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

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 $( x )$ .