Large Data Sets from Appendix B
In Exercises 25–28, use the data set from Appendix B to test the given claim. Use the P-value method unless your instructor specifies otherwise.


Los Angeles Commute Time Use the 1000 Los Angeles Commute times listed in Data Set 31 “Commute Times” to test the claim that the mean Los Angeles commute time is less than 35 minutes. Use a 0.01 significance level. Compare the sample mean to the claimed mean of 35 minutes. Is the difference between those two values statistically significant? Does the difference between those two values appear to have practical significance?

Verified step by step guidance

1. Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is H₀: μ = 35 (the mean commute time is 35 minutes), and the alternative hypothesis is H₁: μ < 35 (the mean commute time is less than 35 minutes). This is a left-tailed test.
2. Step 2: Identify the significance level (α). The problem specifies a significance level of 0.01, which means there is a 1% risk of rejecting the null hypothesis when it is true.
3. Step 3: Calculate the test statistic. Use the formula for the z-test for a population mean: z = (x̄ - μ) / (σ / √n), where x̄ is the sample mean, μ is the claimed population mean (35), σ is the population standard deviation (if known, or use the sample standard deviation if not), and n is the sample size (1000 in this case).
4. Step 4: Determine the P-value. Using the calculated z-test statistic, find the corresponding P-value from the standard normal distribution table. The P-value represents the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true.
5. Step 5: Compare the P-value to the significance level (α). If the P-value is less than 0.01, reject the null hypothesis (H₀). If the P-value is greater than or equal to 0.01, fail to reject the null hypothesis. Then, assess practical significance by considering whether the difference between the sample mean and the claimed mean (35 minutes) is meaningful in the context of commute times.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1). In this case, the null hypothesis would state that the mean commute time is 35 minutes, while the alternative hypothesis would claim it is less than 35 minutes. The goal is to determine whether there is enough evidence to reject the null hypothesis at a specified significance level.

P-value

The P-value is a measure that helps determine the strength of the evidence against the null hypothesis. It represents the probability of obtaining a test statistic as extreme as, or more extreme than, the one observed, assuming the null hypothesis is true. A smaller P-value indicates stronger evidence against the null hypothesis. In this scenario, if the P-value is less than the significance level of 0.01, it suggests that the mean commute time is statistically significantly less than 35 minutes.

Statistical vs. Practical Significance

Statistical significance refers to the likelihood that a result or relationship is caused by something other than mere random chance, typically assessed through P-values. Practical significance, on the other hand, considers whether the size of the effect is large enough to be of real-world importance. In this context, even if the difference in commute times is statistically significant, it is essential to evaluate whether the difference is meaningful in practical terms, such as its impact on daily commuting experiences.