# slide8 - Inference on the Variance So if the test is The...

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1 Inference on the Variance ± So if the test is: ± H 0 : σ = σ 0 ± H 1 : σ≠σ 0 ± The test statistic then becomes ± ± which follows a chi-square distribution with n –1 degrees of freedom. 2 0 2 2 S ) 1 n ( X σ = 2 Rejection region for the χ 2 -test ± For a two-tailed test: ± Reject if χ 2 > χ 2 α /2,n–1 or χ 2 < χ 2 1– α /2,n–1 ± For an upper-tail test: ± Reject if χ 2 > χ 2 α ,n–1 ± For an lower-tail test: ± Reject if χ 2 < χ 2 1- α , n–1

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3 Example: Jen and Barry’s ± Jen and Barry’s uses an automatic machine to box their ice-cream. A sampling of 20 containers results in a sample variance of 0.0153. If the variance of fill volume exceeds 0.01, an unacceptable proportion of containers will be under- and over-filled. Is there evidence to suggest that there is a problem at the 5% level? 4 Type II Error in a χ 2 -test ± To look up the characteristic curves for the chi-square test, we need ± The abscissa parameter 0 σ λ =
5 Example: Jen and Barry’s ± If the variance exceeds 0.01, too many containers will be underfilled.

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slide8 - Inference on the Variance So if the test is The...

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