Test Statistic

A test statistic is a single numerical value that quantifies the difference between the observed sample data and what you would expect to observe if the null hypothesis of the test is true.

In general, larger test statistics are indicative of stronger evidence against the null hypothesis being tested.

Z-Statistic

The #\boldsymbol{Z}#-statistic for a #Z#-test for a population mean #\mu# is obtained by transforming the sample mean #\bar{X}# into a #Z#-score:

\[Z=\cfrac{\bar{X}-\mu_0}{\sigma_{\bar{X}}} =\cfrac{\bar{X}-\mu_0}{\sigma/\sqrt{n}}\]
If the population from which the sample is drawn is normally distributed, then the sampling distribution of the #Z#-statistic is the Standard Normal Distribution. That is #Z\sim N(0,1)#.

If the population is not normally distributed, but the sample size is large (#n>30#), the Central Limit Theorem allows us to proceed as if #Z\sim N(0,1)#.

A lowercase #z# will be used to denote the measured value of #Z# after the data has been collected. The value of #z# is used to assess the strength of the evidence against the null hypothesis.

As the difference between the observed sample mean #\bar{X}# and the hypothesized population mean #\mu_0# increases, the value of #z# becomes larger and the evidence against the null hypothesis becomes stronger.