Problem
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 22 people who drank ethanol and another group of 22 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 0.87 . 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. \[ H_{1}: \sigma_{1}^{2}<\sigma_{2}^{2} \] \[ \begin{array}{l} H_{0}: \sigma_{1}^{2}=\sigma_{2}^{2} \\ H_{1}: \sigma_{1}^{2} \neq \sigma_{2}^{2} \end{array} \] \[ \text { C. } \begin{aligned} H_{0}: \sigma_{1}^{2} & =\sigma_{2}^{2} \\ H_{1}: \sigma_{1}^{2} & >\sigma_{2}^{2} \end{aligned} \] D. \[ \begin{array}{l} H_{0}: \sigma_{1}^{2} \neq \sigma_{2}^{2} \\ H_{1}: \sigma_{1}^{2}=\sigma_{2}^{2} \end{array} \] Identify the test statistic. 7.61 (Round to two decimal places as needed.) Use technology to identify the P-value. (Round to three decimal places as needed.)
Solution
Step 1 ：Define the null and alternative hypotheses as follows: \n\n\[\begin{array}{l}H_{0}: \sigma_{1}^{2}=\sigma_{2}^{2} \H_{1}: \sigma_{1}^{2} \neq \sigma_{2}^{2}\end{array}\]
Step 2 ：Calculate the test statistic. The test statistic is \(\boxed{7.61}\).
Step 3 ：Calculate the P-value using the chi-square distribution with degrees of freedom equal to the sample size minus 1. The P-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. The specific value of the P-value is not provided in the question.
