One-Sample Z-Test Statistic

A z-test asks whether a sample mean is further from a claimed value than sampling noise can comfortably explain. The numerator is the raw discrepancy; the denominator is the standard error, the typical size of that discrepancy under the claim. A machine supposed to fill to 100 ml with known σ = 15 produces 36 bottles averaging 104: z = 4/(15/6) = 1.6. That is inside the usual two-sided 5% cutoff of ±1.96, so the evidence is not strong enough to say the machine has drifted, which is not the same as saying it hasn't.

Use this only when σ is genuinely known from long production history or theory; if you estimated it from the same sample, you want the t-test instead. The deeper trap is confusing statistical and practical significance. With n large enough, the standard error shrinks until any difference at all becomes "significant": the same 4 ml drift measured on 3,600 bottles gives z = 16, an overwhelming result about a discrepancy that no customer would ever notice. Always read the size of x̄ − μ alongside the test statistic.