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 $\alpha = .01$ .

---

Definitions:

True Self-Managing Work Team

A team that operates without direct supervision, managing its own processes and tasks to achieve set objectives.

Selecting New Team Members

The process of evaluating and choosing individuals to form or join a team based on skills, compatibility, and potential contributions.

Evaluating Performance

The process of assessing the effectiveness and efficiency of an individual's or organization's work outcomes and methods.

Forming Stage

Focuses around the initial entry of members to a team.