In a random sample of 200 men, 125 said they used seat belts. In a random sample of 300 women, 65 said they used seat belts. Test the claim that women are less safety-conscious than men at $\alpha = 0.01$. Use the $P$-value method.
A) There is not enough information to draw a conclusion.
B) Yes. There is enough evidence to support the claim that women are less safety-conscious than men.
C) No. There is not enough evidence to support the claim that women are less safety-conscious than men.

Question

Quizplus · Accepted Answer

To test the claim, we use a two-proportion z-test. The null hypothesis $H_0$ is that the proportions of men and women who use seat belts are equal, and the alternative hypothesis $H_a$ is that the proportion of women who use seat belts is less than that of men.1. Calculate the test statistic:$$ p_1 = \frac{125}{200}, \quad p_2 = \frac{65}{300} $$$$ \hat{p} = \frac{125 + 65}{200 + 300} = \frac{190}{500} $$$$ z = \frac{p_1 - p_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}} $$Plugging in the numbers:$$ z = \frac{0.625 - 0.2167}{\sqrt{0.38(1-0.38)(\frac{1}{200} + \frac{1}{300})}} \approx 6.96 $$2. Find the $P$-value:The $P$-value is the probability of observing a test statistic as extreme as, or more extreme than, the observed value under the null hypothesis. For $z = 6.96$, the $P$-value is very small, much less than the significance level of $\alpha = 0.01$.Conclusion:Since the $P$-value is less than $\alpha$, we reject the null hypothesis and conclude that there is enough evidence to support the claim that women are less safety-conscious than men, based on the use of seat belts.

In a Random Sample of 200 Men, 125 Said They α=0.01\alpha = 0.01α=0.01

In a Random Sample of 200 Men, 125 Said They $\alpha = 0.01$