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

question 86

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,does the quadratic term appear to be significant in the response curve relating the demand (Y) and the price (X) at 10% level of significance?

• A) No, sine the value of ?₂ is near 0.
• B) No, since the p-value for the test is greater than 0.10.
• C) Yes, since the p-value for the test is less than 0.10.
• D) Yes, since the value of ?₂ is positive.

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Definitions:

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Family

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A broad term referring to the societal structure, cultural norms, and social behaviors observed within the United States.