Example 16.3 Estimating Total Cost for Several Products

Example 16.3

Estimating Total Cost for Several Products

Thomson/South-Western 2007 ©South-Western/Cengage Learning © 2009Practical Management Science, Revised 3eWinston/Albright

Cost Regression 2.xlsx

• Suppose the company from Example 16.2 now produces three different products, A, B, and C.

• The company has kept track of the number of units produced of each product and the total production cost for the past 15 months.

• These data are shown on the next slide and in the file Cost Regression 2.xlsx.

• What can multiple regression tell us about the relationship between these variables?

• How can multiple regression be used to predict future production costs?



Solution

• The dependent variable Y is again Total Cost, but there are now three potential X’s, Units A, Units B, and Units C.

• We are not required to use all three of these, but we do so here.

• In fact, it is again a good idea to begin with scatterplots of Y versus each X to see which X’s are indeed related to Y.

• We did this and obtained three scatterplots; a typical one is shown on the next slide.



Using Analysis ToolPak

• Unfortunately, when there are multiple X’s, we cannot estimate the multiple regression equation by using Excel’s Trendline option as we did with simple regression.

• Instead, we must use a statistical software package or an Excel add-in. Fortunately, Excel includes such an add-in, called Analysis ToolPak.


Using Analysis ToolPak -- continued

• To run the regression analysis with Analysis ToolPak, select Data Analysis from the Data ribbon, select Regression from the list of Analysis Tools, and fill out the resulting dialog box as shown on the next slide.




• The following options are the most important; you can experiment with the other settings.– The range for the X’s must be contiguous. In other words, the data

for the independent variables must be in adjacent columns. You might have to move the data around to get the X’s next to each other.

– It is useful for reporting purposes to include the variable names (row 3 in our case) in the Y and X’s ranges. If you do this, you should check the Labels box.

– There are a number of options on where to put the output. We chose cell G1 (of the same worksheet as the data) as the upper-left corner of our output range.

– If you check the Residuals box, you automatically get the fitted (predicted) values of the Y’s and the corresponding residuals.



• The resulting output appears on the next slide.• We will not explain all of this output, but we will

focus on the highlights. The most important part is the regression equation itself, which is implied by the values in the H17:H21 range:Predicted Total Cost 20,261 +12.802Units A + 17.691Units B + 15.230Units C

• The interpretation is much like that in simple regression. Each coefficient of a Units variable can be interpreted as a variable cost.




• The other important outputs are R-square, multiple R, the standard error of estimate, the fitted values, and the residuals:– The R-square value is the percentage of variation of Total Cost

explained by the combination of all three explanatory variables. We see that these three Units variables explain about 94.6% of the variation in Total Cost—a fairly high percentage.

– The multiple R, the square root of R-square, is the correlation between the actual Y’s and fitted values. Because R-square is large, the multiple R is also large: 0.973.

– This high value implies that the points in a scatterplot (not shown) of actual Y values versus fitted values are close to a 45° line.



– The standard error of estimate has exactly the same interpretation as before. It is a ballpark estimate of the magnitude of the prediction errors we are likely to make, based on the regression equation. Here, this value is about $1981—not too bad considering that the total costs vary around $50,000.

– The fitted values are found by substituting each set of X’s into the regression equation, and the residuals are the differences between actual total costs and fitted values. Analysis ToolPak calculates these automatically for us in the range H26:G41. As indicated by the standard error of estimate, most of the residuals are no more than about $2000 in magnitude, and quite a few are considerably less than this.



• If we conclude that the fit is good enough to provide useful future predictions, we can then substitute future estimates of production levels into the regression equation.

• We did this for the proposed values of the X’s for months 16 and 17 in rows 17 and 18. Specifically, we enter the formula =$H$17+$H$18*B17+$H$19*C17+$H$20*D17 in cell H44 and copy it to cell H45.

• Of course, this could be done for any assumed production levels in these two months, not just the ones we chose.



• More specifically, it is common to quote a 95% prediction interval for each future prediction.

• We are 95% sure that the actual future value will be inside this interval.

• Although the exact calculation of a 95% prediction interval is somewhat complex, a good and easy approximation is to go out two standard errors of estimate on either side of the predicted value.

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Example 16.3 Estimating Total Cost for Several Products