Here is a simple probability model for multiple-choice tests. Suppose that a student has probability p of correctly answering a question chosen at random from a universe of possible questions. (A good student has a higher p than a poor student.) The correctness of an answer to any specific question doesn't depend on other questions. A test contains n questions. Then the proportion of correct answers that a student gives is a sample proportion from an SRS of size n drawn from a population with population proportion p.
(a) Julie is a good student for whom p = 0.8. Find the probability that Julie scores 77% or lower on a 115 question test.
(b) If the test contains 245 questions, what is the probability that Julie will score 77% or lower? (Use the normal approximation to the sampling distribution to solve this problem.)
(c) How many questions must the test contain in order to reduce the standard deviation of Julie's proportion of correct answers to one-fourth its value for an 100-item test? (Use the sampling approximation to the binomial distribution to solve this problem.)