Question

7. In a large university (45,000 students) it is known that 71% of the students live on campus. a) If we considered samples of 200 students, describe the sampling distribution of the sample proportion of students who live on campus. (Hint – we said there were three things you needed to mention in a full description, and there is check you must perform to be able to describe one of those 3 things!) [3] b) Show the work in finding the probability that in a sample of 200 students at this university more than 75% live on campus. [3]

Answer 1

A) $\mu_{\widehat P}$ = P = 0.71

$\sigma_{\widehat P}$ = sqrt(P(1 - P)/n)

= sqrt(0.71 * (1 - 0.71)/200)

= 0.0321

B) P($\widehat P$ > 0.75)

= P(($\widehat P$ - $\mu_{\widehat P}$ )/$\sigma_{\widehat P}$ > (0.75 - $\mu_{\widehat P}$ )/$\sigma_{\widehat P}$)

= P(Z > (0.75 - 0.71)/0.0321)

= P(Z > 1.25)

= 1 - P(Z < 1.25)

= 1 - 0.8944

= 0.1056