Question

1. Scores for men on the verbal portion of the SAT test are normally distributed with a mean of 509 and a standard deviation of 112. Randomly selected men are given the Columbia Review Course before taking the SAT test. Assume the course has no effect.

a) If 1 of the men is randomly selected, find the probability his score is at least 590

b) If 16 of the men are randomly selected, find the probability that their mean score is at least 590

c) In part b why can the central limit theorem be used even though sample size does not exceed 30

d) If a random sample of 16 men does result in a mean score of 590 is there strong evidence to support the claim the course is effective? Why or why not?

Answer 1

Given,

= 509 , = 112

We convert this to standard normal as

P( X < x) = P( Z < x - / )

Therefore,

P( X >= 590) = P( Z >= 590 - 509 / 112)

= P( Z >= 0.7232)

= 0.2348

b)

Using central limit theorem,

P( < x) = P( Z < x - / / sqrt(n) )

P( >= 590) = P(Z >= 590 - 509 / 112 / sqrt(16) )

= P( Z > 2.8929)

= 0.0019

c)

Since original distribution is normally distributed, we can use central limit theorem in part b) even though

sample size not exceeding 30.

d)

Since probability for score 590 is unusual (below 0.05), we have strong evidence to support the

claim that the course is effective.