Z-test Calculator

A Z-test is a statistical hypothesis test used to determine whether two population means are different when the variances are known and the sample size is large. The core test statistic is calculated using the formula: Z = ((x̄ - μ) / (σ)) * √n, where x̄ is the sample mean, μ is the hypothesized population mean, σ is the population standard deviation, and n is the sample size. Depending on your alternative hypothesis, the calculator evaluates the p-value using cumulative distribution functions to determine whether you can reject the null hypothesis at your chosen significance level.

Imagine a beverage manufacturer claiming that their average soda can volume is 355 ml with a known standard deviation of 2.5 ml. A quality control inspector samples 50 cans and finds a sample mean volume of 354 ml. We want to test if the mean is significantly different at a 5 percent significance level (α = 0.05). First, compute the z-score: Z = ((354 - 355) / 2.5) * √50, which simplifies to (-1 / 2.5) * 7.071 = -2.83. For a two-tailed test, this yields a p-value of approximately 0.0046. Since 0.0046 is less than 0.05, we reject the null hypothesis and conclude the filling machine requires adjustment.

To ensure valid results, always confirm that your sample size is sufficiently large—typically greater than 30—and that your data is randomly sampled from the population. Verify whether your test requires a one-tailed or two-tailed approach before inputting parameters, as selecting the wrong direction will invalidate your p-value interpretation.