The article “The Benefits of Facebook Friends: Social Capital and College Students’ Use of Online Social Network Sites” (Journal of Computer- Mediated Communication [2007]: 1143–1168) describes a study of n = 286 undergraduate students at Michigan State University. Suppose that it is reasonable to regard this sample as a random sample of undergraduates at Michigan State. You want to use the survey data to decide if there is evidence that more than 75% of the students at this university have a Facebook page that includes a photo of themselves. Let p denote the proportion of all Michigan State undergraduates who have such a page.
a. Describe the shape, center, and spread of the sampling distribution of  for random samples of size 286 if the null hypothesis H0: p = 0.75 is true.
b. Would you be surprised to observe a sample proportion of  = 0.83 for a sample of size 286 if the null hypothesis H0: p = 0.75 were true? Explain why or why not.
c. Would you be surprised to observe a sample proportion of  = 0.79 for a sample of size 286 if the null hypothesis H0: p = 0.75 were true? Explain why or why not.
d. The actual sample proportion observed in the study was  = 0.80. Based on this sample proportion, is there convincing evidence that the null hypothesis H0: p = 0.75 is not true, or is  consistent with what you would expect to see when the null hypothesis is true? Support your answer with a probability calculation.