Researchers conducted an experiment to test the effects of alcohol. Errors were recorded in a test of visual and motor skills for a treatment group of 27 people who drank ethanol and another group of 27 people given a placebo. The errors for the treatment group have a standard deviation of 2.40 , and the errors for the placebo group have a standard deviation of 0.75 . Assume that the two populations are normally distributed Use a 0.05 significance level to test the claim that both groups have the same amount of variation among the errors.

Let sample 1 be the sample with the larger sample variance, and let sample 2 be the sample with the smaller sample variance. What are the null and alternative hypotheses?
A. $H_{0}: \sigma_{1}^{2}=\sigma_{2}^{2}$
B. $\mathrm{H}_{0} \cdot \sigma_{1}^{2}=\sigma_{2}^{2}$
$H_{1}: \sigma_{1}^{2}< \sigma_{2}^{2}$
$\mathrm{H}_{1}, \sigma_{1}^{2}> \sigma_{2}^{2}$
c.
\[
\begin{array}{l}
H_{0} \cdot \sigma_{1}^{2}=\sigma_{2}^{2} \\
H_{1} \cdot \sigma_{1}^{2} \neq \sigma_{2}^{2}
\end{array}
\]
D.
\[
\begin{array}{l}
H_{0} \cdot \sigma_{1}^{2} \neq \sigma_{2}^{2} \\
H_{1} \cdot \sigma_{1}^{2}=\sigma_{2}^{2}
\end{array}
\]

Identify the test statistic
(Round to two decimal places as needed.)

Step 1 ：Let sample 1 be the sample with the larger sample variance, and let sample 2 be the sample with the smaller sample variance. The null hypothesis is typically a statement of no effect or no difference. The alternative hypothesis is what you might believe to be true or hope to prove true.

Step 2 ：In this case, we are testing the claim that both groups have the same amount of variation among the errors. Therefore, the null hypothesis would be that the variances are equal, and the alternative hypothesis would be that the variances are not equal. The null and alternative hypotheses are: \n \[ H_{0}: \sigma_{1}^{2}=\sigma_{2}^{2} \] \n \[ H_{1}: \sigma_{1}^{2} \neq \sigma_{2}^{2} \]

Step 3 ：The test statistic for a two-sample test of variance is the ratio of the two sample variances. The larger variance should always go in the numerator to ensure that the test statistic is greater than or equal to 1.

Step 4 ：In this case, the sample with the larger variance is the treatment group (with a standard deviation of 2.40), and the sample with the smaller variance is the placebo group (with a standard deviation of 0.75). Therefore, the test statistic would be \((2.40)^2 / (0.75)^2\).

Step 5 ：Calculate the test statistic: \n std_dev1 = 2.4 \n std_dev2 = 0.75 \n test_statistic = \((std_dev1)^2 / (std_dev2)^2 = 10.24\)

Step 6 ：The test statistic is 10.24. This is the ratio of the variance of the treatment group to the variance of the placebo group.

Step 7 ：The final answer is: \n The null and alternative hypotheses are: \n \[ H_{0}: \sigma_{1}^{2}=\sigma_{2}^{2} \] \n \[ H_{1}: \sigma_{1}^{2} \neq \sigma_{2}^{2} \] \n And the test statistic is \(\boxed{10.24}\).