Reconsider Probs. 20.7-3 and 20.6-1. Since the number of applications for admission submitted to Ivy College has been increasing at a steady rate, causal forecasting can be used to forecast the number of applications in future years by letting the year be the independent variable and the number of applications be the dependent variable.

(a) Plot the data for Years 1, 2, and 3 on a two-dimensional graph with the year on the horizontal axis and the number of applications on the vertical axis.

(b) Since the three points in this graph line up in a straight line, this straight line is the linear regression line. Draw this line.

T (c) Find the formula for this linear regression line.

(d) Use this line to forecast the number of applications for each of the next five years (Years 4 through 8).

(e) As these next years go on, conditions change for the worse at Ivy College. The favorable ratings in the national surveys that had propelled the growth in applications turn unfavorable. Consequently, the number of applications turn out to be 6,300 in Year 4 and 6,200 in Year 5, followed by sizable drops to 5,600 in Year 6 and 5,200 in Year 7. Does it still make sense to use the forecast for Year 8 obtained in part (d)? Explain.

(f) Plot the data for all seven years. Find the formula for the linear regression line based on all these data and plot this line. Use this formula to forecast the number of applications for Year 8. Does the linear regression line provide a close fit to the data? Given this answer, do you have much confidence in the forecast it provides for Year 8? Does it make sense to continue to use a linear regression line when changing conditions cause a large shift in the underlying trend in the data?

(g) Apply exponential smoothing with trend to all seven years of data to forecast the number of applications in Year 8. Use initial estimates of 3,900 for the expected value and 700 for the trend, along with smoothing constants of α = 0.5 and ß = 0.5. When the underlying trend in the data stays the same, causal forecasting provides the best possible linear regression line (according to the method of least squares) for making forecasts. However, when changing conditions cause a shift in the underlying trend, what advantage does exponential smoothing with trend have over causal forecasting?

Probs. 20.7-3

Three years ago, the admissions office for Ivy College began using exponential smoothing with a smoothing constant of 0.25 to forecast the number of applications for admission each year. Based on previous experience, this process was begun with an initial estimate of 5,000 applications. The actual number of applications then turned out to be 4,600 in the first year. Thanks to new favorable ratings in national surveys, this number grew to 5,300 in the second year and 6,000 last year. (Use hand calculations below rather than an Excel template)

(a) Determine the forecasts that were made for each of the past three years.

(b) Calculate MAD for these three years.

(c) Calculate MSE for these three years.

(d) Determine the forecast for next year.

Look ahead at the scenario described in Prob. 20.7-3. Notice the steady trend upward in the number of applications over the past three years-from 4,600 to 5,300 to 6,000. Suppose now that the admissions office of Ivy College had been able to foresee this kind of trend and so had decided to use exponential smoothing with trend to do the forecasting. Suppose also that the initial estimates just over three years ago had been expected value = 3,900 and trend = 700. Then, with any values of the smoothing constants, the forecasts obtained by this forecasting method would have been exactly correct for all three years. Illustrate this fact by doing the calculations to obtain these forecasts when the smoothing constant is α = 0.25 and the trend smoothing constant is ß = 0.25. (Use hand calculations rather than an Excel template)

Probs. 20.7-3

Three years ago, the admissions office for Ivy College began using exponential smoothing with a smoothing constant of 0.25 to forecast the number of applications for admission each year. Based on previous experience, this process was begun with an initial estimate of 5,000 applications. The actual number of applications then turned out to be 4,600 in the first year. Thanks to new favorable ratings in national surveys, this number grew to 5,300 in the second year and 6,000 last year. (Use hand calculations below rather than an Excel template)

(a) Determine the forecasts that were made for each of the past three years.

(b) Calculate MAD for these three years.

(c) Calculate MSE for these three years.

(d) Determine the forecast for next year.