Ch 10 - 94E

QUESTION:

Suppose that \(Y_{1}, Y_{2}, \ldots, Y_{n}\) constitute a random sample from a normal distribution with known mean \(\mu\) and unknown variance \(\sigma^{2}\). Find the most powerful \(\alpha\) -level test of \(H_{0}: \sigma^{2}=\sigma_{0}^{2}\) versus \(H_{\alpha}: \sigma^{2}=\sigma_{1}^{2}\), where \(\sigma_{1}^{2}>\sigma_{0}^{2}\). Show that this test is equivalent to a \(\chi^{2}\) test. Is the test uniformly most powerful for \(H_{\alpha}: \sigma^{2}>\sigma_{1}^{2}\)?

Equation Transcription:

Text Transcription:

Y_1, Y_2, …., Y_n

mu

sigma^2

alpha

H_{0}: sigma^{2} = sigma_{0}^{2}

H_{alpha}: sigma^{2} = sigma_{1}^{2}

sigma_{1}^{2} > sigma_{0}^{2}

chi^{2}

H_{alpha}: sigma^{2} >sigma_{1}^{2}