Question: 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 Y = b₀ + b₁x₁ + b₂x₂ + e, 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 in the data (AUCTION.xls).

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: b₂ = 0 against H₁: b₂ > 0. Interpret your finding.

e) Use a 95% confidence interval to estimate b₁. Interpret the p-value corresponding to the estimate b₁. Does the confidence interval support your interpretation in b)?

f) Determine the coefficient of multiple determination r²_Y.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 a = 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 be included in the model? If so, what is the multiple regression equation?