Question

The Information Please Almanac provides data on the ages at inauguration and of death for the presidents of the United States. We give those data on the WeissStats site for those presidents who are not still living at the time of this writing.

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. From the MINITAB output, the standard error of the estimate is \(9.705\).

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 death and inauguration.

Step-by-step solution

Part a. Step 1. Given information

Given,

INAUGURATION	DEATH		49	49
57	67		51	57
61	90		47	71
57	83		55	67
57	85		55	71
58	73		54	58
57	80		42	60
61	78		51	72
54	79		56	67
68	68		55	57
51	71		51	60
49	53		54	90
64	65		51	63
50	74		60	88
48	64		62	78
65	77		43	46
52	56		55	64
56	66		56	81
46	63		61	93
54	70		69	93

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 Death.

Step 3: In Predictors, enter the column Inauguration.

Step 4: Click OK.

MINITAB output:

From the MINITAB output, the standard error of the estimate is \(9.705\).

Interpretation:

The predicted scores in the sample differ on average from the observed scores by \(9.705\).

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 Death.

Step 3: In Predictors, enter the column Inauguration.

Step 4: In Graphs, enter the columns Inauguration 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 Death.

Step 3: In Predictors, enter the column Inauguration.

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:

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 death and inauguration.