(a) State the null and alternative hypotheses:
Null Hypothesis (H0): The proportion of drivers wearing seat belts in the South (ps) is equal to or less than the proportion of drivers wearing seat belts in the Northeast (pn).
Alternative Hypothesis (Ha): The proportion of drivers wearing seat belts in the South (ps) is greater than the proportion of drivers wearing seat belts in the Northeast (pn).
(b) Set the significance level (alpha):
Given that a = 0.06 (or 6%), the significance level (alpha) is 0.06.
(c) Calculate the pooled proportion (p):
p = (x1 + x2) / (n1 + n2), where x1 and x2 are the number of drivers wearing seat belts in the South and Northeast, respectively, and n1 and n2 are the sample sizes.
p = (392 + 272) / (460 + 350) = 664 / 810 ≈ 0.820
(d) Calculate the standard error (SE):
SE = sqrt[p * (1 - p) * ((1/n1) + (1/n2))]
SE = sqrt[0.820 * (1 - 0.820) * ((1/460) + (1/350))] ≈ 0.0243
(e) Calculate the test statistic (z):
z = (ps - pn) / SE
z = (392/460 - 272/350) / 0.0243 ≈ 4.166
Now, we need to find the critical value for a one-tailed test at a significance level of 0.06. Consulting the z-table or using a statistical calculator, the critical value is approximately 1.555 (rounded to three decimal places).
Since the calculated test statistic (4.166) is greater than the critical value (1.555), we reject the null hypothesis. There is sufficient evidence to support the claim that the proportion of drivers who wear seat belts is greater in the South than in the Northeast at the significance level of 0.06.