Question

a simple random sample of front - seat occupants involved in car crashes is obtained. among 2765 occupants not wearing seat belts, 31 were killed. among 7891 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. complete parts (a) through (c) below.
identify the test statistic.
z = 6.92
(round to two decimal places as needed)
identify the p - value.
p - value = 0
(round to three decimal places as needed)
what is the conclusion based on the hypothesis test?
the p - value is less than the significance level of α = 0.05, so reject the null hypothesis. there is sufficient evidence to support the claim that the fatality rate is higher for those not wearing seat belts.
b. test the claim by constructing an appropriate confidence interval
the appropriate confidence interval is 0.006 < (p₁ - p₂) < 0.013
(round to three decimal places as needed)
what is the conclusion based on the confidence interval?
because the confidence interval limits for those not wearing seat belts.
, it appears that the two fatality rates are
because the confidence interval limits include values, it appears that the fatality rate is

Explanation:

Step1: Identify Hypotheses

The claim is that seat belts reduce fatalities, so \( p_1 \) (fatal rate for not wearing) > \( p_2 \) (fatal rate for wearing). Null hypothesis \( H_0: p_1 = p_2 \), alternative \( H_1: p_1 > p_2 \). So option E.

Step2: Test Statistic

Given \( z = 6.92 \) (calculated from proportions: \( \hat{p}_1=\frac{31}{2765} \), \( \hat{p}_2=\frac{12}{7891} \), pooled \( \hat{p}=\frac{31 + 12}{2765+7891} \), then \( z=\frac{\hat{p}_1-\hat{p}_2}{\sqrt{\hat{p}(1 - \hat{p})(\frac{1}{n_1}+\frac{1}{n_2})}} \approx 6.92 \)).

Step3: P - value

For \( z = 6.92 \), P - value is \( P(Z>6.92)\approx0 \) (since standard normal table for \( z > 3.5 \) is nearly 0).

Step4: Hypothesis Test Conclusion

Since P - value (\( 0 \)) < \( \alpha = 0.05 \), reject \( H_0 \). Sufficient evidence for \( H_1 \).

Step5: Confidence Interval

Confidence interval for \( p_1 - p_2 \) is \( 0.006 < (p_1 - p_2) < 0.013 \), which does not include 0, so \( p_1 - p_2>0 \), meaning \( p_1 > p_2 \).

Answer:

• Hypotheses: E. \( H_0: p_1 = p_2 \), \( H_1: p_1 > p_2 \)
• Test Statistic: \( z = 6.92 \)
• P - value: \( 0 \)
• Hypothesis Test Conclusion: Reject \( H_0 \), sufficient evidence.
• Confidence Interval Conclusion: Not equal (since interval doesn't include 0), \( p_1 > p_2 \)