Question

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

2989

occupants not wearing seat​ belts, 29 were killed. Among 7785 occupants wearing seat​ belts, 12 were killed. Use a 0.05 significance level to test the claim that seat belts are effective in reducing fatalities

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. Identify the test statistic.

C. Identify the​ P-value.

D. What is the conclusion based on the hypothesis​ test?

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

F. What is the conclusion based on the confidence​ interval?

G. What do the results suggest about the effectiveness of seat​ belts?

Answer 1

For sample 1, we have that the sample size is N_1= 2989 , the number of favorable cases is X_1 = 29 , so then the sample proportion is

For sample 2, we have that the sample size is N_2 = 7785 , the number of favorable cases is X_2 = 12 , so then the sample proportion is

The value of the pooled proportion is computed as

Also, the given significance level is α=0.05.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: p_1 = p_2

Ha: p_1 > p_2

This corresponds to a right-tailed test, for which a z-test for two population proportions needs to be conducted.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a right-tailed test is z_c = 1.64

(3) Test Statistics

The z-statistic is computed as follows:

Z = 6.16

(4) Decision about the null hypothesis

Since it is observed that z = 6.16 >zc​=1.64, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p = 0 , and since p = 0 <0.05, it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population proportion p1​ is greater than p2​, at the 0.05 significance level.

The critical value for α=0.05 is z_c = 1.96. The corresponding confidence interval is computed as shown below:

CI = (0.005, 0.012)  

hence both the calculation shows that seat belts prevent number of deaths