Western Steakhouses, a fast-food chain, opened 15 years ago. Each year since then the number of steakhouses in operation, y, was recorded. An analyst for the firm wishes to use these data to predict the number of steakhouses that will be in operation next year. The data are given in Figure 15.30(a) on the next page, and a plot of the data is given in Figure 15.30(b). Examining the data plot, we see that the number of steakhouse openings has increased over time at an increasing rate and with increasing variation. A plot of the natural logarithms of the steakhouse values versus time (see Figure 15.30(c)) has a straight-line appearance with constant variation. Therefore, we consider the model
ln y1 = β0 + β1t + εt
If we use MINITAB, we find that the least squares point estimates of β0 and β1 are b0 = 2.07012 and bt = .256880. We also find that a point prediction of and a 95 percent prediction interval for the natural logarithm of the number of steakhouses in operation next year (year 16) are 6.1802 and [5.9945, 6.3659J. See the MINITAB output in Figure 15.33 on page 677.
a. Use the least squares point estimates to verify the point prediction.
b. By exponentiating the point prediction and prediction interval-that is, by calculating e6.1802 and [e5.9945, e6.3659]-find a point prediction of and a 95 percent prediction interval for the number of steakhouses in operation next year.
c. The model In y1 = β0 + β1t + εt is called a growth curve model because it implies that

where α0 = eβ0, α1 = eβ1, and η1 = eε1. Here α1 = eβ1 is called the growth rate of the y values. Noting that the least squares point estimate of β1 is b1 = .256880, estimate the growth rate a,. Also, interpret this growth rate by using the fact that yt = αʹαt1ηt = (α0α1t-1)α1ηt ‰ˆ (yt-1)α1ηt. This says that yt is expected to be approximately α1 times yt-1.

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