Activity 27

MATH 216: Statistical Thinking

Nonparametric Testing 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 nonparametric tests and when should they be used instead of parametric tests like t-tests?
  1. Explain the key differences between the sign test and the Wilcoxon signed-rank test. When would you choose one over the other?
  1. Why do nonparametric tests use medians instead of means for hypothesis testing, and what advantages does this provide?

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

Apply nonparametric testing framework to real-world scenarios:

  1. Sign Test - Environmental Monitoring: Agency tests if median pollution level exceeds regulatory limit (\(n=12\))

    • Sample: 18, 22, 25, 19, 30, 16, 28, 21, 24, 17, 26, 23 ppm
    • Regulatory limit: 20 ppm, test \(H_a: \eta > 20\)
    • Calculate sign test statistic and interpret results

    S = (number of observations > 20)

    P-value interpretation:

  2. Wilcoxon Signed-Rank Test - Medical Research: Study tests if treatment changes patient pain scores (\(n=10\))

    • Baseline: 6, Sample: 4, 5, 3, 7, 2, 6, 4, 5, 3, 5
    • Test \(H_a: \eta \neq 6\)
    • Calculate \(W^+\) statistic and interpret results

    \(W^+\) = (sum of positive ranks)

    Statistical conclusion:

  3. Davis Weight Data Analysis: First 15 observations from Davis dataset

    • Population median: 57 kg, Sample median: 68 kg
    • Test \(H_a: \eta \neq 57\)
    • Calculate sign test statistic and p-value

    S = (number of observations > 57)

    P-value =

Show your work for one complete calculation:

Part 3: Decision Making and Interpretation (3 minutes)

Make statistical decisions and interpret nonparametric test results:

  1. For the environmental monitoring case (S = 8, \(n\) = 12, p-value = 0.1938), what is your statistical conclusion? What does this mean for environmental regulation enforcement?
  1. For the medical research case (\(W^+\) = 40, p-value < 0.05), what is your statistical conclusion? What are the clinical implications of this finding?
  1. For the Davis weight data (S = 12, \(n\) = 15, p-value = 0.035), what is your statistical conclusion? Why might the t-test give a different result in this case?

Critical Thinking: Why are nonparametric tests particularly valuable when working with small sample sizes and non-normal data distributions?