Overview
This lecture reviews the relationship between confidence intervals and hypothesis tests when analyzing means, focusing on conditions for when both methods provide the same conclusion.
Conditions for Matching Results
- Confidence intervals and hypothesis tests yield the same result if two conditions are met.
- Condition 1: The alternative hypothesis (Ha) uses the "not equal to" (≠) symbol.
- Condition 2: The confidence level (%) plus the significance level (α) adds up to 100%.
Practical Example
- Hypothesis: Null (H₀): μ = 80; Alternative (Ha): μ ≠ 80; where μ is the average height of male college basketball players.
- Significance level is set to 5% (α = 0.05).
- The p-value calculated is 0.02.
- Since p-value < significance level (0.02 < 0.05), reject the null hypothesis.
- Since both conditions are satisfied (Ha uses ≠ and 95% + 5% = 100%), results of the hypothesis test and confidence interval will match.
Interpreting Results
- Confidence interval defines plausible values for μ (the population mean).
- If the null value (80) is rejected in the hypothesis test, it will not be contained in the 95% confidence interval.
- Rejection of the null hypothesis corresponds to the null value not appearing in the confidence interval.
Key Terms & Definitions
- Null Hypothesis (H₀) — The default claim that there is no effect or difference (e.g., μ = 80).
- Alternative Hypothesis (Ha) — The claim we test for, often stating μ ≠ value.
- Significance Level (α) — The threshold for rejecting H₀, typically 0.05 (5%).
- Confidence Interval — A range of plausible values for the population parameter at a specific confidence level.
- P-value — Probability of observing data at least as extreme as the sample, assuming H₀ is true.
Action Items / Next Steps
- Review Sections 9.3 and 9.4 on confidence intervals and hypothesis testing.
- Understand how to apply the two conditions for matching results.
- Practice identifying whether a confidence interval will contain the null value based on test results.