### 9.11: Testing a Claim about Mean: Known Population SD

A complete procedure of testing the hypothesis about a population mean is explained here.

Estimating a population mean requires the samples to be distributed normally. The data should be collected from the randomly selected samples having no sampling bias. The sample size needed to be higher than 30, and most importantly, the population standard deviation should be already known.

In most realistic situations, the population standard deviation is often unknown, but in rare circumstances, when it is known, the claim about the population mean can be tested easily using the normality assumption and the *z* distribution.

The hypothesis (null and alternative) should be stated clearly and then expressed symbolically. The null hypothesis is a neutral statement stating population mean is equal to some definite value. The alternative hypothesis can be based on the mean claimed in the hypothesis with an inequality sign. The right-tailed, left-tailed, or two-tailed hypothesis test can be decided based on the sign used in the alternative hypothesis.

As the method requires normal distribution, the critical value is calculated using the *z* distribution (*z* table). It is calculated at the desired confidence level, most commonly at 95% or 99%. As per the traditional method, the z statistic calculated from the sample data is compared with the *z* score. The *P*-value is calculated based on the data as per the *P*-value method. Both these methods help conclude the hypothesis test.