A collector of antique grandfather clocks believes that the price (in dollars) received for the clocks at an antique auction increases with the age of the clocks and with the number of bidders. Thus the model is hypothesized is where Y = auction price, x1 = age of clock (years) and x2 = number of bidders. A sample of 32 auction prices of grandfather clocks, along with their ages and the number of bidders, is given below. a) State the multiple regression equation. b) Interpret the meaning of the slopes b1 and b2 in the model. c) Interpret the meaning of the regression coefficient b0. d) Test H0: β2 = 0 against H1: β2 > 0. Interpret your finding. e) Use a 95% confidence interval to estimate β2. Interpret the p-value corresponding to the estimate β2. Does the confidence interval support your interpretation in d)? f) Determine the coefficient of multiple determination r2Y.12 and interpret its meaning. g) Perform a residual analysis on your results and determine the adequacy of the fit of the model. i) At α ± = 0.05, is there evidence of positive autocorrelation in the residuals? j) Suppose the collector, having observed many auctions, believes that the rate of increase of the auction price with age will be driven upward by a large number of bidders. In other words, the collector believes that the age of clock and the number of bidders should interact. Is there evidence to support his claim that the rate of change in the mean price of the clocks with age increases as the number of bidders increases? Should the interaction term (x1 x2) be included in the model? If so, what is the multiple regression equation?
Age (x1) Bidders (x2) Price (y) 127 13 1235 115 12 1080 127 7 845 150 9 1522 156 6 1047 182 11 1979 156 12 1822 132 10 1253 137 9 1297 113 9 946 137 15 1713 117 11 1024 137 8 1147 153 6 1092 117 13 1152 126 10 1336 170 14 2131 182 8 1550 162 11 1884 184 10 2041 143 6 854 159 9 1483 108 14 1055 175 8 1545 108 6 729 179 9 1792 111 15 1175 187 8 1593 111 7 785 115 7 744 194 5 1356 168 7 1262
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