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

question 29

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; a portion of the printout follows:
$\begin{array}{lrrrr}\hline &\text { PARAMETER }& \text {STANDARD }& \text { T FOR 0: }\\\text { VARIABLES } & \text { ESTIMATE } & \text { ERROR } & \text { PARAMETER }=0 & \text { PROB }>|T|\\\text { INTERCEPT } & 286.42 & 9.66 & 29.64 & .0001 \\\mathrm{X} & -.31 & .06 & -5.14 & .0016 \\\mathrm{X} \cdot \mathrm{X} & .000067 & .00007 & .95 & .3600 \\\hline\end{array}$

Find the $p$ -value for testing $H _ { 0 } : \beta _ { 2 } = 0$ against $H _ { \mathbf { a } } : \beta _ { 2 } > 0$ .

• A) .18
• B) .36
• C) .0016
• D) .05

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