Central Limit Theorem

Question

A manufacturer knows that their items have a normally distributed length, with a mean of 7.2 inches, and standard deviation of 1.7 inches.

If 12 items are chosen at random, what is the probability that their mean length is less than 8.2 inches?

Answer 1

xbar = mean length

mu = population mean = 7.2

sigma = population standard deviation = 1.7

n = sample size = 12

use transformation: z = (xbar - mu)/(sigma/sqrt(n))

P(xbar < 8.2) = P(z < (8.2 - mu)/(sigma/sqrt(n)) = P(z < (8.2 - 7.2)/(1.7/sqrt(12)) = P(z < 2.04) = 0.9793