Question

On the WeissStats site are data on home size (in square feet) and assessed value (in thousands of dollars) for the same homes as in Exercise \(14.37\).

a. obtain and interpret the standard error of the estimate.

b. obtain a residual plot and a normal probability plot of the residuals.

c. decide whether you can reasonably consider Assumptions \(1-3\) for regression inferences met by the two variables under considerations.

Short answer

Part a. The standard error of the estimate is \(102.423\)

Part b.

Part c. The assumption \(3\) for the regression inferences is violated for the variables value and home size.

Step-by-step solution

Part a. Step 1. Given information

Data on the home size and assessed value

Part a. Step 2. Calculation

Find the standard error of the estimate by using MINITAB.

MINITAB procedure:

Step 1: Choose Stat > Regression > Regression.

Step 2: In Response, enter the column VALUE.

Step 3: In Predictors, enter the columns HOME SIZE.

Step 4: Click OK.

MINITAB output:

Regression Analysis: VALUE versus HOME SIZE

Model Summary

S	R-sq	R-sq(adj)	R-sq(pred)
\(102.423\)	\(47.3%\)	\(46.0%\)

Coefficients

Term	Coef	SE coef	T-value	p-value
Constant	\(111.56\)	\(56.86\)	\(1.96\)	\(0.056\)
HOME SIZE	\(0.11159\)	\(0.01818\)	\(6.14\)	\(0.000\)

The regression equation is

VALUE \(=112+0.112\) HOME SIZE

From the MINITAB output, the standard error of the estimate is \(102.423\)

Interpretation:

The predicted value in the sample differs on average from the observed value by \(102.423\).

Part b. Step 1. Calculation

Construct the residual plot by using MINITAB.

MINITAB procedure:

Step 1: Choose Stat > Regression > Regression.

Step 2: In Response, enter the column VALUE

Step 3: In Predictors, enter the columns HOME SIZE.

Step 4: In Graphs, enter the columns HOME SIZE variables under Residuals versus the variables.

Step 5: Click OK.

MINITAB output:

Construct the normal probability plot of residuals by using MINITAB.

MINITAB procedure:

Step 1: Choose Stat > Regression > Regression.

Step 2: In Response, enter the column VALUE

Step 3: In Predictors, enter the columns HOME SIZE.

Step 4: In Graphs, select Normal probability plot of residuals.

Step 5: Click OK.

MINITAB output:

Part c. Step 1. Calculation

The assumption for regression inferences is given below:

Population regression line:

The conditional mean of the response variable \((Y)\) is \(\beta _{0}+\beta _{1}X\), for each value \(X\) of predictor variable.

Equal standard deviation:

The standard deviation for the response variable \((Y)\) is same for the standard deviation for the explanatory variable \((X)\). The standard deviation is denoted as \(\sigma\).

Normal populations:

The distribution of the response variable follows normal.

Independent observations:

The observations of the response variable are independent of each other.

Check whether the graph suggests violation of one or more of the assumptions for the regression inferences.

• From the residual plot, it is clear that the residuals are fall in the horizontal band.
• From the normal probability plot of residuals, it is clear that the residuals are in the linear pattern.

Hence, the assumption \(3\) for the regression inferences is violated for the variables value and home size.