22.21 Vote for the best face? We often judge other people by their faces. It appears that some people judge candidates for elected office by their faces. Psychologists showed head-and-shoulders photos of the two main candidates in 32 races for the U.S. Senate to many subjects (dropping subjects who recognized one of the candidates) to see which candidate was rated “more competent’’ based on nothing but the photos. On election day, the candidates whose faces looked more competent won 22 of the 32 contests. If faces don’t influence voting, half of all races in the long run should be won by the candidate with the better face. Is there evidence that the proportion of times the candidate with the better face wins is more than 50%?

  1. (a) Explain in words what the parameter p is in this setting.

  2. (b) What are the null and alternative hypotheses H0 and Ha?

  3. (c) What is the numerical value of the sample proportion ? The P-value is the probability of what event?

  4. (d) The P-value is P = 0.017. Explain carefully why this is reasonably good evidence that H0 may not be true and that Ha may be true.