When deriving the F-statistic on page 541 we include a note that the use of the F-distribution can be simulated with a randomization procedure. That is the purpose of this exercise. Consider the model in Example 10.7 that uses PhotoTime and CostColor to predict inkjet printer prices. The F-statistic in that ANOVA table is F = 6.30. Suppose that we want to estimate how unusual that is, when H0 is true and there is no relationship between either predictor and Price. To simulate this, randomly scramble the 20 printer prices in InkjetPrinters and assign them to the various combinations of PhotoTime and CostColor. For each such randomization, estimate the fitted model and compute the F-statistic. (Technology is a must here!) Repeat many times to obtain a randomization distribution and see how far the original F = 6.30 is in the tail. Compare the results to what you get with the F-distribution with 2 and 17 degrees of freedom.

F-statistic on page 54

To compare the variability explained by the model to the unexplained variability of the error we first need to adjust each by an appropriate degrees of freedom. These are shown in the ANOVA section of the output of Example 9.10 as 1 degree of freedom for the model and 28 degrees of freedom for the error. For the simple linear model the degrees of freedom for the model is always 1 (just one predictor) and n − 2 for the error, adding up to n − 1 degrees of freedom for the total.