Aset of normally distributed data has a mean of 485 and a standard deviation of 11.6. find the probability of randomly selecting 40 values and getting a mean less than 490.

Let denote a data point taken from the distribution, where , and let denote the average.

You want to find



First, let's recall a few things. The PDF of a normal distribution with mean and variance is



Each of the are presumably independently selected, so they are i.i.d. random variables.

The MGF of a normal distribution is





The MGF of a linear combination of i.i.d. random variables is



In this case, each . This product of MGFs reduces to an MGF of a normal distribution because the are i.i.d..




which is indeed the MGF of a normal distribution with mean and variance

So, the PDF of , given that and , is



Now,



or, using the CDF of ,