Z-test for two Means, with Known Population Standard Deviations


Instructions: This calculator conducts a Z-test for two population means (\(\mu_1\) and \(\mu_2\)), with known population standard deviations ( \(\sigma_1\) and \(\sigma_2\)). Please select the null and alternative hypotheses, type the significance level, the sample means, the population standard deviations, the sample sizes, and the results of the z-test will be displayed for you:

Ho: \(\mu_1\) \(\mu_2\)
Ha: \(\mu_1\) \(\mu_2\)
Sample Mean (\(\bar X_1\)) =
Sample Mean (\(\bar X_2\)) =
Pop. St. Dev. (\(\sigma_1\)) =
Pop. St. Dev. (\(\sigma_2\)) =
Sample Size (\(n_1\)) =
Sample Size (\(n_2\)) =
Significance Level (\(\alpha\)) =

The Z-test for Two Means

More about the z-test for two means so you can better use the results delivered by this solver: A z-test for two means is a hypothesis test that attempts to make a claim about the population means (\(\mu\)1 and \(\mu\)2). More specifically, we are interested in assessing whether or not it is reasonable to claim that the two population means the population means \(\mu\)1 and \(\mu\)2 are equal, based on the information provided by the samples. The test has two non-overlapping hypotheses, the null and the alternative hypothesis.

The null hypothesis is a statement about the population means, corresponding to the assumption of no effect, and the alternative hypothesis is the complementary hypothesis to the null hypothesis. The main properties of a one sample z-test for two population means are:

  • Depending on our knowledge about the “no effect” situation, the z-test can be two-tailed, left-tailed or right-tailed

  • The main principle of hypothesis testing is that the null hypothesis is rejected if the test statistic obtained is sufficiently unlikely under the assumption that the null hypothesis is true

  • The p-value is the probability of obtaining sample results as extreme or more extreme than the sample results obtained, under the assumption that the null hypothesis is true

  • In a hypothesis tests there are two types of errors. Type I error occurs when we reject a true null hypothesis, and the Type II error occurs when we fail to reject a false null hypothesis

The formula for a z-statistic for two population means is:

\[z = \displaystyle{\frac{\bar X_1 – \bar X_2}{\sqrt{\displaystyle{\frac{\sigma_1^2}{n_1}} + \displaystyle{\frac{\sigma_2^2}{n_2}} }}} \]

The null hypothesis is rejected when the z-statistic lies on the rejection region, which is determined by the significance level (\(\alpha\)) and the type of tail (two-tailed, left-tailed or right-tailed).




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Z-test for two Means, with Known Population Standard Deviations

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