• What is a p-value?
• How it fits into hypothesis testing
• Simulation example
• Visual interpretation
• Common misunderstandings

• Probability of observing data as extreme as ours if the null hypothesis is true
• Helps assess evidence against \(H_0\)

\[ p = P(\text{data as or more extreme than observed} \mid H_0) \]

Let’s test whether the true mean is greater than 0:

\[ H_0: \mu = 0 \] \[ H_1: \mu > 0 \]

We take a sample (\(n = 30\)) and find: \[ \bar{x} = 0.4 \]

Visualizing the p-value

Then replace your slide chunk with this:

• p = 0.05 means there’s a 5% chance the null is true
• Small p-value does not mean a big effect
• p-values do not tell you the probability data is random
• Results near cutoff (e.g., 0.049 vs 0.051) aren’t drastically different