Question

A bank's loan officer rates applicants for credit. The rating are normally distributed with a mean of 175 and a standard deviation of 40. If a sample of 60 applicants is randomly selected, find the probability that the mean rating is below 180.

Answer 1

Here, μ = 175, σ = 40/sqrt(60) = 5.164 and x = 180. We need to compute P(X <= 180). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (180 - 175)/5.164 = 0.97

Therefore,
P(X <= 180) = P(z <= (180 - 175)/5.164)
= P(z <= 0.97)
= 0.834