Question

We often judge other people by their faces. It appears that some people even judge candidates for elected office by their faces. Researchers 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 solely on 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 a better face. Is there evidence that the candidate with the better face wins more than half the time? State the conditions for the inference to be valid.

Answer 1

Answer:

Given,

Null hypothesis

Ho : p = 0.5

Alternative hypothesis

Ha : p > 0.5

alpha = 0.05

p^ = x/n

= 22/32

p^ = 0.6875

degree of freedom = n - 1

= 32 - 1

= 31

Now the corresponding t value for t(0.05 , 31) is 1.6955

Now consider,

test statistic t = (p^ - p) / sqrt(pq/n)

substitute values

= (0.6875 - 0.5) / sqrt(0.5*0.5/32)

= 2.12

Here we observe that test statistic > t(1.6955) , so we reject Ho.

We conclude that the candidate with better face wins more that half the time.