🎯 Hypothesis Testing – Propellant & Student Examples

🚀 One-Sided vs. Two-Sided Tests

Two-Sided: Claim: μ ≠ 50
One-Sided: Claim: μ < 50 or μ > 50
Choosing one vs. two-sided depends on the conclusion you want to emphasize.

📦 Bottling Example

A bottler requires bottles to have a minimum mean bursting strength of 200 psi. Which hypotheses to choose depends on the burden of proof:

Option 1 (One-sided): H₀: μ = 200, H₁: μ < 200 → Bottles fail if H₀ is rejected
Option 2 (One-sided): H₀: μ = 200, H₁: μ > 200 → Bottler must prove strength exceeds 200

Rule of thumb: Put the claim you want to support strongly into H₁.

🧪 Propellant Example – One-Sided Test

H₀: μ = 50 cm/sec
H₁: μ < 50 cm/sec (concern is with underperformance)
Sample mean (x̄): 48.5
n = 10, σ = 2.5 → SE = 0.79
z = (48.5 - 50) / 0.79 = -1.90
Critical z (α = 0.05, left-tailed) = -1.645

Since -1.90 < -1.645 → Reject H₀. There is evidence that the mean is less than 50.

📝 Practice Question – Show Answer

A machine is calibrated to produce pins of length 10.0 mm. You suspect the machine is producing pins shorter than specified. A sample of 25 pins has a mean of 9.8 mm, σ = 0.4 mm. Test at α = 0.05.

H₀: μ = 10.0
H₁: μ < 10.0 (left-tailed test)
SE = 0.4 / √25 = 0.08
z = (9.8 - 10) / 0.08 = -2.5
Critical z = -1.645 → -2.5 < -1.645 → Reject H₀. The machine is producing pins shorter than 10.0 mm.