AP Stats Ch. 24 (Matched Pairs t-test and confidence intervals)

Matched Pairs Overview

Definition: Matched pairs involve pairing up data to minimize variability between groups.

Purpose: Used in hypothesis testing and confidence intervals for data that is naturally related.

Examples of Matched Pair Designs

  • Observations collected in pairs, such as:

    • Same subjects measured before and after treatment.

    • Twin studies comparing twins' characteristics.

    • Comparing left and right shoes on the same person.

Motivation: Eliminate variability and improve the accuracy of testing.

Differences and Pairwise Analysis

  • When data is paired, the analysis focuses on the pairwise differences rather than the original datasets.

  • Calculation: Differences are calculated from paired data, and these differences are analyzed using a one-sample t-test.

  • Example: For a left foot/right foot dataset, calculate a third column based on the differences.

Paired T-Test Mechanics

  • Key Components:

    • Null Hypothesis (H0): The mean of the differences (μ_d) equals zero (no effect).

    • Alternative Hypothesis (H1): μ_d is greater than zero (indicates an effect).

    • Formula:

    [ t = \frac{\bar{d} - \mu_0}{\text{SE}} ] Where ( \bar{d} ) = average of the differences, ( \mu_0 ) is typically zero, and ( \text{SE} ) is the standard error of the differences.

    • Degrees of Freedom: ( n - 1 ), where n is the number of pairs.

    • Obtain p-value using the t-distribution.

Assumptions for Paired T-Test

  • Data must be paired: Observations are related.

  • Independence assumption: Differences must be independent.

  • Randomized condition: Randomization in sample selection.

  • Sample size condition: At least ten samples for normal approximation.

  • Normal population assumption: Population of differences should follow a normal distribution.

  • Nearly normal condition: Check with histogram or probability plot of the differences.

Confidence Interval Calculations

  • Formula: ( CI = \bar{d} \pm t_{} \cdot \text{SE} ) Where ( t_{} ) is the critical value from the t-distribution.

Common Pitfalls

  • Avoid using a two-sample t-test for paired data: This increases variability, negating the purpose of pairing.

  • Use paired t-test only when data are actually paired: If not paired, use a two-sample t-test.

  • Watch for outliers: Outliers affect the differences, impacting the test results.

  • Do not use box plots for comparing means of paired groups; this may reintroduce variation that the matching aimed to eliminate.

Example Application (Medicine vs. Placebo)

  • Input data into TI Nspire:

    • Analyze the placebo and medicine data along with the differences.

    • Setup:

      • Null Hypothesis: mean difference is zero (no effect).

      • Alternative Hypothesis: mean difference is greater than zero (indicating effectiveness of medicine).

    • Perform paired t-test using the differences column:

      • If p-value < alpha (0.05), reject the null hypothesis, providing evidence that the medicine works.

Summary

  • The concepts of matched pairs are crucial for reducing variability and accurately detecting effects in comparisons between groups.

Bottom Line:

  • Matched pair designs provide a robust framework for making valid inferences about the effects of treatments, by focusing on the differences within paired observations and minimizing external variability.