Chapter 4.1, Problem 5CP

5. A company test-markets a new soft drink in 22 cities of approximately equal size. The selling price (in dollars) and the number sold per week in the cities are listed as follows:

a. First, the company wants to find the demand curve1: how many it will sell at each potential price. Let P denote price and S denote sales per week. Find the line $S=c_1+c_2$ that best fits the data from the table in the sense of least squares. Find the normal equations and the coefficients $c_1$ and $c_2$ of the least squares line. Plot the least squares line along with the data, and calculate the root mean square error.

b. After studying the results of the test marketing, the company will set a single selling price P throughout the country. Given a manufacturing cost of $\$0.\text{23}$ per unit, the total profit (per city, per week) is $S(P-0.23)$

dollars. Use the results of the preceding least squares approximation to find the selling price for which the company’s profit will be maximized.