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.

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?
oa. ( h_0: p_1
eq p_2 ), ( h_1: p_1 = p_2 )
ob. ( h_0: p_1 = p_2 ), ( h_1: p_1lt p_2 )
oc. ( h_0: p_1 = p_2 ), ( h_1: p_1
eq p_2 )
oe. ( h_0: p_1 = p_2 ), ( h_1: p_1gt p_2 )
od. ( h_0: p_1leq p_2 ), ( h_1: p_1
eq p_2 )
of. ( h_0: p_1geq p_2 ), ( h_1: p_1
eq p_2 )
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 ( alpha = 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 ( 0.0057lt (p_1 - p_2)lt 0.0137 ) (round to three decimal places as needed.)

Explanation:

Step1: Identify Hypotheses

The claim is that seat belts reduce fatalities, so we test if the fatality rate (proportion \( p_1 \) for non - seat - belt users, \( p_2 \) for seat - belt users) is higher for non - seat - belt users. So \( H_0:p_1 = p_2 \), \( H_1:p_1>p_2 \), which corresponds to option E.

Step2: Find Test Statistic

Given \( z = 6.92 \) (from the problem).

Step3: Calculate P - value

For a right - tailed test with \( z = 6.92 \), the P - value is \( P(Z>6.92) \). Since the standard normal distribution table for \( z = 6.92 \) is almost 0 (as \( z \) - scores beyond 3 or 4 have very small probabilities), the P - value is approximately 0.

Step4: Make Conclusion

Since \( \alpha=0.05 \) and P - value \( = 0<0.05 \), we reject \( H_0 \). There is sufficient evidence to support the claim that the fatality rate is higher for those not wearing seat belts.

Step5: Confidence Interval Interpretation

The confidence interval for \( p_1 - p_2 \) is \( 0.0057<(p_1 - p_2)<0.0137 \). Since 0 is not in this interval, it also supports the rejection of \( H_0 \) (that \( p_1 - p_2 = 0 \)) and shows that \( p_1>p_2 \).

Answer:

a. Hypotheses: E. \( H_0:p_1 = p_2 \), \( H_1:p_1>p_2 \); Test statistic \( z = 6.92 \); P - value \( = 0 \); Conclusion: Reject \( H_0 \), sufficient evidence.
b. Confidence interval \( 0.0057<(p_1 - p_2)<0.0137 \) supports the claim.