Because an ANOVA does not presume a linear trend, it can be used to check for deviations from the straight-enough condition. The procedure requires replicated observations; we need several values of y at each value of x. The response is the weekly sales of a beverage in 47 stores, and the explanatory variable is the number of feet of shelf space used to display the product.
(a) Fit the linear regression of sales on number of feet of shelf space. Does the relationship meet the straight-enough condition?
(b) Build six dummy variables to represent the values of the explanatory variable (1, 2, 3, c , 6 with 7 excluded). The dummy variable D1 identifies stores displaying the product on 1 foot of shelf space, D2 identifies those with 2 feet, and so forth. Fit the multiple regression of the residuals from the simple regression in part (a) versus the six variables D1, D2, c , D6. Summarize the fit.
(c) Does the regression of the residuals on the dummy variables explain statistically significant amounts of variation in the residuals? Should it?