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 $( x )$.

Question

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 $( x )$.

Quizplus · Accepted Answer

The p-value of 0.02 is less than the significance level $\alpha = 0.05$, so we reject $H_0$. This indicates there is evidence of upward curvature in the relationship between auction price and number of bidders, as $\beta_2 > 0$.

A Collector of Grandfather Clocks Believes That the Price Received E(y)=β0+β1x+β2x2E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 }E(y)=β0​+β1​x+β2​x2

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