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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.