TABLE 15-3
a Certain Type of Rare Gem Serves as a Status

TABLE 15-3
A certain type of rare gem serves as a status symbol for many of its owners. In theory, for low prices, the demand increases and it 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 owners believe they gain in obtaining the gem. Thus, the model proposed to best explain the demand for the gem by its price is the quadratic model:

$Y=\beta_{0}+\beta_{1} X+\beta_{2} X^{2}+\varepsilon$

where Y = demand (in thousands) and X = retail price per carat.
This model was fit to data collected for a sample of 12 rare gems of this type. A portion of the computer analysis obtained from Microsoft Excel is shown below:
SUMMARY OUTPUT
$\begin{array}{lc}\text { Regression Statistics}\\\hline\text { Multiple R } & 0.994 \\\text { R Square } & 0.988 \\\text { Standard Error } & 12.42 \\\text { Observations } & 12 \\\hline\end{array}$

$ANOVA$
$\begin{array}{lrrrrc}\hline & d f & S S & \text { MS } & F & \text { Signifcance } \\\hline \text { Regression } & 2 & 115145 & 57573 & 373 & 0.0001 \\\text { Residual } & 9 & 1388 & 154 & & \\\text { Total } & 11 & 116533 & & & \\\hline\end{array}$


$\begin{array}{lrccc}\hline & \text { Coeff } & \text { Std Error } & t \text { Stat } & \text { p-value } \\\hline \text { Intercept } & 286.42 & 9.66 & 29.64 & 0.0001 \\\text { Price } & -0.31 & 0.06 & -5.14 & 0.0006 \\\text { Price Sq } & 0.000067 & 0.00007 & 0.95 & 0.3647\end{array}$

-Referring to Table 15-3, a more parsimonious simple linear model is likely to be statistically superior to the fitted curvilinear for predicting sale price (Y).