Predictor Coef St. dev. T P. Analysis of Variance

Constant 87.92 13.96 6.30 0.000

X1 -3.3869 0.9930 -3.41 0.011

X2 2.327 2.264 1.03 0.338

S = 13.77 R-SQ = 78.4% R-SQ(adj) = 72.3%

Analysis of Variance

Source DF SS MS F P

Regression 2 4824.5 2412.3 12.72 0.005

Residual Error 7 1327.9 189.7

Total 9 6152.4

Source DF SEQ. SS

X1 1 4624.1

X2 1 200.4

We now proceed to interpret the results in Figure 4.1 and use them to make further statistical inferences.

a) The equation of the fitted linear regression is

From this equation,

; ;

We can also read the values of these coefficients from the column labeled

COEF in the MINITAB solution of Figure 4.1.

Notice that in this column the coefficients appear with more digits after the decimal point. With these coefficient values, we can write the estimated regression equation as

b) The value of is in the estimated regression equation gives the value of for and . It means that a driver with no experience and no driving violations is expected to pay an auto insurance premium of $87.92 per year. This is the technical interpretation of a.

The value of in the estimated regression model gives the change in for a one unit change in when is held constant. Thus, we can state that a driver with one extra year of experience but with the same number of violations is expected to pay $3.3869 less for the auto insurance premium per year.

The value of in the estimated regression model gives change in for a one unit change in when is held constant. Thus, we can state that a driver with one extra driving violation but with the same years of driving experience is expected to pay $82.327 more per year for the auto insurance premium.

b) is estimated by , so .

The values of the standard deviation of errors, the coefficient of determination and the adjusted coefficient of determination are also given in MINITAB solution. From Figure 4.1 we obtain

; R-SQ ; R-SQ (adj) =72.3%

The value of tells us that the two independent variables included in our model explain 78.4% of the variation in the dependent variable.

The value of is the value of the coefficient of determination adjusted for degrees of freedom. It states that when adjusted for degrees of freedom, the two independent variables explain 72.3% of the variation in the dependent variable.

c) To predict auto premium paid per year by a driver with seven years of experience and four driving violations, we substitute and in the estimated regression model

Note that this value of is a point estimate of the predicted value of y, which is denoted by .

Remark:

In figure 4.1 there is portion of solution in the end reproduced below,

Source DF SEQ. SS

X1 1 4624.1

X2 1 200.4

which we have not used in any of the examples. From figure 4.1 we have

If we estimate the simple linear regression of y on ,

the value of SSR will be 4624.1, which is the value in the row of X1 and the column labeled SEQ.SS. That is, alone will reduce SST by 4624.1

Then, if we add to model above, the SST will further be reduced by 200.4, which is the value in the row of X2 and the column labeled SEQ.SS.

The sum of the two numbers in the column of SEQ.SS is

which is the value of SSR in the Figure 4.1.

Example2:

We are interested in studying the blood pressure y of males in relation to weight and age . Sample of 10 male was selected. The data set listed

below:

Use a computer package to perform a regression analysis using model

Solution:

Using MINITAB, we first enter the data of y, , in three different columns and then use the regression command. The computer executes a multiple regression analysis. We focus our attention on the principal aspects of the output as shown in Table 4.2

Table 4.2

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