Question

A simple random sample of​ front-seat occupants involved in car crashes is obtained. Among

2725

occupants not wearing seat​ belts,

39

were killed. Among

7898

occupants wearing seat​ belts,

11

were killed. Use a

0.01

significance level to test the claim that seat belts are effective in reducing fatalities. Complete parts​ (a) through​ (c) below.

a. Test the claim using a hypothesis test.

Consider the first sample to be the sample of occupants not wearing seat belts and the second sample to be the sample of occupants wearing seat belts. What are the null and alternative hypotheses for the hypothesis​test?

b. Test the claim by constructing an appropriate confidence interval.

The appropriate confidence interval is

​(Round to three decimal places as​ needed.)

Answer 1

Say p1 be the proportion of occupants killed not wearing the seat belts

p2 be the proportion of occupants killed wearing the seat belts

a) Ho: p1 = p2

Ha: p1 > p2

This is a one-tailed difference in proportions test of independent samples.

b) The formula for the confidence interval is:

p1^ = 39/2725 = 0.0143

p2^ = 11/7898 = 0.0014

n1 = 2725

n2 = 7898

Let's insert the values into the formula, we get:

z-critical value at 95% confidence interval (assume) is 1.96

Lower bound: 0.0129 - 1.96*0.0023 = 0.0084

Upper bound: 0.0129 + 1.96*0.0023 = 0.0174

The 95% confidence interval is: (0.0084, 0.0174)

The CI is always positive and does not include 0, hence we can reject the null hypothesis and say that p1>p2. The seat belts are effective in reducing fatalities.