Activity 29

MATH 216: Statistical Thinking

Independent Samples 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 are the key differences between pooled variance t-test and Welch’s t-test, and when should you use each approach?
  1. Explain the assumptions required for valid independent samples t-test inference and how you would check these assumptions in practice.
  1. How does the standard error calculation differ between independent samples tests and paired tests, and why does this matter for statistical power?

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

Apply independent samples t-test framework to real-world scenarios:

  1. Two-Tailed Test - Educational Research: Comparing test scores between traditional and online learning methods

    • Traditional: \(n_1 = 28\), \(\\bar{x}_1 = 78.5\), \(s_1 = 6.2\)
    • Online: \(n_2 = 32\), \(\\bar{x}_2 = 82.3\), \(s_2 = 5.8\)
    • Test \(H_a: \\mu_1 \\neq \\mu_2\) (methods differ)
    • Calculate pooled variance t-statistic and interpret results

    \(s_p^2\) = , t-statistic =

  2. Right-Tailed Test - Clinical Trial: Testing if new medication reduces blood pressure compared to placebo

    • Placebo: \(n_1 = 25\), \(\\bar{x}_1 = 142\), \(s_1 = 8.5\)
    • Treatment: \(n_2 = 22\), \(\\bar{x}_2 = 135\), \(s_2 = 7.2\)
    • Test \(H_a: \\mu_{treatment} < \\mu_{placebo}\) (treatment reduces BP)
    • Calculate Welch’s t-statistic and interpret results

    t-statistic = , degrees of freedom =

  3. Manufacturing Quality Control: Comparing production efficiency between two assembly lines

    • Line A: \(n_1 = 35\), \(\\bar{x}_1 = 48.2\), \(s_1 = 4.1\)
    • Line B: \(n_2 = 40\), \(\\bar{x}_2 = 45.8\), \(s_2 = 3.9\)
    • Test \(H_a: \\mu_1 > \\mu_2\) (Line A more efficient)
    • Calculate pooled variance t-statistic and critical value

    t-statistic = , critical value =

Show your work for one complete calculation:

Part 3: Decision Making and Interpretation (3 minutes)

Make statistical decisions and interpret independent test results:

  1. For the educational research case (t = -2.45, critical = 2.002), what is your statistical conclusion? What does this mean for educational policy decisions?
  1. For the clinical trial case (t = -3.28, critical = -1.684), what is your statistical conclusion? What are the clinical implications for patient treatment?
  1. For the manufacturing case (t = 2.78, critical = 1.671), what is your statistical conclusion? What are the operational implications for production management?

Critical Thinking: Why is it important to choose between pooled variance and Welch’s test based on variance homogeneity rather than automatically using one approach?