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

question 101

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}$

Root MSE $\quad 12.42 \quad$ R-Square $\quad .988$

PARAMETER T for $\mathrm { HO }$ :



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

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

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Definitions:

Social Communication

The process of using language and nonverbal cues to share information, feelings, and ideas with others, crucial for building and maintaining personal and professional relationships.

Anxiety

A feeling of worry, nervousness, or unease, often about an imminent event or something with an uncertain outcome.

Clarification

The process of making a statement or situation less confused and more comprehensible.

Appropriate

Suitable or proper in the circumstances.