Test means against known population variance accurately. Inspect z scores, p-values, and confidence intervals instantly. Visualize rejection regions clearly with flexible tails and significance.
This example shows a process quality sample with a known population standard deviation. You can paste these values into the raw data field.
| Observation | Measured Value | Observation | Measured Value |
|---|---|---|---|
| 1 | 51.2 | 7 | 53.6 |
| 2 | 52.8 | 8 | 50.9 |
| 3 | 54.1 | 9 | 52.4 |
| 4 | 49.8 | 10 | 53.2 |
| 5 | 52.6 | 11 | 54.0 |
| 6 | 51.9 | 12 | 52.1 |
Test statistic: z = (x̄ − μ0) / (σ / √n)
Standard error: SE = σ / √n
Confidence interval: x̄ ± zα/2 × SE
Decision rule: Reject H0 when the observed z enters the rejection region defined by α and the selected tail direction.
Use a one sample z test when the population standard deviation is known and you want to test whether a sample mean differs from a hypothesized population mean. The page calculates z score, p value, critical values, margin of error, confidence interval, and a decision summary.
Use it when you want to compare one sample mean against a target mean and the population standard deviation is known or treated as known.
A z test uses a known population standard deviation. A t test uses the sample standard deviation and a t distribution, especially for smaller samples.
Yes. When raw values are provided, the page automatically calculates the sample mean and sample size, then runs the test with your known population standard deviation.
The p value measures how extreme your sample result looks if the null hypothesis were true. Smaller values provide stronger evidence against the null hypothesis.
The interval is useful as a summary estimate of the sample mean. The formal hypothesis decision still follows the selected one-tailed rejection rule.
It is the standardized mean difference, computed as (x̄ − μ0) divided by the known population standard deviation. It helps describe practical magnitude.
Changing alpha updates the rejection region, critical values, and confidence level. A smaller alpha makes rejection harder and widens the confidence interval.
Your z statistic may not be large enough in the selected direction, or the p value may be greater than alpha. That means evidence is not strong enough.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.