Overview
This lecture explains how to interpret test statistics and critical values by determining if a sample result falls in the "tail" region during hypothesis testing, using examples from t-tests, z-tests, and chi-square tests.
Critical Values and Tails
- The "tail" of a distribution begins at the critical value on the number line.
- The percentage in the tail is set by the significance level (alpha).
- For two-tailed tests, the left tail starts at the negative critical value, and the right tail at the positive critical value.
- Understanding the number line and placement of critical values is essential for hypothesis testing.
Interpreting Test Statistics
- If the test statistic falls in the tail, the sample data significantly disagrees with the null hypothesis.
- If the test statistic does not fall in the tail, the sample data does not significantly disagree with the null hypothesis; any disagreement may be due to sampling variability.
- For a test statistic value larger than the critical value (in the direction of the tail), the result is significant.
Example 1: Two-Tailed t-Test
- Critical values: right tail at +2.045, left tail at –2.045.
- Test statistic: +2.571 falls in the right tail.
- This indicates significant disagreement with the null hypothesis (e.g., sample mean differs from population mean).
Example 2: Left-Tailed z-Test
- Significance level (alpha): 0.01.
- Critical value: –2.327.
- Test statistic: –1.173 does not fall in the tail.
- Conclusion: Sample data does not significantly disagree with the null hypothesis (e.g., sample proportion close to population proportion).
Example 3: Right-Tailed Chi-Square Test
- Significance level: 0.10.
- Critical value: 7.779 (degrees of freedom = 4).
- Test statistic: 11.328 falls in the tail.
- Conclusion: Sample data significantly disagrees with the null hypothesis.
Key Terms & Definitions
- Critical Value — The cutoff point on the number line where the tail of the distribution begins, based on the significance level.
- Tail — The region beyond the critical value where results are considered statistically significant.
- Significance Level (Alpha) — The probability threshold set to determine "rare" or significant results, often 0.05 or 0.01.
- Test Statistic — A calculated value (e.g., t, z, chi-square) from sample data used to test the null hypothesis.
- Null Hypothesis — The default assumption that there is no effect or difference.
Action Items / Next Steps
- Practice drawing number lines and locating critical values and test statistics for different tests.
- Review using technology or tables (e.g., StatKey) to find critical values for various distributions.
- Prepare for a demonstration on calculating critical values with StatKey in the next session.