Computational Details

Under the null hypothesis, the population has mean $\mu_0$ and standard deviation $\sigma$ .

Using either the fact that the data are normal or the fact that the central limit theorem applies, assume that the data are from a Normal$(\mu_0,\sigma)$ distribution.

With this assumption, if the null hypothesis is true, the sample mean $\overline{x}$ will be normally distributed with the following parameters:

   mean$$\;=\;\mu_0$$   standard deviation$$\;=\;\frac{\sigma}{\sqrt{n}}$$

The strategy for testing the hypothesis is to consider the sample mean as a single observation from this distribution, standardize it based on the assumption that its mean is $\mu_0$ and its standard deviation is $\sigma/\sqrt{n}$ , and base our decision to accept or reject the null hypothesis on where the standardized value or $Z$ -score falls on the standard normal bell curve.

We calculate the standardized or $Z$ -score for the sample mean as:

$$Z\;=\;\frac{\overline{x}-\mu_0}{\sigma/\sqrt{n}}$$

(This value is in cell B10)

The exact decision rule to accept or reject the null hypothesis depends on whether we want:

• a two-tailed test $(\mu\neq\mu_0)$
• a left-tailed $(\mu < \mu_0)$
• a right-tailed test. $(\mu > \mu_0)$

See the sections below for details on these three cases.