R2/(k — 1)


— R2)/(n — k)

where use is made of the definition R2 = ESS/TSS. Equation (8.5.11) shows how F and R2 are related. These two vary directly. When R2 = 0, F is zero ipso facto. The larger the R2, the greater the F value. In the limit, when R2 = 1, F is infinite. Thus the F test, which is a measure of the overall significance of the estimated regression, is also a test of significance of R2. In other words, testing the null hypothesis (8.5.9) is equivalent to testing the null hypothesis that (the population) R2 is zero.


Source of variation

SS df


Due to regression

R 2(E y,2) 2

R 2(E y,2)/2

Due to residuals

(1 - R 2)(E y,2) n - 3

(1 - R 2)(E y,2)/(n - 3)


E y,2 n -1

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