Overview
This lecture introduces hypothesis testing in statistics, focusing on how to use null and alternative hypotheses, p values, and significance levels to assess differences and avoid common pitfalls in data interpretation.
Perception vs. Reality in Statistics
- Human perceptions are often misleading and need careful inspection.
- Hypothesis testing helps distinguish real differences from imagined ones.
Hypotheses in Statistics
- A hypothesis is a specific, testable claim about a real-world outcome or difference.
- The null hypothesis states that there is no effect or no difference.
- The alternative hypothesis asserts that there is a meaningful effect or difference.
Hypothesis Testing Process
- Collect data relevant to both the null and alternative hypotheses.
- Use statistical tests to compare data to models predicted by the hypotheses.
- Hypotheses should be specific and measurable for reliable testing.
One-Tailed vs. Two-Tailed Tests
- One-tailed test: predicts an effect in a specific direction (e.g., more or less).
- Two-tailed test: tests for any difference, regardless of direction.
The Role of the p Value
- The p value is the probability that data as extreme as observed would occur if the null hypothesis is true.
- A small p value (< significance level) suggests data are unlikely under the null hypothesis.
Significance Levels and Statistical Decisions
- Significance level (often 5% or 1%) is the threshold for rejecting the null hypothesis.
- If p value < significance level, reject the null hypothesis; if not, fail to reject it.
- Failing to reject does not mean the null is true; it means insufficient evidence for the alternative.
Limitations and Pitfalls
- Hypothesis testing canβt prove a hypothesis is true, only provide evidence for/against.
- The significance threshold is somewhat arbitrary.
- A p value is not the probability that the hypothesis is true.
Key Terms & Definitions
- Null Hypothesis (Hβ) β a statement that there is no effect or difference.
- Alternative Hypothesis (Hβ) β a statement that there is an effect or difference.
- p Value β probability of obtaining observed data, or more extreme, if the null hypothesis is true.
- Significance Level (Ξ±) β chosen probability threshold for rejecting the null hypothesis.
Action Items / Next Steps
- Review definitions of null and alternative hypotheses, p values, and significance levels.
- Practice forming specific, measurable hypotheses.
- Understand how to interpret p values in context with significance levels.