Chapter 14: Q. 14.37 (page 562)

The document Arizona Residential Property Valuation System published by the Arizona Department of Revenue, describes how country assessors use computerized systems to value single-family residential properties for property tax purposes. On the WeissStats site are data on lot size (in acres) and assessed value (in thousands of dollars) for a sample of homes in a particular area.
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

Expert verified

Part a. From the MINITAB output, the standard error of the estimate is \(139.012\).

Part b. The residual plot and normal probability plot are obtained.

Part c. The assumptions for the regression inferences are not violated for the variables value and lot size.

Step by step solution

Part a. Step 1. Given information

Given,

LOT SIZE	VALUE	2.02	317
2.37	396	2.05	387
2.09	355	2.09	462
2.21	586	2.32	678
2.21	254	2.02	309
2.1	646	2.9	648
2.06	278	2.2	295
2.03	279	2.03	548
2.44	748	2	527
2.14	546	2.01	590
2.63	338	2.63	505
2.02	409	2.95	716
2.09	382	2.08	385
2.98	443	2.28	314
2.05	610	2.09	483
2.17	394	2.61	468
2.18	340	3.08	418
2.13	208	2.47	419
2.18	350	2.11	296
2.56	687	3.62	375
2	496	2.28	587
2.4	568
2.47	321
2.58	328

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 Lot Size.

Step 4: Click OK.

MINITAB output:

From the MINITAB output, the standard error of the estimate is \(139.012\).

Interpretation:

The predicted scores in the sample differ on average from the observed scores by \(139.012\) thousand dollars.

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 Lot Size.

Step 4: In Graphs, enter the columns Lot Size 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 Lot Size.

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

Step 5: Click OK.

MINITAB output:

Thus, the residual plot and normal probability plot are obtained.

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 assumptions for the regression inferences are not violated for the variables value and lot size.