Textbook Problem

In Example 7.5, we implied that each of the five observations was from one period of time, such as a particular week. Suppose instead that each is an average over several weeks. For example, the 4.7 million exposures corresponding to one ad might really be an average over 15 different weeks where one ad was shown in each of these weeks. Similarly, the 90.3 million exposures corresponding to 50 ads might really be an average over only three different weeks where 50 ads were shown in each of these weeks. If the observations are really averages over different numbers of weeks, then simply summing the squared prediction errors doesn’t seem appropriate. For example, it seems more appropriate that an average over 15 weeks should get five times as much weight as an average over only three weeks. Assume the five observations in the example are really averages over 15 weeks, 10 weeks, 4 weeks, 3 weeks, and 1 week, respectively. Devise an appropriate fitting function, to replace sum of squared errors or RMSE, and use it to find the best fit.

EXAMPLE 7.5 ESTIMATING AN ADVERTISING RESPONSE FUNCTION

Recall that the General Flakes Company from Example 4.1 of Chapter 4 sells a brand of low-fat breakfast cereal that appeals to people of all age groups and both genders. The company has advertised this product in various media for a number of years and has accumulated data on its advertising effectiveness. For example, the company has tracked the number of exposures to young men from ads placed on a particular television show for five different time periods. In each of these time periods, a different number of ads was used. Specifically, the numbers of ads were 1, 8, 20, 50, and 100. The corresponding numbers of exposures (in millions) were 4.7, 22.1, 48.7, 90.3, and 130.5. What type of nonlinear response function might fit these data well?