Consider the normal distribution with unknown mean and unknown variance 2, and suppose that it is desired to test the following hypotheses H0 0, H1 0 Suppose that it is possible to observe only a single value of X from this distribution, but that an independent random sample of n observations Y1, , Yn is available from the normal distribution with known mean 0 and the same variance 2 as for X Show how to carry out a test of the hypotheses H0 and H1 based on the t distribution with n degrees of freedom

Question: Consider the normal distribution with unknown mean and unknown variance 2, and suppose that it is desired to test the following hypotheses: H0:

Consider the normal with unknown mean μ and unknown variance σ2, and suppose that it is desired to test the following hypotheses:
H0: μ ≤ μ0,
H1: μ>μ0.
Suppose that it is possible to observe only a single value of X from this but that an independent random sample of n observations Y1, . . . , Yn is available from the normal with known mean 0 and the same variance σ2 as for X. Show how to carry out a test of the hypotheses H0 and H1 based on the t with n degrees of freedom.

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