A student answers all 48 questions on a multiple-choice test by guessing. Each question has four possibleanswers, only one of which is correct. Find the probability that the student gets exactly 15 correct answers. Usethe normal distribution to approximate the binomial distribution.
A) 0.0823
B) 0.8577
C) 0.7967
D) 0.0606

Question

Quizplus · Accepted Answer

The normal distribution can be used to approximate the binomial distribution when np and n(1-p) are both greater than or equal to 10. In this case, np = 48 * 0.25 = 12 and n(1-p) = 48 * 0.75 = 36, so the normal distribution can be used. The mean of the binomial distribution is np = 12 and the standard deviation is sqrt(np(1-p)) = sqrt(12 * 0.75) = 3. The probability of getting exactly 15 correct answers is the area under the normal distribution curve between z = (15 - 12) / 3 = 1 and z = (15 - 12) / 3 = 1. The area under the normal distribution curve between z = 1 and z = 1 is 0.0823, so the probability of getting exactly 15 correct answers is 0.0823.

A Student Answers All 48 Questions on a Multiple-Choice Test