Overview
This lecture covers comparing gasoline prices across three cities using two-sample t-tests, including how to apply the Bonferroni correction for multiple comparisons.
Number of Comparisons
- With three samples (cities), there are three pairwise comparisons: Denver vs Houston, Houston vs Cleveland, Denver vs Cleveland.
Bonferroni Correction
- The overall significance level (alpha) is 0.05.
- The Bonferroni-corrected significance level per comparison is 0.05 ÷ 3 = 0.0167.
Sample Means
- Denver sample mean: 2.738.
- Houston sample mean: 2.698.
- Cleveland sample mean: 2.9.
Hypotheses for t-tests
- Null hypothesis (H₀): All city means are equal (μ₁ = μ₂ = μ₃); for each test, H₀: μ₁ = μ₂.
- Alternative hypothesis (H₁): The means are not equal (μ₁ ≠ μ₂).
Test Statistics and p-values
- Denver vs Houston: t = 2.7091, p = 0.0389.
- Houston vs Cleveland: t = -5.0754, p = 0.0038.
- Denver vs Cleveland: t = -4.278, p = 0.0116.
Statistical Conclusions
- Denver vs Houston: p > 0.0167 → Fail to reject H₀ (not significantly different).
- Houston vs Cleveland: p < 0.0167 → Reject H₀ (significantly different).
- Denver vs Cleveland: p < 0.0167 → Reject H₀ (significantly different).
- Cleveland has a significantly higher average price than both Denver and Houston.
Key Terms & Definitions
- Bonferroni Correction — Adjustment to the significance level when conducting multiple comparisons to reduce Type I error.
- Two-sample t-test — Statistical test comparing the means of two independent samples.
- Null Hypothesis (H₀) — Assumes no difference between group means.
- Alternative Hypothesis (H₁) — Assumes a difference between group means.
- p-value — Probability of obtaining results as extreme as observed, under H₀.
Action Items / Next Steps
- Write down whether you rejected or failed to reject the null hypothesis for each pair.
- Try constructing Bonferroni-corrected confidence intervals for each city comparison.