Question

Here is a simple probability model for multiple-choice tests. Suppose that each student has probability p of correctly answering a question chosen at random from a universe of possible questions. (A strong student has a higher p than a weak student.) The correctness of answers to different questions are independent. Jodi is a good student for whom p = 0.83.

(a) Use the Normal approximation to find the probability that Jodi scores 78% or lower on a 100-question test. (Round your answer to four decimal places.)


(b) If the test contains 250 questions, what is the probability that Jodi will score 78% or lower? (Use the normal approximation. Round your answer to four decimal places.)


(c) How many questions must the test contain in order to reduce the standard deviation of Jodi's proportion of correct answers to half its value for a 100-item test?
questions

Answer 1

The total number of questions Jordi answers out of is a Binomial random variable with . Its normal approximation is valid when is close to 1/2 or when is large. Thus, .

a) The probability that Jodi scores 78% or lower on a 100-question test is equivalent to getting 78 correct out of 100.

b) The probability that Jodi scores 78% or lower on a 250-question test is equivalent to getting correct out of 250.

c) Let the test contains questions in order to reduce the standard deviation of Jodi's proportion of correct answers to half its value for a 100-item test.Then