Here is a simple probability model for multiple-choice tests. Suppose that each
ID: 3335766 • Letter: H
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 = 0.76.
(a) Use the Normal approximation to find the probability that Jodi scores 72% 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 72% 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
Explanation / Answer
Answer to all the question is as follows:
p = .76
a. P(X<=.72) = P(Z<= (.72-.76)/sqrt(.76*.24/100) = P(Z<-.94) = .1763
b. P(X<=.72) = P(Z<= (.72-.76)/sqrt(.76*.24/250) = P(Z<-1.48) = .0693
c. In order to reduce the stdev to half for a 100 item test we must have 400 questions
The Stdev relates to sample size n , in the following way: Stdev = sqrt(p*p'/n)
So, if increase n by 4 times, stdev decreases by half