A sleep center hypothesizes that people who sleep only fourhours will score lower than people who sleep for eight hours on acognitive skills test. The center recruited 20 participants andsplit them into two groups, giving one group 8 hours of sleep andthe other only 4 hours. The following morning, the CAT (CognitiveAbility Test) was conducted, with scores ranging from 1-9, 9 beingthe best score. Use this information to answer questions . CATScores Group X: Eight hrs sleep 4 7 9 4 3 3 8 6 3 7 Group Y: Fourhrs sleep 7 8 1 4 2 3 5 2 7 4 Conduct the following hypothesistest: - A one-tail T-test for a two-sample difference in means atthe 95% confidence level - with Null Hypothesis that the Group Xmean CAT score is equal to the Group Y mean CAT score - and withAlternate Hypothesis that the Group X mean CAT score is greaterthan the Group Y mean CAT score a). Calculate the mean and standarddeviation of the scores for each group. (10%)

b)Using the correct degrees of freedom (df = group X size +group Y size ? # of groups), the correct number of tails, and atthe correct confidence level, determine the critical value of t.(10%)

c). Explain under which scenarios using a pooled variance beinadvisable, then, calculate the pooled variance (formula for S2 ison page 379) for the groups. (10%)

d). Calculate the test statistic, Ttest (formula for t is onpage 380). (10%)

e). The sleep center’s statistician tells you that the p-valuefor the test is 0.1535. Summarize the result of the study. Comparethe mean scores in each group. Compare the test statistic to thecritical value. Compare the p-value to alpha. Do you find astatistically significant difference between Group X and Group Y oncognitive test performance? Is there a meaningful/practicaldifference? Explain your decisions and Justify your claims