# Question

Reconsider Probs. 27.7-3 and 27.6-1. 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.

(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 year’s 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 turns 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 β = 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?

(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.

(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 year’s 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 turns 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 β = 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?

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