Two worlds. H₀ says the population mean is μ₀ = 50. Reality might be different. Draw a sample, see where the sample mean lands, and ask: "would this be surprising if H₀ were true?"
The logic: Pretend H₀ is true → sample means would follow N(μ₀, σ/√n) → compute z = (x̄ − μ₀) / (σ/√n) → p-value = probability of seeing |z| this large by chance → if p < α, reject H₀.
Sample mean x̄—
Test statistic z—
p-value—
Decision at α=0.05—
A. Population & sample
B. Sampling distribution of x̄ under H₀
55.0
30
What changes what: Move true μ away from 50 to make reality different from H₀ — larger samples (n ↑) make the sampling distribution narrower, so even tiny differences become "extreme." That's why with huge n, almost any effect is "significant" — statistical significance ≠ practical importance.