Question

A multiple-choice examination consists of

85

questions, each having possible choices a, b, c, d, and e. Approximate the probability that a student will get at most

18

answers correct if she randomly guesses at each answer. (Note that, if she randomly guesses at each answer, then the probability that she gets any one answer correct is 0.2.) Use the normal approximation to the binomial with a correction for continuity.

Answer 1

The probability that she will answer a question correctly if she randomly guesses among the possible choices = 1/5 = 0.2

Let X be the random variable denoting the number of correct choices out of 85 questions.

Hence, X ~ Bin(85, 0.2)

E(X) = 85 * 0.2 = 17, Var(X) = 85 * 0.2 * 0.8 = 13.6

Using normal approximation, X ~ N(17, 13.6) i.e. (X - 17)/3.6878 ~ N(0,1)

The probability that the student will get at most 18 answers correct = P(X 18.5) [Using continuity correction for Binomial to Normal approximation] = P[(X - 17)/3.6878 (18.5 - 17)/3.6878] = (0.4067) = 0.6579.

[(.) is the cdf of N(0,1)]