# Sampling Distribution Question

I've added a screenshot (https://imgur.com/a/isQXZ) and the text below for your convenience.

Please show steps if possible to help my understanding. Thank you.

Consider a random variable that is Normally distributed, with population mean mu = E [X] and population variance sigma^2 = var [X]. Assume that we have a random sample of size N from this distribution. Let x-bar be the usual sample average.

(a) Using the properties of the Normal distribution, derive explicitly the sampling distribution of the random variable: Z = (sqrt(N)) ((x-bar - mu)/(sigma))

(b) Assume that we know sigma^2, but not mu. For the Null hypothesis H0 : mu = mu0, derive a test statistic, characterize its distribution under H0, and describe a test with the property that the probability of committing a type I error is alpha.

(c) How do you need to modify the results in (b) if sigma^2 is unknown.

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I've added a screenshot (https://imgur.com/a/isQXZ) and the text below for your convenience.

Please show steps if possible to help my understanding. Thank you.

Consider a random variable that is Normally distributed, with population mean mu = E [X] and population variance sigma^2 = var [X]. Assume that we have a random sample of size N from this distribution. Let x-bar be the usual sample average.

(a) Using the properties of the Normal distribution, derive explicitly the sampling distribution of the random variable: Z = (sqrt(N)) ((x-bar - mu)/(sigma))

(b) Assume that we know sigma^2, but not mu. For the Null hypothesis H0 : mu = mu0, derive a test statistic, characterize its distribution under H0, and describe a test with the property that the probability of committing a type I error is alpha.

(c) How do you need to modify the results in (b) if sigma^2 is unknown.
Are you using a textbook? If so, all you need should be in there.

You are asking us to take the time to write things out and explain them to you, but many other people have already done that and put it into books and on web pages. So: I suggest you start looking elsewhere first, and if you have made a genuine effort---and are still confused about very specific issues---then come back here. General questions like yours will never be well received by most helpers..