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 the page margin, and a plot of the data is given in Figure 13.30. 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 has a straight-line appearance with constant variation. Therefore, we consider the model ln yt = β0 + β1t + εt. If we use MINITAB, we find that the least squares point estimates of β0 and β1 are β0 = 2.07012 and b1 = .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, 6.3659]. See the MINITAB output in Figure.
a. Use the least squares point estimates to calculate 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 ln yt = β0 + β1t + εt is called a growth curve model because it implies that
where α0 = eβ0, α1 = eβ1, and ηt = eεt. 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 α1.
d. We see that yt = α0αt1ηt = (α0αt-11)α1ηt ≈ (yt–1) α1ηt. This says that y t is expected to be approximately α1 times yt-1. Noting this, interpret the growth rate of part (c).