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• The ANOVA table for a linear regression gives the degrees of freedom, sum of squares, and mean squares for the model, error, and total sources of variation. The ANOVA F statistic is the ratio MSM/MSE. Under H0: β1 = 0, this statistic has an F(1, n − 2) distribution and is used to test H0 versus the two-
• The square of the sample correlation can be expressed as
and is interpreted as the proportion of the variability in the response variable y that is explained by the explanatory variable x in the linear regression.
• The standard errors for b0 and b1 are
• The standard error that we use for a confidence interval for the estimated mean response for the subpopulation corresponding to the value x* of the explanatory variable is
• The standard error that we use for a prediction interval for a future observation from the subpopulation corresponding to the value x* of the explanatory variable is
• When the variables y and x are jointly Normal, the sample correlation is an estimate of the population correlation ρ. The test of H0: ρ = 0 is based on the t statistic
which has a t(n − 2) distribution under H0. This test statistic is numerically identical to the t statistic used to test H0: β1 = 0.