In engineering and everyday life, we often compare what we observe to what we expect. Hypothesis testing helps us decide, using data, whether a claim about a population (like an average value) is likely true or not.
🔬 Key Concepts
Null Hypothesis (H₀): The claim we assume true unless data strongly suggest otherwise.
Alternative Hypothesis (H₁): What we consider if the data contradict H₀.
Test Statistic: A number we calculate from our data to evaluate H₀.
p-value: Probability of observing a result as extreme as ours if H₀ is true.
Significance Level (α): The threshold we compare the p-value to (often 0.05).
🎓 Student Life Example
A university claims that students sleep an average of 7 hours per night. You suspect they sleep less. You collect a sample of 30 students and find an average of 6.5 hours with a known standard deviation of 1.2 hours.
📐 Step-by-Step Setup
H₀: μ = 7 hours
H₁: μ < 7 hours (one-sided test)
α: 0.05
🧮 Calculation
We use the z-test because σ (population std dev) is known.
Test Statistic: z = (6.5 - 7) / (1.2 / √30) ≈ -2.28
Critical value for α = 0.05 (left-tailed): -1.645
✅ Decision Rule
If z < -1.645 → reject H₀
Since -2.28 < -1.645 → Reject H₀. Students sleep less than 7 hours.
Practice Question:
A nutritionist claims that students consume 2,000 calories daily. A sample of 25 students shows an average of 1,850 calories/day, with a known σ = 300. Test at α = 0.05. Is the claim valid?
Conclusion: There is statistically significant evidence at the 0.05 level to suggest that students consume a different average amount of calories than 2,000 per day.
Note: If the z-value had fallen between -1.96 and +1.96, we would have failed to reject H₀ and concluded that the data do not provide sufficient evidence to dispute the 2,000 calorie claim.