The p-value is the probability of obtaining a test statistic as extreme as the observed value, assuming the null hypothesis is true. It reflects the strength of the evidence against H₀.
🧪 Example – Propellant Burning Rate
A sample of 16 observations has x̄ = 51.3 cm/sec, σ = 2.5. We test:
H₀: μ = 50
H₁: μ ≠ 50 (two-sided)
SE = 2.5 / √16 = 0.625
z = (51.3 − 50) / 0.625 = 2.08
P-value = 2 × P(Z > 2.08) = 2 × 0.019 = 0.038
📉 Visualization of the P-Value
📌 Interpretation
P-value = 0.038 → If H₀ were true (μ = 50), getting a sample mean as far as 51.3 or further is rare (~3.8%).
P-values are uniformly distributed between 0 and 1
About 5% fall below 0.05 → Type I error rate matches α = 0.05
✅ Case 2: H₀ is False (μ = 1)
P-values bunch up near 0 → more small values → more rejections
Some p-values still > 0.05 → Type II errors occur
✅ Case 3: H₀ is Very False (μ = 2)
P-values concentrate near 0 → test becomes more powerful
Almost all p-values < 0.05 → fewer Type II errors
🎯 Final Takeaways:
Scenario
P-Value Pattern
Type of Error
Power
H₀ true (μ = 0)
Uniform (flat)
Type I (α ≈ 5%)
Low
H₀ false, μ = 1
Skewed toward 0
Type II (some missed)
Medium
H₀ false, μ = 2
Strong skew toward 0
Very few Type II
High
❓ Does α Still Matter When H₀ is False?
YES: α = 0.05 still defines your rejection rule (e.g., reject if p < 0.05)
But when H₀ is false, you are no longer at risk of a Type I error
Instead, you worry about Type II error — failing to reject H₀
Scenario
Can Type I Happen?
Is α Used?
Main Error Risk
H₀ is true
✅ Yes
✅ Yes
Type I
H₀ is false
❌ No
✅ Yes
Type II
Conclusion: You always use α to set your rejection rule, but it only governs Type I error risk when H₀ is true. When H₀ is false, your main concern is β (Type II error) and power.
📝 Practice Question – Show Answer
A chemical process is designed to produce concentration μ = 8. A sample of 25 measurements gives x̄ = 7.6, σ = 1. What is the p-value for testing H₀: μ = 8 vs H₁: μ ≠ 8?