The following estimated regression equation is given: Y = 7.62 - 0.16x1 + 1.23x2

Question

The following estimated regression equation is given:

Y = 7.62 - 0.16x1 + 1.23x2

R2 = 0.37         Sb1 = 0.008      Sb2 = 0.496

where:

y – price-earnings ratio

x1 – size of insurance company assets

x2 – dummy variable taking the value of 1 for regional companies and 0 for national companies.

1. Interpret the regression coefficients.

2. Test whether the regression coefficients (in the population) of the independent variables are significantly different from zero. Use a sample size of 30. Make sure to state the null and the alternative hypotheses and the decision rule.

3. Interpret the coefficient of determination.

Explanation / Answer

(1)

The value of the intercept is 7.62 which is the value of Y when X1 and X2 are 0

The value of the regression coefficient of X1 is -0.16 which implies that for each unit of increase in X1, Y deceases by 1 unit. Therefore X1 have a negative effect on Y. Thus, as the size of insurance company assets goes up, price-earnings ratio goes down.

The value of the regression coefficient of X2 is 1.23 which implies that for each unit of increase in X2, Y increases by 1 unit. Therefore X2 have a positive effect on Y. Thus if it is a regional company, then it is likely to hve a price-earnings ratio than a national company.

(2)

H0: 1 = 0 i.e. the coefficient of X1 is not significantly different from 0
Ha: 1 0 i.e. the coefficient of X1 is significantly different from 0.

We use the test statistic t = B1/Sb1 to test the above above hypothesis. Therefore the value of the test statistic is:

t = B1/Sb1 = -0.16/0.008 = -20.

At (n-2) = (30-2) = 28 d.f. the corresponding p value is: 0. Therefore as the p value is less than the significance level (0.05) we reject our null hypothesis and conclude that the coefficient of X1 is significantly different from 0.

Similarly let us consider the case of X2.

H0: 2 = 0 i.e. the coefficient of X2 is not significantly different from 0
Ha: 2 0 i.e. the coefficient of X2 is significantly different from 0.

We use the test statistic t = B2/Sb2 to test the above above hypothesis. Therefore the value of the test statistic is:

t = B2/Sb2 = 1.23/0.496 = 2.479839.

At (n-2) = (30-2) = 28 d.f. the corresponding p value is: 0.01942847. Therefore since the p value is less than the significance level (0.05) we reject our null hypothesis and conclude that the coefficient of X2 is also significantly different from 0.

(3)

The coefficient of determination = 0.37 implies that X1 and X2 accounts for only 37% of changes in Y. The rest 63% of variation in Y is explained by some other factors.

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