Activity 28

MATH 216: Statistical Thinking

Paired t-Test Framework

Time Allocation: 15 minutes total (5 min reading, 10 min individual work)

Part 1: Conceptual Understanding (3 minutes)

Instructions: Answer the following questions based on the lecture content:

  1. What is the key advantage of using paired data in statistical comparisons, and how does this affect the test statistic calculation?
  1. Explain the difference between the null and alternative hypotheses in a paired t-test context, and why we focus on mean differences rather than individual group means.
  1. What are the key assumptions for valid paired t-test inference, and how do we check these assumptions in practice?

Part 2: Real-World Paired Testing Applications (4 minutes)

Apply paired t-test framework to real-world scenarios:

  1. Right-Tailed Test - Medical Study: Testing if new drug increases blood pressure (\(n=15\))

    • Before: 120, 118, 122, 119, 121, 117, 123, 119, 120, 118, 121, 119, 122, 120, 118
    • After: 125, 122, 128, 124, 126, 120, 130, 125, 127, 123, 129, 124, 128, 126, 122
    • Test \(H_a: \mu_d > 0\) (drug increases blood pressure)
    • Calculate mean difference and t-statistic

    \(\bar{d}\) = , t-statistic =

  2. Two-Tailed Test - Educational Research: Testing if teaching method affects test scores (\(n=12\))

    • Traditional: 78, 82, 85, 79, 88, 81, 83, 86, 80, 84, 87, 82
    • New Method: 85, 88, 90, 83, 92, 86, 89, 91, 84, 87, 93, 88
    • Test \(H_a: \mu_d \neq 0\) (methods differ)
    • Calculate mean difference and t-statistic

    \(\bar{d}\) = , t-statistic =

  3. Training Program Comparison: Testing effectiveness of two training methods (\(n=20\))

    • Method A: 85, 88, 90, 92, 91, 89, 93, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107
    • Method B: 83, 89, 87, 84, 92, 90, 85, 91, 98, 94, 100, 101, 99, 111, 111, 106, 109, 103, 111, 114
    • Test \(H_a: \mu_d > 0\) (Method A better than Method B)
    • Calculate mean difference and t-statistic

    \(\bar{d}\) = , t-statistic =

Show your work for one complete calculation:

Part 3: Decision Making and Interpretation (3 minutes)

Make statistical decisions and interpret paired test results:

  1. For the medical study case (\(\bar{d} = 4.8\), t = 8.85, critical = 1.761), what is your statistical conclusion? What are the clinical implications of this finding?
  1. For the educational research case (\(\bar{d} = 4.3\), t = 8.27, critical = 2.201), what is your statistical conclusion? What does this mean for educational practice?
  1. For the training program comparison (\(\bar{d} = 3.2\), t = 2.15, critical = 1.729), what is your statistical conclusion? What are the practical implications for training program selection?