Overview
This lecture explains how to calculate p-values in hypothesis testing using both simulation (randomization) and traditional (test statistic) methods, and how p-values relate to statistical significance.
Calculating p-values by Simulation
- Simulating sample proportions more extreme than the observed statistic helps calculate the p-value.
- In a two-tailed test, add the proportions from both tails to find the total p-value.
- The two-tailed p-value is always twice the one-tailed p-value for symmetric distributions.
- Computers automate simulation and p-value calculation in modern practice.
Calculating p-values by Test Statistic (Traditional Method)
- Before computers, statisticians used test statistics (like t or z scores) and theoretical distributions.
- The p-value is the probability in the tail beyond the test statistic in the distribution.
- For right-tailed tests, the p-value is the area to the right; for left-tailed, to the left; for two-tailed, sum both sides.
- This approach uses lookup tables or calculus to find areas under the curve.
Comparing Methods and Their Relation to Significance
- Both simulation and traditional methods yield similar p-values.
- Four key elements: test statistic, critical value, p-value, significance level.
- If the test statistic falls in the tail (beyond the critical value), the p-value will be less than the significance level.
- If the test statistic does not fall in the tail, the p-value is greater than the significance levelβno statistical significance.
- Both graphs and calculations help visualize the relationship between these concepts.
Key Terms & Definitions
- p-value β Probability of observing data as extreme as the sample, assuming the null hypothesis is true.
- Test statistic β A standardized value calculated from sample data for hypothesis testing.
- Critical value β The threshold value that defines the boundary of the tail(s) in a distribution.
- Significance level (Ξ±) β The cutoff probability (often 0.05) for determining statistical significance.
- Two-tailed test β A test looking for deviations in both directions from the null hypothesis.
Action Items / Next Steps
- Review examples of calculating p-values using both simulation and test statistic methods.
- Prepare to use computer software for p-value calculation in future lessons.