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.
Age (x1) Bidders (x2) Price (y) Age (x1) Bidders (x2) Price (y)
127 13 1235 170 14 2131
115 12 1080 182 8 1550
127 7 845 162 11 1884
150 9 1522 184 10 2041
156 6 1047 143 6 854
182 11 1979 159 9 1483
156 12 1822 108 14 1055
132 10 1253 175 8 1545
137 9 1297 108 6 729
113 9 946 179 9 1792
137 15 1713 111 15 1175
117 11 1024 187 8 1593
137 8 1147 111 7 785
153 6 1092 115 7 744
117 13 1152 194 5 1356
126 10 1336 168 7 1262
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: ?2 = 0 against H1: ?2 > 0. Interpret your finding.
e) Use a 95% confidence interval to estimate ?2. Interpret the p-value corresponding to the estimate ?2. 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.
h) Plot the residuals against the prices. Is there evidence of a pattern in the residuals? Explain.
i) At ? = 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?