# Minimum Sample Size Required Calculator - Estimating the Population Mean

**Instructions:** This calculator finds the minimum sample size required to estimate the population mean (\(\mu\)) within a specified margin of error. Please select type the the significance level (\(\alpha\)), the population standard deviation \(\sigma\) (or the approximated pop. standard deviation. If not known, the sample standard deviation can be used), and the required margin of error (E), and the solver will find the minimum sample size required:

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 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 claim, 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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