John M. answered • 05/16/19

Analytical assistance -- Writing, Math, and more

This problem is an application of the central limit theorem (CLT)

The CLT says that for a normally distributed population, the distribution of the sampling means of sample size n from that population is normally distributed with a mean equal to the population mean and a standard deviation equal to the population standard deviation, divided by the square root of n.

So if we draw a random sample (X) of 192, its mean will be as if it was drawn from a sampling distributions means which has a normal distribution with a mean = 73, and a standard deviation of 42.8/√(192)=3.0888. Accordingly, the P(74.8<X<76.7)

= P([[74.8-73]/3.0888] < Z <[[76.7-73]/3.0888]) = P(.583 < Z <1.198).

The rest is looking up those values on a z-table (or using a calculator) to compute the probability. If you have questions on how to do that let me know, and I can walk you through that procedure as well.