Answer: We can use a hypothesis test to determine whether there is a statistically significant difference in seat-belt use between the two age groups. Let p1 be the proportion of drivers 25-34 years old who use seat belts, and let p2 be the proportion of drivers 50-59 years old who use seat belts. The null hypothesis is that there is no difference in seat-belt use between the two age groups, or H0: p1 - p2 = 0. The alternative hypothesis is that there is a difference in seat-belt use between the two age groups, or Ha: p1 - p2 ≠ 0.
We can use a two-sample z-test to test this hypothesis. The test statistic is calculated as:
z = (p1 - p2) / sqrt(p_hat * (1 - p_hat) * (1/n1 + 1/n2))
where p_hat = (x1 + x2) / (n1 + n2) is the pooled sample proportion, x1 and x2 are the number of drivers who use seat belts in each age group, and n1 and n2 are the sample sizes.
For the 25-34 age group, we have x1 = 0.28 * 1800 = 504 and n1 = 1800. For the 50-59 age group, we have x2 = 415 and n2 = 1500. The pooled sample proportion is:
p_hat = (x1 + x2) / (n1 + n2) = (504 + 415) / (1800 + 1500) ≈ 0.257
The test statistic is:
z = (0.28 - 0.277) / sqrt(0.257 * (1 - 0.257) * (1/1800 + 1/1500)) ≈ 0.233
Using a two-tailed test at the 1% significance level, the critical value of z is ±2.58. Since the calculated test statistic is less than the critical value in absolute value (|0.233| < 2.58), we fail to reject the null hypothesis. Therefore, we do not have sufficient evidence to conclude that there is a difference in seat-belt use between drivers 25-34 years old and those 50-59 at the 1% significance level.