Z-Score de Valor

One-sample z-test with sample mean

The sample z-statistic tells you how many standard errors separate your sample mean from the value you're testing against: z = (x̄ − μ₀) / (σ / √n). It's the right tool when the population standard deviation σ is already known and either the sample is large (n > 30) or the population is roughly normal. If |z| comes out larger than the critical value z_crit (1.96 for α = 0.05, two-tailed), you reject H₀. The two-tailed p-value works out to 2·(1 − Φ(|z|)). Take x̄ = 105, μ₀ = 100, σ = 15, n = 36: that gives z = 2 and p ≈ 0.046, so H₀ falls at the 5% level.

Applications

You'll see it in statistical process control and Six Sigma, where the question is whether a process mean has drifted off target. It also turns up in psychometrics with standardized scores, in large-sample A/B tests on continuous metrics, and in manufacturing QC where σ is already pinned down from historical data.

FAQ

z-test or t-test? Reach for z when σ is known and n is large. Switch to t when σ is unknown and you have to estimate it from the sample s.

What does the denominator σ/√n mean? That's the standard error of the mean, the standard deviation of the sampling distribution of x̄. It gets smaller as n grows.

One-tailed or two-tailed? The default, two-tailed, looks for a difference in either direction. Go one-tailed only if the research question already commits to which way the effect should go.