Hypothesis Testing

Motivation

  • Point estimation and sample variation discussed.

  • Sample mean may not equal population mean.

Hypothesis Testing Framework

  • Aim to determine if SPH smokers smoke differently than Texas average (21.8 cigs/day).

  • Two parts:

    • Null Hypothesis (H0): No difference (H0: μ = 21.8)

    • Alternative Hypothesis (H1): Difference exists (H1: μ ≠ 21.8)

Research Question

  • Are SPH smokers smoking a different number of cigs/day than Texan smokers in 2012?

Steps for Hypothesis Testing

  1. Determine research question.

  2. Formulate H0 and H1 based on population parameters.

  3. Identify test statistic.

  4. Define decision rule: Rejection region based on significance level (α) and distribution assumptions.

  5. Perform test; decide to reject or fail to reject H0.

  6. Conclude in scientific terms.

Decision Making Methods

  • Test statistic vs critical value.

  • P-value comparison vs significance level.

  • Confidence interval analysis.

P-value Concept

  • Importance in evaluating significance; computed from sample mean.

One-sided vs Two-sided Tests

  • One-sided: Tests a specific direction (H0: μ = k vs H1: μ > k or H1: μ < k).

  • Two-sided: Tests for any difference (H0: μ = k vs H1: μ ≠ k).

Decision Rule (P-value)

  • Reject H0 if p-value < α; Fail to reject if p-value ≥ α.

Critical Values

  • One-sided tests use specific critical values based on significance levels (e.g., 0.05).

  • Critical values determine rejection region.

Examples of Hypothesis Testing

  • Example 3b: Two-sided test with sleep study; if p-value > 0.05, fail to reject H0.

Statistical vs Practical Significance

  • Statistical significance indicates rejection of null hypothesis.

  • Practical significance relates to real-world importance of results.

Final Recap of Hypothesis Testing

  1. Identify research question and parameter.

  2. Decide on test type (one/two-sided).

  3. Establish H0 and H1.

  4. Choose significance level and method for testing (p-value, critical value).

  5. State conclusion in relatable terms.

Important Notes on P-value

  • A smaller p-value indicates stronger evidence against H0.

  • Context matters; statistical significance does not guarantee practical significance.