#6 Hypothesis Testing

🧮 Hypothesis Testing Lecture Notes

Statistical Analysis for Civil Engineering Students
90-Minute Classroom Delivery | Concrete, Soil & Structural Applications

📢 Lecture Introduction

"Good morning, Class!"

Today we learn Hypothesis Testing – the statistical tool that helps civil engineers make decisions under uncertainty.

Real Civil Engineering Question: "Is this concrete batch strong enough for the bridge? Is this soil variance too high for foundation design?"

Hypothesis Testing Answer: We test if sample evidence rejects the "null hypothesis" (status quo/design specification).

🎯 Module 1: Concepts of Hypothesis Testing

1.1 What is Hypothesis Testing?

H₀: Null Hypothesis (Status Quo) = "Concrete meets 30 MPa spec"
H₁: Alternative Hypothesis = "Concrete < 30 MPa"

1.2 Types of Errors

	H₀ True	H₀ False
Accept H₀	Correct	Type II Error (β)
Reject H₀	Type I Error (α)	Correct (1-β = Power)

α = Reject good concrete (5% risk)
β = Accept bad concrete

1.3 5-Step Test Procedure

1. H₀: μ = μ₀ vs H₁: μ ≠/>/<
2. Choose test statistic (t, z, F, χ²)
3. Compute p-value or critical value
4. Decision: If p < α, Reject H₀
5. Engineering Conclusion

📊 Module 2: Tests for Means

2.1 Z-Test (Large Sample, σ known)

z = (x̄ - μ₀) / (σ/√n)
|z| > z_α/2 → Reject H₀

Worked Example: 50 trucks: x̄=420kN, σ=60kN, μ₀=400kN
z = (420-400)/(60/√50) = 2.36 > 1.645
REJECT H₀: "Bridge loads exceed design"

2.2 T-Test (Small Sample)

t = (x̄ - μ₀) / (s/√n), df = n-1

Concrete Test: 8 cubes: x̄=29.625, s=1.53, μ₀=30
t = -0.693 < 1.895 → Concrete meets spec

📈 Module 3: Tests for Variance

3.1 Chi-Square Test

χ² = (n-1)s² / σ₀², df = n-1

Soil Test: 12 samples, s²=4.5, σ₀²=3.0
χ² = 16.5 (between 3.816-21.92)
FAIL TO REJECT: "Soil variability acceptable"

3.2 F-Test (Two Variances)

F = s₁²/s₂² → Compare to F_critical

🏗️ Module 4: Civil Engineering Applications

Test	Situation	Statistic	Civil Application
Z-test	n≥30, σ known	z-score	Traffic loads
T-test	n<30 or σ unknown	t-score	Concrete strength
χ²-test	Single variance	χ²	Soil variability
F-test	Two variances	F-ratio	Batch comparison

🎯 Module 5: P-Values (Modern Method)

p-value = P(Type I error | H₀ true)
If p < α → Reject H₀

✍️ In-Class Exercises

Exercise 1: Concrete: x̄=28.5MPa, s=2.1, spec=30MPa
Solution: t=-2.36, p≈0.02 < 0.05 → REJECT BATCH

Exercise 2: Soil: s²=2.8, spec=2.0 → ACCEPTABLE

🧠 Classroom Quiz (5 Questions)

1. H₀ always contains: a) Equality
2. α=0.05 means: a) 5% Type I error risk
3. Concrete test alternative: b) μ < 30
4. Smaller p-value: a) Stronger evidence vs H₀

🚀 Key Takeaways

1. Hypothesis Testing = Decision making from samples
2. Civil Applications: Concrete ✓ Soil ✓ Loads ✓ Quality Control ✓
3. Always state: H₀, H₁, α, Decision + Engineering Conclusion
4. p < 0.05 → ACTION REQUIRED

Homework: Analyze your lab concrete data vs M20 spec

Next Class: Confidence Intervals + Regression for Load Prediction

👨‍🏫 Prepared for Undergraduate Civil Engineering Students | March 2026