Assume in a survey of N = 400 students, 215 responded that they prefer small classes and 185 responded that they prefer large classes. Let p denote the fraction of all students who preferred small classes at the time of the survey, and ˆp be the fraction of survey respondents who preferred small classes. (Hint: X is distributed as a Bernoulli random variable)
1. Show that E(ˆp) = p and Var(ˆp) = p(1 ? p)/N.
2. Use the survey result to estimate p, and determine the standard error of your estimator. (c) Determine the p-value for the test H0: p = 0.5 vs. H1: p 6= 0.5? Will you reject the null at the 5% significance level?
3. How can construct a 95% confidence interval for p.