Psychologists showed head-and-shoulders photos of the two main candidates in 32 U.S. Senate races to many subjects (excluding anyone who recognized a candidate). For each race, they asked which candidate's face looked more competent. On election day, the candidate whose face looked more competent won 22 of the 32 contests. Let p be the true proportion of races in which the "more-competent-looking" candidate wins. Tasks (a) Parameter & hypotheses. Define p and state appropriate hypotheses to test whether facial "competence" predicts winners: Ho p=? VS Hap?? Briefly justify your direction for Ha. (b) Conditions & test. Check conditions for a one-sample z test for a proportion. Compute the test statistic and p-value using the data = 22, n = 32. Conclude at a = 0.05 in context. (c) Type I and Type II errors (in context). Write one sentence defining each error in this study and give a plausible consequence of each. (d) Prospective power analysis. Suppose a follow-up study will use a = 0.01. Pilot work suggests the true effect could be p = 0.65. For this design, the probability of a Type II error is = 0.50. 1. Compute the power of the test. 2. On a single horizontal axis for the sampling distribution of p, sketch and label the null distribution (centered at pm) and the alternative distribution (centered at p = 0.65). Mark the critical value implied by a=0.01. Shade and label the regions corresponding to a, , and power. (e) Improving power. Is the power from part (d) adequate? Give three practical ways to increase power for this study and, for each, provide a specific value or change (e.g., change a to increase n from to switch to a one- sided test, reduce variability by using ratings protocol, etc.).