Example 16.1

1. Example 16.1 Forecasting Sales at Best Chips

2. Exponential Growth.xlsx The Best Chips Company produces and sells potato chips throughout the country. Its sales have been growing steadily over the past 10 years, as shown on the next slide and in this file. The company wants to predict its sales for the next couple of years, assuming that the upward trend it has observed in the past 10 years will continue in the future. How should the company proceed?

4. Solution We begin by using the Chart Wizard to create the X-Y plot of Sales versus Year shown on the next slide. Sales are clearly increasing through time, but it is not absolutely clear whether they are increasing at a constant rate, which would favor a linear trendline, or at an increasing rate, which would favor an exponential trendline. Therefore, we try fitting both of these.

6. Solution -- continued To superimpose a linear trendline on any scatterplot, select the chart and then More Trendline Options from the Trendline dropdown on the Chart Tools Layout ribbon. This brings up the dialog box shown here.

7. Solution -- continued You can select any of six types of trendlines. For now, select the default Linear option. Also, click on the Options tab and check the Display equation box. The result is shown on the next slide. This figure shows the best-fitting straight line to the points, and it indicates that the equation of this straight line is y = 92,091x +1,168,200.

9. Solution -- continued Here, y corresponds to sales and x corresponds to year. The most important part of this equation is the coefficient of x, 92,091. It implies that sales are increasing by $92,091 per year—if we believe that the linear trendline provides a good fit. To obtain an exponential trendline, we go through the same procedure except that we select the Exponential option in the dialog box.

10. Solution -- continued The resulting curve appears on the next slide. The equation for the curve is y 1.2278e0.0541x. The most important part of this equation is the coefficient in the exponent, 0.0541. It implies that sales are increasing by approximately 5.4% per year.

12. Solution -- continued • Which of these trendlines provides the better fit? • We can proceed in two ways. • First, we can “eyeball” it. Looking at the superimposed trendlines, it appears that the exponential fit is slightly better. • The typical way to measure fits to a trendline through time is to calculate the historical predictions from each curve and the corresponding absolute percentage errors (APEs). • We find the predictions by plugging the year indexes (1 to 10) into the trendline equations.

13. Solution -- continued We then calculate the APE for each year from the following equation. A measure of goodness-of-fit is then the average of these APE values, denoted by MAPE (mean absolute percentage error). This measure is quite intuitive.

14. Solution -- continued • All of this is implemented and shown on the next slide. • To create the predictions, APEs, and MAPEs, proceed as follows: • Predictions. Calculate the predictions from the linear trendline by entering the formula =1168200+92091*A53 in cell D3 and copying it down to cell D14. Similarly, calculate the predictions from the exponential trendline by entering the formula =1227762*EXP(0.0541*A3) in cell E3 and copying it down to cell E14.

16. Solution -- continued • APE values. Calculate all of the APE values at once by entering the formula =ABS($B3-D53/$B3in cell F3 and copying it to the range F3:G12. • MAPE values. Calculate the MAPE for each trendline by entering the formula =AVERAGE(F3:F12) in cell F16 and copying it to cell G16. • The MAPE values confirm that the exponential trendline is slightly better than the linear trendline.

17. Solution -- continued Exponential trendlines are often used in predicting sales and other economic quantities. However, we urge caution with such predictions. It is difficult for any company to sustain a given percentage increase year after year.