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

question 104

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 $\alpha = .01$ .

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Definitions:

Hypothesis

A proposed explanation for a phenomenon, made as a starting point for further investigation, which can be tested and falsified through research and experimentation.

Case Study

A detailed analysis of a particular individual, group, event, or situation over a specific period of time, used for research purposes.

Case Study Method

An in-depth examination of one person or one group.

Replication

The process of copying or reproducing something, such as a study or an experiment, to ensure accuracy or verify results.