Testing H₀: μ = μ₀ against H₁: μ = μ₀ + d (one-tailed, upper rejection region) with a known population SD, the sample mean has a normal sampling distribution under each hypothesis, with standard error:
SE = σ / √n (σ = 1 assumed)
The critical value is set from α using the standard-normal quantile z1−α, then converted back into sample-mean units:
x_crit = z(1−α) · SE
Reject H0 when sample mean > x_crit
Because the alternative distribution is just the null shifted by d, its probability mass past x_crit is the statistical power, and the mass that falls short is β (a miss):
Power = 1 − Φ( z(1−α) − d·√n )
β = Φ( z(1−α) − d·√n )
- Teal bars (back row) — the null sampling distribution. The red slice past the critical plane is α, the chance of a false positive (Type I error) even though H₀ is true.
- Orange bars (front row) — the alternative sampling distribution, shifted right by d·√n standard errors. The blue slice below the critical plane is β, a miss (Type II error); the green slice is the power — correctly detecting a real effect.
- n slider — a larger sample shrinks SE, narrowing both curves and pulling them apart in z-units, which is why bigger studies detect smaller effects reliably.
- d slider — the true size of the effect being tested for; larger d separates the two curves and raises power directly.
- α select — moving the critical plane left (larger α) trades a higher false-positive rate for higher power — the classic α/power trade-off behind every "underpowered study" warning.