The F-distribution is then used to determine the significance of this ratio.