Question

according to a certain government traffic safety agency for a large country, the proportion of fatal traffic accidents in the country in which the driver had a positive blood alcohol concentration (bac) is 0.34. suppose a random sample of 107 traffic fatalities in a certain region with a large population results in 45 that involved a positive bac. does the sample evidence suggest that the region has a higher proportion of traffic fatalities involving a positive bac than the country at the α = 0.05 level of significance? next, interpret the test statistic. the sample proportion is \\(\square\\) standard error(s) \\(\square\\) the proportion stated in the null hypothesis (round to two decimal places as needed.) find the p - value. p - value = \\(\square\\) (round to three decimal places as needed.) choose the correct conclusion for this hypothesis test. \\(\bigcirc\\) a. since p - value < α, do not reject the null hypothesis and conclude that there is not sufficient evidence that the region has a higher proportion of traffic fatalities involving a positive bac than the country \\(\bigcirc\\) b. since p - value < α, reject the null hypothesis and conclude that there is sufficient evidence that the region has a higher proportion of traffic fatalities involving a positive bac than the country

Explanation:

Step1: Define Hypotheses

Null hypothesis \( H_0: p = 0.34 \), Alternative hypothesis \( H_1: p > 0.34 \) (since we test if region has higher proportion). Sample size \( n = 107 \), number of successes \( x = 45 \). Sample proportion \( \hat{p}=\frac{45}{107}\approx0.4206 \).

Step2: Calculate Test Statistic

Standard error \( SE=\sqrt{\frac{p_0(1 - p_0)}{n}}=\sqrt{\frac{0.34(1 - 0.34)}{107}}\approx\sqrt{\frac{0.34\times0.66}{107}}\approx\sqrt{\frac{0.2244}{107}}\approx0.0457 \). Test statistic \( z=\frac{\hat{p}-p_0}{SE}=\frac{0.4206 - 0.34}{0.0457}\approx\frac{0.0806}{0.0457}\approx1.76 \). So sample proportion is 1.76 standard errors above the null proportion.

Step3: Find P - value

For right - tailed test, \( P - value = P(Z > 1.76) \). From standard normal table, \( P(Z\leq1.76)=0.9608 \), so \( P - value=1 - 0.9608 = 0.0392\approx0.039 \).

Step4: Conclusion

\( \alpha = 0.05 \), since \( P - value=0.039<0.05=\alpha \), we reject the null hypothesis. So the correct conclusion is option B.

Answer:

The sample proportion is \( 1.76 \) standard error(s) above the proportion stated in the null hypothesis.
\( P - value=\boldsymbol{0.039} \)
The correct conclusion is B. Since \( P - value < \alpha \), reject the null hypothesis and conclude that there is sufficient evidence that the region has a higher proportion of traffic fatalities involving a positive BAC than the country.