# Question: For the likelihood ratio statistic of Exercise 12 22 show that

For the likelihood ratio statistic of Exercise 12.22, show that –2 ∙ ln λ approaches t2 as n → ∞.

In exercise

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.

In exercise

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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