Final answer:
To test if the mean commute time in the U.S. is less than half an hour, a one-sample t-test is used with a null hypothesis that the mean is 30 minutes. If the calculated t-value is less than the critical value at 5% significance level, we can reject the null hypothesis. The assumption of normality should be considered due to the skewed distribution.
Step-by-step explanation:
To determine if the mean commute time in the U.S. is less than half an hour based on the sample data provided, we can perform a one-sample t-test. The null hypothesis (H0) is that the mean commute time is equal to 30 minutes, and the alternative hypothesis (H1) is that the mean commute time is less than 30 minutes.
Using the sample mean (μ) of 28.0 minutes, standard deviation (s) of 19.1 minutes, and sample size (n) of 510, the test statistic (t) is calculated as follows:
t = (Sample mean - Population mean) / (Sample standard deviation / sqrt(n))
t = (28.0 - 30.0) / (19.1 / sqrt(510))
After calculating the t-value, we compare it to the critical value from the t-distribution table at the 5% significance level (α = 0.05) with 509 degrees of freedom (since n-1 = 510-1).
If the calculated t-value is less than the critical value, we reject the null hypothesis and conclude that the mean commute time is significantly less than half an hour. However, if the t-value is greater than or equal to the critical value, we fail to reject the null hypothesis, which means we do not have sufficient evidence to claim that the mean commute time is less than half an hour.
It should also be noted that the assumption of normality is especially important in this case, as the skewed distribution could affect the validity of the t-test results.