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The response (y) variable is weight (in pounds), and the predictor (x) variables are HT (height in inches), WAIST (waist circumference in cm), and CHOL (cholesterol in mg).

Predictor (x) VariablesP-valueR2Adjusted R2Regression EquationHT, WAIST, CHOL0.0000.8800.870y^=199+2.55HT+2.18WAIST0.00534CHOLHT, WAIST0.0000.8770.870y^=206+2.66HT+2.15WAISTHT, CHOL0.0020.2770.238y^=148+4.65HT+0.00589CHOLWAIST, CHOL0.0000.8040.793y^=42.8+2.41WAIST0.0106CHOLHT0.0010.2730.254y^=139+4.55HTWAIST, CHOL0.0000.7900.785y^=44.1+2.37WAISTCHOL0.8740.0010.000y^=1730.00233CHOL\begin{matrix} \text{Predictor (x) Variables} & \text{P-value} & \text{}{\boldsymbol{R}^{2}} & \text{Adjusted }{\boldsymbol{R}^{2}} & \text{Regression Equation}\\ \text{HT, WAIST, CHOL} & \text{0.000} & \text{0.880} & \text{0.870} & \text{}{\hat{y}=-199+2.55 HT+2.18 WAIST-0.00534 CHOL}\\ \text{HT, WAIST} & \text{0.000} & \text{0.877} & \text{0.870} & \text{}{\hat{y}=-206+2.66 HT+2.15 WAIST}\\ \text{HT, CHOL} & \text{0.002} & \text{0.277} & \text{0.238} & \text{}{\hat{y}=-148+4.65 HT+0.00589 CHOL}\\ \text{WAIST, CHOL} & \text{0.000} & \text{0.804} & \text{0.793} & \text{}{\hat{y}=-42.8+2.41 WAIST-0.0106 CHOL}\\ \text{HT} & \text{0.001} & \text{0.273} & \text{0.254} & \text{}{\hat{y}=-139+4.55 HT}\\ \text{WAIST, CHOL} & \text{0.000} & \text{0.790} & \text{0.785} & \text{}{\hat{y}=-44.1+2.37 WAIST}\\ \text{CHOL} & \text{0.874} & \text{0.001} & \text{0.000} & \text{}{\hat{y}=173-0.00233 CHOL}\\ \end{matrix}

If only one predictor (x) variable is used to predict weight, which single variable is best? Why?

Question

Refrigerator prices are affected by characteristics such as whether or not the refrigerator is on sale, whether or not it is listed as a Sub-Zero brand, the number of doors (one door or two doors), and the placement of the freezer compartment (top, side, or bottom). The table below shows the regression output from a regression model using the natural log of price as the dependent variable. The model was developed by the Bureau of Labor Statistics. (a) Write the regression model, being careful to exclude the base indicator variable. (b) Find the p-value for each coefficient, using 319 degrees of freedom. Using an α=.01\alpha=.01, which predictor variable(s) are not significant predictors? (c) By how much does the natural log of refrigerator price decrease from a two door, side freezer model to a two door, top freezer model? (d) Which model demands a higher price: the side freezer or the one door with freezer model?

 Variable  Coefficient  Standard Error t Statistic  Intercept 5.4840920.1308130941.923 Sale price 0.07330.023388263.134 Sub-Zero brand 1.119620.146156997.660 Total capacity (in cubic ft) 0.069560.0053510312.999 Two door, freezer on bottom 0.0465690.08085690.576 Two door, side freezer  Base  Two door, freezer on top 0.3432460.035958739.546 One door with freezer 0.7095580.130970475.418 One door, no freezer 0.8819810.149139925.914\begin{array}{lccr} \hline \text { Variable } & \text { Coefficient } & \text { Standard Error } & \boldsymbol{t} \text { Statistic } \\ \hline \text { Intercept } & 5.484092 & 0.13081309 & 41.923 \\ \text { Sale price } & -0.0733 & 0.02338826 & -3.134 \\ \text { Sub-Zero brand } & 1.11962 & 0.14615699 & 7.660 \\ \text { Total capacity (in cubic ft) } & 0.06956 & 0.00535103 & 12.999 \\ \text { Two door, freezer on bottom } & 0.046569 & 0.0808569 & 0.576 \\ \text { Two door, side freezer } & \text { Base } & & \\ \text { Two door, freezer on top } & -0.343246 & 0.03595873 & -9.546 \\ \text { One door with freezer } & -0.709558 & 0.13097047 & -5.418 \\ \text { One door, no freezer } & -0.881981 & 0.14913992 & -5.914 \\ \hline \end{array}

Solution

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a)

Write the regression model using the values from the table in the Coefficient column.

First let's write the variable names for our predictors:

Sale=Sale priceBrand=Sub-Zero brandCapacity=Total capacityTwoDFzBot=Two door, freezer on bottomTwoDFzTop=Two door, freezer on topOneDFz=One door with freezerOneDNoFz=One door, no freezer\begin{align*} Sale&=\text{Sale price}\\ Brand&=\text{Sub-Zero brand}\\ Capacity&=\text{Total capacity}\\ TwoDFzBot&=\text{Two door, freezer on bottom}\\ TwoDFzTop&=\text{Two door, freezer on top}\\ OneDFz&=\text{One door with freezer}\\ OneDNoFz&=\text{One door, no freezer}\\ \end{align*}

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