SECTION 10.2 SUMMARY

• • 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-sided alternative.

• • The square of the sample correlation can be expressed as

  $$r^2=\frac{\text{SSM}}{\text{SST}}$$

  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

  $$\begin{array}{lll}\hfill {\displaystyle {\text{SE}}_{b_0}}& {\displaystyle =}\hfill & {\displaystyle s\sqrt{\frac{1}{n}+\frac{{\overline{x}}^2}{\sum {\left(x_i-\overline{x}\right)}^2}}}\hfill \\ \hfill {\displaystyle {\text{SE}}_{b_1}}& {\displaystyle =}\hfill & {\displaystyle \frac{s}{\sqrt{\sum {\left(x_i-\overline{x}\right)}^2}}}\hfill \end{array}$$

• • 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

  $${\text{SE}}_{\widehat{\mu }}=s\sqrt{\frac{1}{n}+\frac{{\left(x*-\overline{x}\right)}^2}{\sum {\left(x_i-\overline{x}\right)}^2}}$$

• • 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

  $${\text{SE}}_{\widehat{y}}=s\sqrt{1+\frac{1}{n}+\frac{{\left(x*-\overline{x}\right)}^2}{\sum {\left(x_i-\overline{x}\right)}^2}}$$

• • 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

  $$t=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}$$

  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.