Based on a random sample of size n from N(mu, sigma^2) when both the mean/variance are UNKNOWN, we want to test the null

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Based on a random sample of size n from N(mu, sigma^2) when both
the mean/variance are UNKNOWN, we want to test the null hypothesis
H_0: mu = mu_0 versus H_1: mu # mu_0.
Derive the LRT - showing all details - start from the definition
of LRT - work out the numerator and denominator separately and
simplify.
Show that the LRT rejects H_0 when the |t| statistic is large.
Here t stand for the familiar Student's t statistic as defined in
the class.
Explain how you can determine the cut-off point.
As an application, assume n=16, sample mean xbar = 20, sample
standard deviation = 5. What is your conclusion at alpha = 5%
for testing H_0: mu = 25?