# GR5205 p44-p50 Annotated.pdf - P ROPOSITION 2.8 Let X1 X2...

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PROPOSITION2.8LetX1, X2, . . . , Xmbe a random sample from a normal distribution withvarianceσ21, letY1, Y2, . . . , Ynbe a random sample from another normal distribution with vari-anceσ22, and letS21andS22denote the two sample variances. Also assume the random samplesare independent of each other. Then the random variableF=S21/σ21S22/σ22(2.13)has an F-distribution with degrees of freedom1=m-1and2=n-1.PROPOSITION2.9LetTbe distributed student’s t-distribution with degrees of freedom. Thenthe random variableT2has an F-distribution with degrees of freedom1and. NamelyT2F(1,)Example 2 continuedRcode:model <- lm(yx)anova(model)Table 2: Simple linear regression analysis of varianceRoutputDfSum SqMean SqF valuePr(>F)x1549097.62549097.62130.580.0001Residuals521026.104205.2244
2.8General Linear TestThe analysis of variance test ofH0:β1= 0versusHA:β16= 0is an example of thegeneraltest for a linear statistical model.