Setting prices is rarely an easy task. A low price usually results in higher sales, but there will be less profit per sale. A higher price produces higher profit per sale, but sales will be lower overall. Usually, a firm wants to choose the price that will maximize the total profit, but there is considerable uncertainty about the demand. Table 12.5.16 shows hypothetical results of a study of profits in comparable test markets of equal sizes, where only the price was changed.
a. Find the regression equation of the form Predicted Profit = a + b(Price).
b. Test to see whether or not the regression is significant. Is this result reasonable?
c. To within approximately how many dollars can profit be predicted from price in this way?
d. Examine a diagnostic plot to see if there is any further structure remaining that would help you explain profit based on price. Describe the structure that you see. e. Create another X variable using the squared price values and find the multiple regression equation to predict profit from price and squared price.
f. To within approximately how many dollars can profit be predicted from price using these two X variables?
g. Test to see whether a significant proportion of the variation in profit can be explained by price and squared price taken together.
h. Find the price at which the predicted profit is maximized. Compare this to the price at which the observed profit was the highest.

TABLE 12.5.16 Price and Profit in Test Markets Price 58 Profit $6,486 10,928 15,805 13,679 12,758 9,050 5,702 -109 10 12 13 14 15