Question

A final exam in Math 160 has a mean of 73 with standard deviation 7.8. If 24 students are randomly selected, find the probability that the mean of their test scores is less than 70.

Answer 1

A sample of size n taken from a population with mean = µ and standard deviation = σ has sample mean = µ and sample s.d. = σ/√n.

If the final exam scores are normally distributed, and X is a random variable for the mean of a sample, then

Pr[X < 70] = Pr[(X - 73) / (7.8/√24) < (70 - 73) / (7.8/√24)]

… ≈ Pr[Z < -1.8842]

… ≈ 0.0298

(where Z is normally distributed with µ = 0 and σ = 1).