Question

A random sample of size n from a normal population with unknown mean and variance is to be used to test the null hypothesis µ = µ0 against the alternative µ ≠ µ0. Using the simultaneous maximum likelihood estimates of µ and σ2 obtained in Example 10.18 on page 300, show that the values of the likelihood ratio statistic can be writ-ten in the form
Where t =  – µ0 / s/√n. The likelihood ratio test can thus be based on the t distribution.

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