Notice that this gives an equivalent method for testing

Chapter 10, Problem 81E

(choose chapter or problem)

From two normal populations with respective variances \(\sigma_{1}^{2}\) and \(\sigma_{2}^{2}\), we observe independent sample variances \(S_{1}^{2}\) and \(S_{2}^{2}\), with corresponding degrees of freedom \(v_{1}=n_{1}-1\) and \(v_{2}=n_{2}-1\). We wish to test \(H_{0}: \sigma_{1}^{2}=\sigma_{2}^{2}\) versus \(H_{\alpha}: \sigma_{1}^{2} \neq \sigma_{2}^{2})

a Show that the rejection region given by

\(\left\{F>F_{v_{2} \alpha / 2}^{v_{1}} \text { orF } \alpha\left(F_{v . \alpha / 2}^{v_{2}}\right)^{-1}\right\}\)

where \(F=S_{1}^{2} / S_{2}^{2}\), is the same as the rejection region given by

\(\left\{S_{1}^{2} / S_{2}^{2}>F_{\nu_{2} \sigma / 2}^{\nu_{1}} \text { orS }_{2}^{2} / S_{1}^{2}>F_{v_{1} \cdot \sigma / 2}^{v_{2}}\right\}\)

b Let \(S_{L}^{2}\) denote the larger of \(S_{1}^{2}\) and \(S_{2}^{2}\) and let \(S_{S}^{2}\) denote the smaller of \(S_{1}^{2}\) and \(S_{2}^{2}\). Let \(v_{L}\) and \(v_{S}\) denote the degrees of freedom associated with \(S_{L}^{2}\) and \(S_{S}^{2}\), respectively. Use part (a) to show that, under ,

\(p\left(S_{L}^{2} / S_{S}^{2}>F_{v y . \alpha / 2}^{v t}\right)=\alpha\)

Notice that this gives an equivalent method for testing the equality of two variances.

Equation transcription:

Text transcription:

S_{1}^{2}

S_{2}^{2}

\sigma_{1}^{2}

\sigma_{2}^{2}

v_{1}=n_{1}-1

v_{2}=n_{2}-1

H_{0}: \sigma_{1}^{2}=\sigma_{2}^{2}

H_{\alpha}: \sigma_{1}^{2} \neq \sigma_{2}^{2}

\left\{F>F_{v_{2} \alpha / 2}^{v_{1}} \text { orF } \alpha\left(F_{v . \alpha / 2}^{v_{2}}\right)^{-1}\right\}

F=S_{1}^{2} / S_{2}^{2}

\left\{S_{1}^{2} / S_{2}^{2}>F_{\nu_{2} \sigma / 2}^{\nu_{1}} \text { orS }_{2}^{2} / S_{1}^{2}>F_{v_{1} \cdot \sigma / 2}^{v_{2}}\right\}

S_{L}^{2}

S_{S}^{2}

v_{L}

v_{S}

p\left(S_{L}^{2} / S_{S}^{2}>F_{v y . \alpha / 2}^{v t}\right)=\alpha