Researchers conducted an experiment to lest the effects of alochol. Errors were recorded in a lest of visual and motor skills for a treatment group of 29 people who drank ethanol and another group of 29 people given a placebo. The errors for the treatment group have a standard deviation of 2.10 , and the errors for the placebo group have a standard deviation of 0.81 . 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. $H_{0}: \sigma_{1}^{2}=\sigma_{2}^{2}$ $H_{1}: \sigma_{1}^{2} \neq \sigma_{2}^{2}$ $H_{1}: \sigma_{1}^{2}>\sigma_{2}^{2}$ C. $H_{0}: \sigma_{1}^{2} \neq \sigma_{2}^{2}$ D. $H_{0}: \sigma_{1}^{2}=\sigma_{2}^{2}$ $H_{1}: \sigma_{1}^{2}=\sigma_{2}^{2}$ $H_{1}: \sigma_{1}^{2}<\sigma_{2}^{2}$ Identity the test statistic. (Round lo two decimal places as needed.) ew an example Get more help. Clearall Check answer
Solution
Step 1 ：Define the null and alternative hypotheses. The null hypothesis is that the variances of the two groups are equal, and the alternative hypothesis is that the variances are not equal. So, we have: $H_{0}: \sigma_{1}^{2}=\sigma_{2}^{2}$ $H_{1}: \sigma_{1}^{2} \neq \sigma_{2}^{2}$
Step 2 ：Calculate the variance for each group by squaring the standard deviation. For the treatment group, the variance is $2.1^2 = 4.41$. For the placebo group, the variance is $0.81^2 = 0.6561$.
Step 3 ：Calculate the test statistic, which is the F statistic. This is the ratio of the two sample variances. So, we have $F = \frac{\sigma_{1}^{2}}{\sigma_{2}^{2}} = \frac{4.41}{0.6561} = 6.72$
Step 4 ：The final answer is that the null and alternative hypotheses are $H_{0}: \sigma_{1}^{2}=\sigma_{2}^{2}$ and $H_{1}: \sigma_{1}^{2} \neq \sigma_{2}^{2}$, and the test statistic (F statistic) is \(\boxed{6.72}\)
