Sophie, the editor of the student newspaper, is going to conduct a survey of students to determine the level of support for the current president of the students' association. She needs to determine her prior distribution for , the proportion of students who support the president. She decides her prior mean is .5 , and her prior standard deviation is .15 .

(a) Determine the \(\operatorname{beta}\left(\begin{array}{ll}a & b\end{array}\right)\) prior that matches her prior belief.

(b) What is the equivalent sample size of her prior?

(c) Out of the 68 students that she polls, \(y=21\) support the current president. Determine her posterior distribution.