One-Tailed Z-Test p-Value

One-tailed p-value (z)

The p-value answers one question: if the null hypothesis H₀ were true, how likely would you be to see a statistic at least as extreme as the one you got? For a right-tailed test, p = P(Z ≥ z_obs) = 1 − Φ(z_obs), where Φ is the standard-normal CDF. The two-tailed version is p = 2 · min(p, 1 − p). You will usually see thresholds of α = 0.05 (significant) and α = 0.01 (highly significant). Worked example: z_obs = 2.33 → p = 1 − Φ(2.33) ≈ 0.0099 → reject H₀ at α = 0.05 and α = 0.01.

Applications

It turns up in A/B testing (control vs variant in CRO), in clinical trials weighing whether a treatment works, across scientific research, in ML model validation when you compare accuracy, on the factory floor for quality control, and in economics or finance (Granger causality, anomaly tests).

FAQ

Does p < 0.05 prove the hypothesis? No. The 2016 ASA statement is blunt about this: a p-value on its own does not settle anything. Read it alongside the effect size, a confidence interval, and whether the result replicates. All p < 0.05 tells you is that the data would be unlikely under H₀, not that the alternative is "proven".

What is p-hacking? It is choosing your analysis after the fact until p < 0.05 falls out: running test after test, dropping outliers when it suits you, fishing around for a subgroup that comes back significant. HARKing (Hypothesizing After Results are Known) is the cousin of this. Pre-registering the analysis is what cuts it off.

One-tailed or two-tailed? Reach for one-tailed only if you fixed the direction before any data came in (say, a new drug has to be better, not merely different). When in doubt, go two-tailed, since it is the more conservative call.