Question 1
(a) The estimated equation corresponding to the regression suggested is as shown below.
Y= β_1+ α_1 x_1+ α_2 x_2+ α_3 x_3+α_4 x_4
Where Y = F – fatal injuries per million man-hours worked;
β_1 = A constant;
x_1 = T – percent of output that was mechanically loaded;
x_2 = S – average number of workers per mine;
x_3 = O – tons of coal produced per man-hours;
x_4 = DU – dummy equal to 1 from 1955 when mining safety and health measures became mandatory.
Figure 1: Regression results
Thus, the fitted regression equation using the coefficients is as follows;
F= 3.541- 0.0243 T- 0.0156 S+ 0.0032 O+0.0719 DU
(b) The coefficients show the significant change in Y-variable when there is a significant change in the factors of study. The constant value shows that Ceteris Paribus the fatal injuries per million-man hours worked in the coal industry in UK between the period of 1940 to 1965 was 3.541 injuries. A significant change in the percent of output that was mechanically loaded by 1-unit Ceteris Paribus leads to a decrease in the fatal injuries per million-man hours worked by 0.0243 injuries. Subsequently, a significant change in average number of workers per mine by 1-unit Ceteris Paribus would lead to a decrease in the fatal injuries per million-man hours worked by 0.0156 injuries. By contrast, a significant change in both tons of coal produced per man-hours and the dummy equal to 1 from 1955 when mining safety and health measures became mandatory leads to an increase of the fatal injuries per million man-hours worked Ceteris Paribus by 0.0032 injuries and 0.0719 injuries respectively.
(c) to check for the relevant individuality to explain F I check the significance of the coefficients estimates and perform a wald test for each predictor factor with the null hypothesis that the predictor effect is equal to zero. All these tests are validated at 5% significance level.
From the figure 1 above, we have the significance tests for the coefficient estimates of T, S and O. At 5% significance level, T and S are significant factors and therefore can individually explain F i.e. Prob = 0.000 and Prob = 0.0164. This is used to test the hypothesis that both factors are not significant factors and, in this case, 0.000 & 0.0164 < 0.05 thus the null hypothesis is rejected. For O, at 5% significance level, Prob = 0.9844 > 0.05 thus, the null hypothesis is accepted in this case, and we conclude O cannot individually explain F.
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