Activity 27

MATH 216: Statistical Thinking

Activity 1: Quick Hypothesis Check

Scenario: A company claims their meditation app reduces stress (median stress score < 20). 12 users’ scores:

data <- c(18, 22, 19, 17, 25, 16, 21, 19, 20, 18, 23, 17)

Tasks:

  1. Calculate test statistic \(S\)
  2. Write null hypothesis (\(H_0: m \geq 20\))
  3. Calculate binomial probability for \(S\) successes
  4. R code: SIGN.test(data, md=20, alternative="less")

Activity 2: P-Value Understanding

Question: For n=15 non-zero differences with 12 positive signs (testing \(H_1: m > m_0\)), which formula calculates the one-tailed p-value?

Options:

  1. \(P(X \leq 12)\)
  2. \(P(X \geq 12)\)
  3. \(2 \times P(X \geq 12)\)
  4. \(P(X = 12)\)

Activity 3: Test Statistic Calculation

Data: 10 paired observations with 7 positive differences, 1 negative, 2 zeros.

Question: What is the value of the test statistic \(S\)?

Activity 4: R Code Interpretation


    One-sample Sign-Test

data:  c(3, 5, -2, 0, 1)
s = 2, p-value = 1
alternative hypothesis: true median is not equal to 2
93.75 percent confidence interval:
 -2  5
sample estimates:
median of x 
          1

Question: What is being tested?

P-Value Discussion Key Points:

  1. Under \(H_0\), probability of “+” signs follows binomial\((n,0.5)\)

  2. For \(S\) observed “+” signs:

    • Left-tailed: \(P(X \leq S)\)
    • Right-tailed: \(P(X \geq S)\)
    • Two-tailed: \(2 \times \min[P(X \leq S), P(X \geq S)]\)

  3. In R, SIGN.test() automatically handles:

    • Exclusion of zero differences
    • Normal approximation for large n
    • Exact binomial p-value for small n

Activity 5: Full Analysis

Scenario: 14 patients tried a diet. Weight changes:

differences <- c(-2, -1, 0, -3, -2, -1, -4, 0, -3, -2, -1, -2, 0, -1)

Tasks:

  1. State \(H_0\) (\(m \geq 0\)) vs \(H_1\) (\(m < 0\))
  2. Calculate \(S\) = number of positive differences
  3. Exact p-value = pbinom(S, 12, 0.5)
  4. Conclusion at α=0.05: