A computer manufacturer hired a market research firm to investigate the relationship between the likelihood a family
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A computer manufacturer hired a market research firm to investigate the relationship between the likelihood a family

13.5. Home computers. A computer manufacturer hired a market research firm to investigate the relationship between the likelihood a family will purchase a home computer and the price of the home computer. The data that follow are based on replicate surveys done in two similar cities. One thousand heads of households in each city were randomly selected and asked if they would be likely to purchase a home computer at a given price. Eight prices (X, in dollars) were studied, and 100 heads of households in each city were randomly assigned to a given price. The proportion likely to purchase at a given price is denoted by Y.


No location effect is expected and the data are to be treated as independent replicates at each of the 8 prices. The following exponential model with independent normal error terms is deemed to be appropriate:

a. To obtain initial estimates of Gamma 0, Gamma 1, and Gamma 2, note that f(x, Gamma) approaches a lower asymptote Gamma 0 as X increases without bound. Hence, let g(0)0 = 0 and observe that when we ignore the error term, a logarithmic transformation then yields Y'i = Beta 0 + Beta 1 Xi, where Y'i = loge, Yi, Beta 0 = loge Gamma 2, and Beta 1= - Gamma 1. Therefore, fit a linear regression function based on the transformed data and use as initial estimates g(0)0 = 0, g(0)1  = –b1, and g(0)2  = exp(b0).
b. Using the starting values obtained in part (a), find the least squares estimates of the parameters Gamma 0, Gamma 1, and Gamma 2.
Hint
"A linear regression models the relationship between two variables (explanatory & dependent) by fitting a linear equation to observed data.The least squares method best fits a linear regression model by minimizing the sum of the squares of the vertical deviations from each data point to the line. "...
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