Question

Given X has a normal distribution, could you find a probability for x bar if n was only 10, why or why not?

Answer 1

Yes, we can find the probability for x bar

From Central Limit Theorem, if the population follows Normal distribution, the sampling distribution of the sample mean also follows Normal distribution irrespective of the sample size

Thus, x bar will have mean equal to the population mean and standard deviation equal to where is the population standard deviation

Now, we can calculate the Corresponding z value and get the required probability from Standard normal distribution table