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

question 106

Multiple Choice

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.

$\begin{array} { l r r r r } \text { VARIABLES } & \begin{array} { r } \text { PARAMETER } \\ \text { ESTIMATES } \end{array} & \text { STD. ERROR } & \text { PARAMETER } = 0 & \text { PR > |T| } \\ \text { INTERPCEP } & 286.42 & 9.66 & 29.64 & .0001 \\ \mathrm { X } & - .31 & .06 & - 5.14 & .0006 \\ \mathrm { X } \cdot \mathrm { X } & .000067 & .00007 & .95 & .3647 \end{array}$
Does there appear to be upward curvature in the response curve relating $y$ (demand) to $x$ (retail price) ?

• A) Yes, since the value of \(\beta _ { 2 }\) is positive.
• B) Yes, since the \(p\)-value for the test is less than .01.
• C) No, since the value of \(\beta _ { 2 }\) is near 0 .
• D) No, since the \(p\)-value for the test is greater than .10.

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Definitions:

Decisions

Choices made by individuals or organizations concerning options that affect processes, strategies, or outcomes.

Split-Off Point

That point in the manufacturing process where some or all of the joint products can be recognized as individual products.

Joint Products

Two or more products that are produced from a common input.

Manufacturing Process

A sequence of operations that transforms raw materials into finished products, involving both human labor and machinery.