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

question 39

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:
$\begin{array} { l r r r r r } \text { SOURCE } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { PR > F } \\ \text { Model } & 2 & 115145 & 57573 & 373 & .0001 \\ \text { Error } & 9 & 1388 & 154 & & \\ \text { TOTAL } & 11 & 116533 & & & \end{array}$
$\begin{array} { l l l l } \text { Root MSE } & 12.42 & \text { R-Square } & .988 \end{array}$ $A certain type of rare gem serves as a status symbol for many of its owners. In theory, for low prices, the demand decreases as the price of the gem increases. However, experts hypothesize that when the gem is valued at very high prices, the demand increases with price due to the status the owners believe they gain by obtaining the gem. Thus, the model proposed to best explain the demand for the gem by its price is the quadratic model E ( y ) = \beta _ { 0 } + \beta _ { 1 } ^ { x + \beta _ { 2 } x ^ { 2 } } where y = Demand (in thousands) and x = Retail price per carat (dollars). This model was fit to data collected for a sample of 12 rare gems. A portion of the printout is given below: \begin{array} { l r r r r r } \text { SOURCE } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { PR > F } \\ \text { Model } & 2 & 115145 & 57573 & 373 & .0001 \\ \text { Error } & 9 & 1388 & 154 & & \\ \text { TOTAL } & 11 & 116533 & & & \end{array} \begin{array} { l l l l } \text { Root MSE } & 12.42 & \text { R-Square } & .988 \end{array} Is there sufficient evidence to indicate the model is useful for predicting the demand for the gem? Use \alpha = .01 .$

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

Understand the basics of hypothesis testing, including reading t and z statistics from tables.

Identify and interpret Type I and Type II errors in hypothesis testing.

Determine the correct hypothesis setup for given scenarios.

Recognize the relationship between the level of significance (α), p-values, and the decision to reject or fail to reject the null hypothesis.

Definitions:

Outgroups

Groups to which an individual does not belong or identify with, often perceived as different or oppositional to one's ingroup.

Negative Stereotypes

Oversimplified and unfavorable beliefs about a group of people that do not account for individual differences and complexities.

Stereotypes

Oversimplified and generalized beliefs about a group of people or type of individual.

Group Membership

The affiliation or association with a specific group of people who share common interests or characteristics.

A Certain Type of Rare Gem Serves as a Status E(y)=β0+β1x+β2x2E ( y ) = \beta _ { 0 } + \beta _ { 1 } ^ { x + \beta _ { 2 } x ^ { 2 } }E(y)=β0​+β1x+β2​x2​

Definitions:

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