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

question 71

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: Does the quadratic term contribute useful information for predicting the demand for the gem? Use $\alpha = .10$ .

$\begin{array}{lrrrrr}\text { SOURCE } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { PR > F } \\\text { Model } & 2 & 115145 & 57573 & 373 & .0001 \\\text { Error } & 9 & 1388 & 154 & & \\\text { TOTAL } & 11 & 116533 & & &\end{array}$


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


$\begin{array}{lrrrr} & \text { PARAMETER } & \text { T for HO: } \\\text { VARIABLES } & \text { ESTIMATES } & \text { STD. ERROR } & \text { PARAMETER }=0 & \text { PR > }>\mid \\\text { INTERPCEP } & 286.42 & 9.66 & 29.64 & .0001 \\\text { X } & -.31 & .06 & -5.14 & .0006 \\\text { X.X } & .000067 & .00007 & .95 & .3647\end{array}$

Does the quadratic term contribute useful information for predicting the demand for the gem? Use $\alpha=10$ .

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Definitions:

Business

The practice of engaging in commerce, trade, and dealings by individuals or organizations to generate profit.

Expert Systems

Artificial intelligence programs designed to solve complex problems by mimicking the decision-making ability of a human expert.

CEO Position

The Chief Executive Officer position, the highest-ranking executive in a company, responsible for making major corporate decisions and managing overall operations.

Cloud Computing

A technology in which tasks are performed by computers physically removed from the user and accessed over a network, in particular the Internet.