In a regression analysis of on-the-job head injuries of warehouse labourers caus

Question

In a regression analysis of on-the-job head injuries of warehouse labourers caused by falling objects, Y is a measure of severity of the injury, X_1 is an index reflecting both the weight and the distance it felt, and X_2 and X_3 are indicator variables for nature of head protection worn at the time of the accident, coded as follows: Type of protection X_2 X_3 Hard hat 1 0 Bump cap 0 1 None 0 1 The response function to be used in the study is E(Y) = beta0+beta1*X1+beta2*X2+beta3*X3 A. Develop the response function for each type of protection category B. For each of the following questions specify the alternatives H_0 and H_1 for the appropriate test: (1) with X1 fixed, does wearing a bump cap reduce the expected severity of injury as compared with wearing no protection, (2) with X1 fixed, is the expected severity of injury the same when wearing a hard hat as whwn wearing a bump cap?

Explanation / Answer

(1) X1 is fixed and we want to check if wearing a bump cap reduces the expected severity
So, for bump cap, X2 = 0 and X3 = 1.
The regression equation then is : Y = beta0 + beta1*X1 + beta3*X3
So we need to check if beta3 = 0 or not to compare with bump cap vs no protection
H0 : beta3 = 0
H1 : beta3 not equal to 0

(2) With X1 fixed and wearing a hard hat, X2 = 1 and X3 = 0
The regression equation then for hard hat is : Y = beta0 + beta1*X1 + beta2*X2
and for bump cap is : Y = beta0 + beta1*X1 + beta3*X3
So, to compare the expected severity is both cases the appropriate test is :
H0 : beta2 = beta3
H1: beta2 not equal to beta3

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