Reject H₀? Let the curve decide.
Drag, slide and toggle your way through rejection regions, p-values, confidence-interval duality and power.
The Rejection Zone
H₀: μ₁ ≤ μ₂ vs. H_A: μ₁ > μ₂ (directional). Drag z into the coral zone to reject H₀.
- z
- 1.86
- z_crit
- 1.645
- p-value
- 0.0314
P = P(Z ≥ 1.86) = 0.0314 is ≤ α = 0.05. Same verdict as comparing z to the critical value.
Direction: P = 0.0314 in favor of a larger μ₁. Try the two-sided toggle: at z = 1.86 it doubles to 0.0628, which is no longer ≤ 0.05.
Type I vs. Type II
Tap any cell. Rejecting H₀ supports H_A more convincingly than failing to reject supports H₀.
A false alarm. Its probability is the significance level α, also called the size of the test.
Simple or Composite?
A simple hypothesis names a single value; a composite one names more than one.
H: μ₁ − μ₂ ≤ 0
From Data to Decision
State how the data were collected (e.g. a random sample) and, for a parametric test, the distribution assumed. Nonparametric tests relax the distributional assumption — that is the whole point of this course.
Confidence Interval ⇄ Hypothesis Test
- 1. Suppose your sample gave a 95% confidence interval of [60, 80] (the green band).
- 2. Someone claims the true value is θ₀. Drag the dot to try different claims.
- 3. Inside the band → the data are consistent with the claim, so you fail to reject H₀. Outside → reject H₀ (two-sided test, α = 0.05).
Statistical Power Dial
Power rises with a larger n, a larger α (at the cost of more Type I error), and a larger true effect. ✅ Meets the conventional 80% power target.
Relative efficiency calculator
For the same H₀, H_A, α and β, the relative efficiency of test A to test B is nB / nA. ARE (Pitman efficiency) is the limit as n → ∞.
RE(A to B) = 105 / 110 = 0.955 — B is more efficient: it needs the smaller sample.
Significant… but does it matter?
Imagine a drug that lowers blood pressure by only 0.3 mm Hg on average (σ = 10). The true difference never changes, only the sample size does.
Very large samples detect tiny, useless differences; small samples can miss practically important ones. Only someone knowledgeable in the field can judge practical significance.
✅ “There is a significant difference between the sample means.”
❌ “The population means are significantly different.”
§1.3 Estimation Lab →