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One-Sample z Hypothesis Test Calculator — result sheet
Calculate a one-sample z statistic and tail-specific p-value when the population standard deviation is supplied.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: z = (sample mean − null mean) ÷ (σ/√n); p-value is the selected one- or two-sided standard-normal tail.
This page implements the known-σ one-sample z-test: it reports the standard error, standardized statistic, selected-tail p-value, and the comparison with alpha. The alternative hypothesis is an input because a right-tail, left-tail, and two-sided question produce different p-values from the same sample.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Null-hypothesis mean μ₀ — minimum -1000000000000; maximum 1000000000000
- Observed sample mean — minimum -1000000000000; maximum 1000000000000
- Known population standard deviation σ — minimum 1.0E-6; maximum 1000000000000
- Sample size n — minimum 1; maximum 1000000000
- Significance level α — %; minimum 1.0E-6; maximum 99.999999
- Alternative hypothesis — 3 choices
Worked example
| Input | Value |
|---|---|
| Null-hypothesis mean μ₀ | 40 |
| Observed sample mean | 45 |
| Known population standard deviation σ | 2 |
| Sample size n | 20 |
| Significance level α | 5 |
| Alternative hypothesis | greater |
z ≈ 11.1803, the right-tail p-value is effectively 0%, and the null is rejected at 5% in this model.
Assumptions and limits
- The population standard deviation is known or the normal approximation is justified; if σ is estimated from a small sample, use a t-test instead.
- The observations are a suitable random sample and the sample mean is measured on one consistent scale.
- The p-value describes evidence under the null model and does not prove a claim or measure practical importance.
- The result does not adjust for multiple testing, power, effect size, nonresponse, or study design.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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