statistics probability

Question

A manufacturer knows that their items have a normally distributed length, with a mean of 5.4 inches, and standard deviation of 0.9 inches.

If 25 items are chosen at random, what is the probability that their mean length is less than 4.9 inches?

Answer 1

The population has a normal distribution, so the sample means should have a nearly normal distribution with the same mean and with a standard error equal to the standard deviation divided by the square root of the number of items in the sample.

Get a z score from

CORRECTED 12:40 pm 3/7/22:

z = (mean-mu) / (sigma/sqrt(n))

mean = sample mean

mu = populaiton mean

sigma = population standard deviation

sigma/sqrt(n) = standard error of the mean

here,

z = (4.9-5.4) / [(0.9)/(sqrt(25)]

z = (-0.5)/(0.18) = -2.77

The probability their average would be 4.9 or less is equal to the cumulative normal distribution evaluated at 2.77 standard deviations below the mean, giving 0.0028 or 0.28%.

If z = 0, there would have been a 50% probability the sample mean would be less than the population mean.

If we had a sample standard deviation, s, to work with instead of a known population standard deviation, sigma, we would have used Student's t-test instead.