Question

Data 9.1 on page 577 introduces the dataset InkjetPrinters, which includes information on all-in-one printers. Two of the variables are Price (the price of the printer in dollars) and CostColor (average cost per page in cents for printing in color). Computer output for predicting the price from the cost of printing in color is shown: $$ \begin{aligned} &\text { The regression equation is Price }=378-18.6 \text { CostColor }\\\ &\begin{array}{lrrrrr} \text { Analysis of Variance } & & & & \\ \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \\ \text { Regression } & 1 & 57604 & 57604 & 13.19 & 0.002 \\ \text { Residual Error } & 18 & 78633 & 4369 & & \\ \text { Total } & 19 & 136237 & & & \end{array} \end{aligned} $$ (a) What is the predicted price of a printer that costs 10 cents a page for color printing? (b) According to the model, does it tend to cost more or less (per page) to do color printing on a cheaper printer? (c) Use the information in the ANOVA table to determine the number of printers included in the dataset. (d) Use the information in the ANOVA table to compute and interpret \(R^{2}\). (e) Is the linear model effective at predicting the price of a printer? Use information from the computer output and state the conclusion in context.

Short answer

(a) The predicted price of a printer that costs 10 cents a page for color printing is $558. (b) It tends to cost less per page to do color printing on a cheaper printer. (c) There are 19 printers in the dataset. (d) The \(R^{2}\) value is 0.42, which means that 42% of the variance in the printer's price can be explained by the cost of color printing. (e) The model is statistically significant (P=0.002) and it explains a fair proportion (42%) of the variance in the printer's price.

Step-by-step solution

Predict the Price

To predict the price of a printer that costs 10 cents a page for color printing, substitute the value of CostColor (10) into the regression equation. So, Price = 378 - 18.6*10.

Interpret the Model

The model suggests that for each additional cent it costs to print in color, the price of the printer decreases by 18.6 dollars. Hence, it tends to cost less per page to do color printing on a cheaper printer.

Determine the Number of Printers in the Dataset

The total degrees of freedom (DF) is 19. Given that the regression has 1 degree of freedom, the degrees of freedom of the residual error (which corresponds to the number of printers - 1) would be 18. So, the number of printers in the dataset is 19.

Compute and Interpret \(R^{2}\)

The formula for \(R^{2}\) is \(R^{2}\) = SS Regression / SS Total = 57604 / 136237. The interpretation of \(R^{2}\) is that it represents the proportion of the variance in the price that is explained by the cost of color printing.

Interpret the Effectiveness of the Model

The effectiveness of the model is indicated by the p-value (P) and the \(R^{2}\) value. A low p-value (less than 0.05) indicates a statistically significant effect of CostColor on Price. The value of \(R^{2}\) indicates the proportion of variance in the Price explained by CostColor. Thus, one can make conclusions about the model's effectiveness based on these values.