The Concept of Least-Squares Regression

Fortunately, computers are adept at carrying out the computations required by the least-squares regression formulas. The data—the observed values of X and Y—are entered into the computer, and software does the rest. In the case of the Brentline Hospital maintenance cost data, a statistical software package on a personal computer can calculate the following least-squares regression estimates of the total fixed cost (a) and the variable cost per unit of activity (b):

Therefore, using the least-squares regression method, the fixed element of the maintenance cost is $3,431 per month and the variable portion is 75.9 cents per patient-day.

In terms of the linear equation Y = a + bX, the cost formula can be written as

where activity (X) is expressed in patient-days.

Appendix 2A  discusses how to use Microsoft Excel to perform least-squares regression calculations. For now, you only need to understand that least-squares regression analysis generally provides more accurate cost estimates than the high-low method because, rather than relying on just two data points, it uses all of the data points to fit a line that minimizes the sum of the squared errors. The table below compares Brentline Hospital’s cost estimates using the high-low method and the least-squares regression method:

When the least-squares regression method is used to create a straight line that minimizes the sum of the squared errors, it results in a Y-intercept that is $31 higher than the Y-intercept derived using the high-low method. It also decreases the slope of the straight line resulting in a lower variable cost estimate of $0.759 per patient-day rather than $0.80 per patient-day as derived using the high-low method.

IN BUSINESS: THE ZIPCAR COMES TO COLLEGE CAMPUSES

Zipcar is a car sharing service based in Cambridge, Massachusetts. The company serves 13 cities and 120 university campuses. Members pay a $50 annual fee plus $7 an hour to rent a car. They can use their iPhones to rent a car, locate it in the nearest Zipcar parking lot, unlock it using an access code, and drive it off the lot. This mixed cost arrangement is attractive to customers who need a car infrequently and wish to avoid the large cash outlay that comes with buying or leasing a vehicle.

Source: Jefferson Graham, “An iPhone Gets Zipcar Drivers on Their Way,” USA Today, September 30, 2009, p. 3B.