Question

Using the above data, the following multi-linear regression model was fitted. Y=12.8800+1.9360X1 +1.6090X2 -2.3960X3 +0.1640X4 (3.847) (1.1260) (1.6680) (2.3770) (0.8520) where values in brackets represent the standard error of the corresponding S.E(j=0,1, 2, 3, 4 R=0.8640 Required:

a) Estimate the total expenditure for X12.5.X21.5, X3 3.0 and X4 = 5.0

b) Test the significance of the overall model at 5% level of significance.

c) Test the null hypothesis that interest payments have no influence on Total Expenditure at 5% level of significance.

d) Test the alternative hypothesis that security expenditure has positive influence on Total expenditure at 10% level of significance

e) Test the significance of the regression coefficient 4 at 5% level of significance and interpret your results​

Answer 1

a) To estimate the total expenditure (\(Y\)) for given values of \(X_1, X_2, X_3,\) and \(X_4\), substitute these values into the regression equation:

\[ Y = 12.8800 + 1.9360X_1 + 1.6090X_2 - 2.3960X_3 + 0.1640X_4 \]

For \(X_1 = 2.5, X_2 = 1.5, X_3 = 3.0, X_4 = 5.0\), plug these values into the equation to find the estimated total expenditure.

b) To test the significance of the overall model, you can use the F-test. The null hypothesis (\(H_0\)) is that all regression coefficients are equal to zero. The test statistic is calculated as the ratio of the explained variance to the unexplained variance. Compare this to the critical F-value at a 5% level of significance.

c) To test the null hypothesis that interest payments (\(X_3\)) have no influence on total expenditure, you can use the t-test for the corresponding coefficient. The null hypothesis (\(H_0\)) is that the coefficient for \(X_3\) is equal to zero. Compare the t-statistic to the critical t-value at a 5% level of significance.

d) To test the alternative hypothesis that security expenditure (\(X_4\)) has a positive influence on total expenditure, use a one-sided t-test. The null hypothesis (\(H_0\)) is that the coefficient for \(X_4\) is less than or equal to zero. Compare the t-statistic to the critical t-value at a 10% level of significance.

e) To test the significance of the regression coefficient for \(X_4\), you can use a t-test. The null hypothesis (\(H_0\)) is that the coefficient for \(X_4\) is equal to zero. Compare the t-statistic to the critical t-value at a 5% level of significance. If the p-value is less than 0.05, you reject the null hypothesis and conclude that \(X_4\) is a significant predictor of total expenditure.