A Certain Type of Rare Gem Serves as a Status $$E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 }$$

Essay

A certain type of rare gem serves as a status symbol for many of its owners. In theory, for low prices, the demand decreases as the price of the gem increases. However, experts hypothesize that when the gem is valued at very high prices, the demand increases with price due to the status the owners believe they gain by obtaining the gem. Thus, the model proposed to best explain the demand for the gem by its price is the quadratic model $$E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 }$$
where $y =$ Demand (in thousands) and $x =$ Retail price per carat (dollars).
This model was fit to data collected for a sample of 12 rare gems. A portion of the printout is given below:
$\begin{array} { l r r r r r } \text { SOURCE } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { PR >F } \\ \text { Model } & 2 & 115145 & 57573 & 373 & .0001 \\ \text { Error } & 9 & 1388 & 154 & & \\ \text { TOTAL } & 11 & 116533 & & & \end{array}$

$\begin{array} { l l l l } \text { Root MSE } & 12.42 & \text { R-Square } & .988\end{array}$

$\begin{array} { l l l l l }&\text {PARAMETER}&&\text {T for \(H O\) :}\\ \text { VARIABLES } & \text { ESTIMATES } & \text { STD. ERROR } & \text { PARAMETER } = 0 & \text { PR } > | T | \end{array}$

$\begin{array}{lrrrr}\text { INTERPCEP } & 286.42 & 9.66 & 29.64 & .0001 \\X & -.31 & .06 & -5.14 & .0006 \\X \cdot X & .000067 & .00007 & .95 & .3647\end{array}$


Is there sufficient evidence to indicate the model is useful for predicting the demand for the gem? Use $\alpha = .01$ .