World organization Random sample

Question

A world organization of academics claims tht the mean IQ score of its members is 112, with a standard deviation of 15. A randomly selected group of 40 members of this organization is tested, and the results reveal that the mean IQ score in the sample is 114.4. If the organizations claim is correct, what is the probability of having a sample mean of 114.4 or less for a random sample of this size?

Answer 1

The sample mean follows a normal distribution with mean μ = 112 and standard deviation σ/√n = 15/√40 ≈ 2.37.

In R we can find the probability the sample mean is less than or equal to 114.4 with:

pnorm(114.4, 112, 2.37, lower.tail = T)

0.8443883

If you don't have R or a graphing calculator with stats functions, you can find the z score of (x̄-x₀)/(σ/√n) = (114.4-112)/(15/√40) ≈ 1.01, and then look up in a z-score table the area to the left of 1.01.