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. 549 City A i: 1 2 3 4 5 6 7 8 Xi: 200 400 800 1200 1600 2000 3000 4000 Yi: .65 .46 .34 .26 .17 .15 .06 .04 i: 9 10 11 12 13 14 15 16 Xi: 200 400 800 1200 1600 2000 3000 4000 Yi: .63 .50 .30 .24 .19 .12 .08 .05 City B 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: Yi = 0 + 2 exp(-1Xi) + i a. To obtain initial estimates of 0, 1, and 2, note that f (X, ) approaches a lower asymptote 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 = 0 + 1 Xi, where Y'i = loge Yi, 0 = loge 2, and 1 = 1. Therefore, fit a linear regression function based on the transformed data and use as initial estimates g 0(0) = 0, g0(1) = b1, and g0(2) = exp(b0). b. Using the starting values obtained in part (a), find the least squares estimates of the parameters 0, 1, and 2