Question

A simple random sample of front-seat occupants involved in car crashes is obtained. Among 2833 occupants not wearing seat belts, 31 were killed. Among 7765 occupants wearing seat belts, 15 were killed.

(a) Construct a 95% confidence interval for the difference of fatality rate between those who wear seat belts and those who don’t.

(b) Based on the confidence interval in (a), can we conclude that seat belts are effective in reducing fatalities? Why?

Answer 1

Part a)

(p̂1 - p̂2) ± Z(α/2) * √( ((p̂1 * q̂1)/ n1) + ((p̂2 * q̂2)/ n2) )
Z(α/2) = Z(0.05 /2) = 1.96
Lower Limit = ( 0.0109 - 0.0019 )- Z(0.05/2) * √(((0.0109 * 0.9891 )/ 2833 ) + ((0.0019 * 0.9981 )/ 7765 ) = 0.0051
upper Limit = ( 0.0109 - 0.0019 )+ Z(0.05/2) * √(((0.0109 * 0.9891 )/ 2833 ) + ((0.0019 * 0.9981 )/ 7765 )) = 0.013
95% Confidence interval is ( 0.0051 , 0.013 )
( 0.0051 < ( P1 - P2 ) < 0.013 )

Part b)

Since the values in the confidence interval is positive, hence seat belts are effective in reducing fatalities.