Question

Assume SAT math scores have a mean of 420 and a standard deviation of 110. (a) If a sample of 100 students are selected randomly, find the probability that the sample mean is above 500.

(b) Why can the central limit theorem be applied in part a?

Answer 1

Part a

We are given

µ = 420

σ = 110

n = 100

We have to find P(Xbar>500)

P(Xbar>500) = 1 – P(Xbar<500)

Z = (Xbar - µ)/[σ/sqrt(n)]

Z = (500 - 420)/(110/sqrt(100))

Z = 80/11

Z = 7.272727

P(Z<7.272727) = P(Xbar<500) = 1.000

P(Xbar>500) = 1 – P(Xbar<500)

P(Xbar>500) = 1 – 1

P(Xbar>500) = 0.0000

Required probability = 0.0000

Part b

Central limit theorem can be applied in part a, because we know that the sampling distribution of the sample means follows approximately normal distribution.