A simple random sample of front-seat occupants inv

Questions: A simple random sample of front-seat occupants involved in car crashes is obtained. Among 2996 occupants not wearing seat belts, 31 were killed. Among 7649 occupants wearing seat belts, 11 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? A. H0: p1 >= p2 H1: p1 != p2 B. H0: p1 != p2 D. H0: P1=p2 E. H0: P1 <= P2 C. H0: P1=P2 H1: P1>P2 H1: p1 != p2 H0: P1=P2 H1: P1<P2

Transcript text: A simple random sample of front-seat occupants involved in car crashes is obtained. Among 2996 occupants not wearing seat belts, 31 were killed. Among 7649 occupants wearing seat belts, 11 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? A. $H_{0}: p_{1} \geq p_{2}$ $H_{1}: p_{1} \neq p_{2}$ B. $H_{0}: p_{1} \neq p_{2}$ D. $H_{0}: P_{1}=p_{2}$ E. $H_{0}: P_{1} \leq P_{2}$ C. $H_{0}: P_{1}=P_{2}$ $H_{1}: P_{1}>P_{2}$ $H_{1}: p_{1} \neq p_{2}$ $H_{0}: P_{1}=P_{2}$ $H_{1}: P_{1}

Solution

Solution Steps

Step 1: State the Hypotheses

We are testing the effectiveness of seat belts in reducing fatalities in car crashes. The hypotheses are defined as follows:

Null Hypothesis ($H_0$): $p_1 = p_2$ (The proportion of fatalities is the same for both groups)
Alternative Hypothesis ($H_1$): $p_1 > p_2$ (The proportion of fatalities is higher for those not wearing seat belts)

Step 2: Calculate the Test Statistic

The test statistic $Z$ is calculated using the formula:

\[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \]

Where:

$\hat{p} = \frac{31}{2996} \approx 0.01035$ (proportion of fatalities for occupants not wearing seat belts)
$p_0 = \frac{11}{7649} \approx 0.00144$ (proportion of fatalities for occupants wearing seat belts)
$n = 2996$ (sample size for occupants not wearing seat belts)

Substituting the values, we find:

\[ Z \approx 12.8683 \]

Step 3: Determine the P-value

The P-value associated with the test statistic $Z = 12.8683$ is calculated to be:

\[ \text{P-value} = 0.0 \]

Step 4: Compare with the Critical Value

For a significance level of $\alpha = 0.05$ in a right-tailed test, the critical value is:

\[ Z_{critical} \approx 1.6449 \]

Since $Z = 12.8683 > 1.6449$, we reject the null hypothesis.

Step 5: Conclusion

Given that the P-value is significantly less than $\alpha$ and the test statistic exceeds the critical value, we conclude that there is sufficient evidence to support the claim that seat belts are effective in reducing fatalities.