Bonferroni-Corrected Confidence Intervals

Jul 25, 2025

Overview

This lecture covers how to construct and interpret Bonferroni-corrected confidence intervals for multiple two-sample t-tests, comparing mean gas prices across three cities.

Bonferroni Correction and Confidence Level

  • The confidence level (C-level) for Bonferroni correction is 1 minus adjusted alpha.
  • For three comparisons, adjusted alpha is 0.0167, so C-level is 0.9833 (or 98.33%).

Computing Two-Sample T-Intervals

  • Use the CLT (Central Limit Theorem) since sample sizes are sufficient.
  • Perform a two-sample t-interval (2-SampTInt) for each pair of cities using calculator:
    • Denver vs. Houston: interval is (-0.0104, 0.09038).
    • Houston vs. Cleveland: interval is (-0.3422, -0.0618).
    • Denver vs. Cleveland: interval is (-0.3079, -0.0161).
  • Each interval estimates μ₁ - μ₂ (mean difference between groups).

Interpreting Confidence Intervals

  • If the interval contains zero, there’s no significant difference between groups (e.g., Denver vs. Houston).
  • If the interval is entirely negative, the second city's mean is higher (Cleveland is higher than both Houston and Denver).
  • The interval bounds (e.g., 6 to 34 cents, or 2 to 31 cents) indicate the estimated average price difference.

Comparing to Previous Results

  • Conclusions from Bonferroni intervals match results from prior examples.
  • Confidence intervals are harder to interpret than hypothesis tests with two samples.

Key Terms & Definitions

  • Bonferroni Correction — Technique to adjust alpha when making multiple comparisons to control the overall error rate.
  • Two-Sample t-Interval (2-SampTInt) — Confidence interval for the difference in means between two independent groups.
  • C-level (Confidence Level) — Probability that the interval contains the true parameter value.
  • μ₁ - μ₂ — The difference between the means of group 1 and group 2.

Action Items / Next Steps

  • Complete any written interpretations by filling in the specific interval values as needed.
  • Review differences between confidence intervals and hypothesis testing for two-sample comparisons.

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