A Certain Type of Rare Gem Serves as a Status $$Y = B _ { 0 } + B _ { 1 } X + B_ { 2 } X ^ { 2 } + \varepsilon$$

question 80

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 hypothesise that when the gem is valued at very high prices, the demand increases with price due to the status owners believe they gain in obtaining the gem. Thus, the model proposed to best explain the demand for the gem by its price is the quadratic model:
$$Y = B _ { 0 } + B _ { 1 } X + B_ { 2 } X ^ { 2 } + \varepsilon$$
where Y = demand (in thousands) and X = retail price per carat.
$$\begin{array} { | l | l | l | l | l | l | } \hline \text { SUMMARY } & \text { OUTPUT } & & & & \\\hline \text { Regression } & \text { Statistics } & & & & \\\hline \text { Multiple R } & & 0.994 & & & \\\hline \text { R Square } & & 0.988 & & & \\\hline \text { Std. Error } & & 12.42 & & & \\\hline \text { Observations } & & 12 & & & \\\hline & & & & & \\\hline \text { ANOVA } & & & & & \\\hline & \text { df } & \text { SS } & \text { MS } & \text { F } & \text { Signif F } \\\hline \text { Regression } & 2 & 115,145 & 57,573 & 373 & 0.0001 \\\hline \text { Residual } & 9 & 1,388 & 154 & & \\\hline \text { Total } & 11 & 116,533 & & & \\\hline & & & & & \\\hline & \text { Coeff } & \text { Std. Error } & \text { t Stat } & P \text {-Value } & \\\hline \text { Intercept } & 286.42 & 9.66 & 29.64 & 0.0001 & \\\hline \text { Price } & - 0.31 & 0.06 & - 5.14 & 0.0006 & \\\hline \text { Price Sq } & 0.000067 & 0.00007 & 0.95 & 0.3647 & \\\hline\end{array}$$ This model was fit to data collected for a sample of 12 rare gems of this type. A portion of the computer analysis obtained from Microsoft Excel is shown below:
-Referring to Instruction 16-1,what is the p-value associated with the test statistic for testing whether the quadratic term is necessary in fitting the response curve relating the demand (Y) and the price (X) ?

• A) 0.0001
• B) 0.0006
• C) 0.3647
• D) None of the above

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Definitions:

Monopoly

A market structure characterized by a single seller who has exclusive control over a particular good or service, leading to limited consumer choice.

Uninsurable Risks

Risks that are not covered under standard insurance policies due to their high probability or inability to be quantified.

National Income

The total value of all goods and services produced by a country's economy over a specified time period, including labor and capital provided by citizens.

Wages and Salaries

Compensation received by employees for their labor, including hourly pay, salaries, bonuses, and commissions.