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# Z-test for One Mean, with Known Population Standard Deviation

Instructions: This calculator conducts a Z-test for one population mean ($$\mu$$), with known population standard deviation ($$\sigma$$). Please select the null and alternative hypotheses, type the hypothesized mean, the significance level, the sample mean, the population standard deviation, and the sample size, and the results of the z-test will be displayed for you:

Ho: $$\mu$$     $$\mu_0$$
Ha: $$\mu$$     $$\mu_0$$
Hypothesized Mean ($$\mu_0$$)
Sample Mean ($$\bar X$$)
Population St. Dev. ($$\sigma$$)
Sample Size (n)
Significance Level ($$\alpha$$)

More about the z-test for one mean so you can better interpret the results obtained by this solver: A z-test for one mean is a hypothesis test that attempts to make a claim about the population mean ($$\mu$$). The test has two non-overlaping hypotheses, the null and the alternative hypothesis. The null hypothesis is a statement about the population mean, under 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 one population mean 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 is

$z = \frac{\bar X - \mu_0}{\sigma/\sqrt{n}}$

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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