Hassan Ijaz
Ai, Web & Design
Hypothesis testing and p-values
Courtroom simulation game where users act as statistical judges, seeing null distribution animations and making accept/reject decisions
Concept Overview
Hypothesis testing is a systematic method for making decisions about population parameters based on sample data. It provides a framework for evaluating claims and controlling error rates.
The Testing Framework
Null Hypothesis (H₀)
- Default position or status quo
- What we assume is true unless proven otherwise
- Often states "no effect" or "no difference"
- Example: μ = μ₀ or p₁ = p₂
Alternative Hypothesis (H₁)
- What we're trying to establish
- Contradicts the null hypothesis
- Can be one-sided (μ > μ₀) or two-sided (μ ≠ μ₀)
- Determines the rejection region
The Testing Procedure
- State Hypotheses: Define H₀ and H₁ clearly
- Choose Significance Level: Set α (typically 0.05)
- Select Test Statistic: Z, t, χ², F, etc.
- Determine Rejection Region: Critical values based on α
- Calculate Test Statistic: From sample data
- Make Decision: Reject or fail to reject H₀
P-Values
P-value = Probability of observing data at least as extreme as what we saw, assuming H₀ is true
- Measures strength of evidence against H₀
- Smaller p-values = stronger evidence against H₀
- Reject H₀ if p-value < α
- Not the probability that H₀ is true!
Common Test Statistics
One Sample Mean
t = (x̄ - μ₀) / (s/√n)
Follows t-distribution with n-1 df
Two Sample Means
t = (x̄₁ - x̄₂) / SE(x̄₁ - x̄₂)
Pooled or Welch's t-test
Proportion
z = (p̂ - p₀) / √[p₀(1-p₀)/n]
Normal approximation for large n
Decision Outcomes
Decision \ Truth
H₀ True
H₁ True
Fail to Reject H₀
Correct (1-α)
Type II Error (β)
Reject H₀
Type I Error (α)
Correct (Power)
Courtroom Analogy: H₀ is "innocent until proven guilty." We need strong evidence (small p-value) to reject innocence and conclude guilt.
The courtroom simulation below lets you act as a statistical judge. See null distribution animations and make accept/reject decisions based on evidence strength.
Interactive Visualization
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Courtroom simulation game where users act as statistical judges, seeing null distribution animations and making accept/reject decisions