Question

The National Safety Council routinely analyzes the benefit of seat belt use on driver safety. Their data showed that among 2823 drivers not wearing seat belts, 31 died as a result of injuries, and among 7765 drivers wearing seat belts 16 were killed. Test the hypothesis (at 95% confidence) that there is no difference in the proportion of deaths between the 2 groups. What do you conclude? Calculate the margin of error (E) at 95%.

Answer 1

Two-Proportion Z test

The following information is provided: (a) Sample 1 - The sample size is N1 = 2823, the number of favorable cases is X1 = 31 and the sample proportion is p^1​=X1/N1​=31/2823​=0.011 (b) Sample 2 - The sample size is N2 = 7765, the number of favorable cases is X2 = 16 and the sample proportion is p^2​=X2/N2​=16/7765​=0.0021 and the significance level is α=0.05 Pooled Proportion The value of the pooled proportion is computed as (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: p1 = p2 Ha: p1 ≠ p2 This corresponds to a Two-tailed test, for which a z-test for two population proportions needs to be conducted. (2a) Critical Value Based on the information provided, the significance level is α=0.05, therefore the critical value for this Two-tailed test is Zc​=1.96. This can be found by either using excel or the Z distribution table. (2b) Rejection Region The rejection region for this Two-tailed test is |Z|>1.96 i.e. Z>1.96 or Z<-1.96 (3) Test Statistics The z-statistic is computed as follows: (4) The p-value The p-value is the probability of obtaining sample results as extreme or more extreme than the sample results obtained, under the assumption that the null hypothesis is true. In this case, the p-value is p =P(|Z|>6.1058)=0 (5) The Decision about the null hypothesis (a) Using traditional method Since it is observed that |Z|=6.1058 > Zc​=1.96, it is then concluded that the null hypothesis is rejected. (b) Using p-value method 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. (6) Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population proportion p1 is different than p2, at the 0.05 significance level.

Margin of error:

Two-Proportion Confidence Interval

We need to construct the 95% Margin of error for the difference between population proportions p1​−p2. We have been provided with the following information: (a) Sample 1 - The sample size is N1 = 2823, the number of favorable cases is X1 = 31 and the sample proportion is p^1​=X1/N1​=31/2823​=0.011 (b) Sample 2 - The sample size is N2 = 7765, the number of favorable cases is X2 = 16 and the sample proportion is p^2​=X2/N2​=16/7765​=0.0021 and the significance level is α=0.05 Critical Value Based on the information provided, the significance level is α=0.05, therefore the critical value is Zc​=1.96. This can be found by either using excel or the Z distribution table. Margin of Error

Let me know in the comments if anything is not clear. I will reply ASAP! Please do upvote if satisfied!