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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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