A Collector of Grandfather Clocks Believes That the Price Received $$E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 }$$

question 19

Multiple Choice

A collector of grandfather clocks believes that the price received for the clocks at an auction increases with the number of bidders, but at an increasing (rather than a constant) rate. Thus, the model proposed to best explain auction price (y, in dollars) by number of bidders (x) is the quadratic model $$E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 }$$
This model was fit to data collected for a sample of 32 clocks sold at auction.
Suppose the $p$ -value for the test of $H _ { 0 } : \beta _ { 2 } = 0$ vs. $H _ { \mathrm { a } } : \beta _ { 2 } > 0$ is $.02$ . What is the proper conclusion?

• A) There is evidence (at \(\alpha = .05\) ) of downward curvature in the relationship between auction price \(( y )\) and number of bidders \(( x )\).
• B) Reject \(H _ { 0 }\) at \(\alpha = .05\); the model is not useful for predicting auction price \(( y )\).
• C) There is evidence (at \(\alpha = .05\) ) of upward curvature in the relationship between auction price (y) and number of bidders \(( x )\).
• D) There is no evidence (at \(\alpha = .05\) ) of upward curvature in the relationship between auction price \(( y )\) and number of bidders \(( x )\).

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