A researcher wanted to test the claim that, "Seat belts are effective in reducing fatalities." A simple random sample of front-seat occupants involved in car crashes is obtained. Among 2823 occupants not wearing seat belts, 31 were killed. Among 7765 occupants wearing seat belts, 16 were killed. If a significance level of 5% is used, which of the following statements gives the correct conclusion?

a. Since p >α we conclude that this data shows that seat belts are effective in reducing fatalities.
b. Since p <α, we conclude that this data shows that seat belts are effective in reducing fatalities U
c. Since p >α we conclude that this data shows that seat belts aren't effective in reducing fatalities.
d. Since p<α, we conclude that this data shes that seat belts aren't effective in reducing fatalities.

This is a test of 2 population proportions. Let 1 and 2 be the subscript for the occupants not wearing seat belts and occupants wearing seat belts. The population proportion of occupants not wearing seat belts and occupants wearing seat belts would be p1 and p2 respectively.

P1 - P2 = difference in the proportion of occupants not wearing seat belts and occupants wearing seat belts.

The null hypothesis is

H0 : p1 = p2

p1 - p2 = 0

The alternative hypothesis is

Ha : p1 > p2

p1 - p2 > 0

it is a right tailed test

Sample proportion = x/n

Where

x represents number of success

n represents number of samples.

For occupants not wearing seat belts,

x1 = 31

n1 = 2823

P1 = 31/2823 = 0.011

For occupants wearing seat belts,

x2 = 16

n2 = 7765

P2 = 16/7765 = 0.0021

The pooled proportion, pc is

pc = (x1 + x2)/(n1 + n2)

pc = (31 + 16)/(2823 + 7765) = 0.0044

1 - pc = 1 - 0.0044 = 0.9956

z = (P1 - P2)/√pc(1 - pc)(1/n1 + 1/n2)

z = (0.011 - 0.0021)/√(0.0044)(0.9956)(1/2823 + 1/7765) = - 0.0089/0.00145462023

z = 6.81

From the normal distribution table,

p < 0.00001

0.00001 < 0.05, we would reject the null hypothesis.

Therefore,

b. Since p <α, we conclude that this data shows that seat belts are effective in reducing fatalities