One Sample Z Test Calculator — Free & Interactive

The One-Sample z-Test. Is a two-milliliter shift unusual when process spread is known?

A one sample z-test evaluates whether a population mean is compatible with a specified value when the population standard deviation sigma is known independently of the current sample. The statistic measures the sample mean’s distance from the null mean in known standard-error units. This is a specialized test, a large sample does not by itself make an estimated sample standard deviation become “known,” and the one sample t-test is usually appropriate when sigma must be estimated.

The null is H naught, mu equals mu naught. The two-sided alternative is mu not equal to mu naught; the directional alternatives are mu greater than mu naught or mu less than mu naught. The statistic z divides x bar minus mu naught by sigma divided by the square root of n, comparing the observed difference with null-model sampling variability. Under the null and model assumptions, z follows the standard normal distribution. The p-value is the probability, assuming H naught and the model, of a statistic at least as incompatible with H naught as the observed statistic in the pre-specified tail or tails. It is not the probability that H naught is true. Use the test for one numerical sample, one pre-specified population mean mu naught, and a defensible population standard deviation supplied by design, calibration, or reliable external knowledge. Observations should be independent under the sampling or assignment process. The sign of z gives direction relative to mu naught; its magnitude gives standardized distance. Degrees of freedom are not needed because sigma is treated as known. The critical value is another representation of the alpha rule, while the p-value reports how extreme the observed statistic is under the null model. If p less than or equal to alpha, reject H naught at that threshold and describe evidence for the stated alternative. If p greater than alpha, fail to reject H naught; do not say the means are equal or that the null has been proved. The current output does not supply a confidence interval, so do not infer or advertise one.

A calibrated process has a known long-run individual-fill standard deviation of sigma equals 4 mL. Ten independently sampled fills are 503, 500, 506, 498, 501, 504, 499, 505, 502, and 502 mL. Before sampling, the analyst specifies a two-sided test of H naught, mu equals 500 at alpha equals 0.05. Enter the ten values, population mean 500, population standard deviation 4, alpha 0.05, and Two-tailed. The sample mean is 502.0 mL and the known standard error is 4 divided by square root of 10, about 1.265 mL. The difference, 502 minus 500, divided by 1.265 gives z approximately 1.58. A two-sided standard-normal tail beyond z equals positive or negative 1.58 produces a p-value around 0.114, which is above 0.05. The calculation therefore fails to reject H naught at the chosen threshold. The observed two-milliliter difference is not sufficiently unusual relative to the known process spread and this sample size under the planned rule.

The data do not provide strong enough evidence, at alpha equals 0.05, that the population mean differs from 500 mL. This is not evidence that the mean is exactly 500 and not a process-capability assessment. The known-sigma claim, sampling independence, measurement calibration, and operational importance of a two-milliliter shift must be considered separately.

Try it free at Distri Scope dot com.