One-sample t-test and z-test

Use the one-sample t-test or z-test to compare the average of a sample to a reference. Do it in Excel using the XLSTAT add-on statistical software.

What are one-sample t- and z-tests

This tool is used to compare the average of a sample represented by µ with a reference value.

When to use the Student's t-test or the z-test

Two parametric tests are possible but they should be used on certain conditions:

The Student's t-test

Use the Student's t-test when the true variance of the population from which the sample has been extracted is unknown; the variance of sample s² is used as variance estimator.

The z-test

Use the z-test when the true variance σ² of the population is known.

The Student's t-test and the z-test are parametric tests

Both the Student's t-test and the z-test are said to be parametric as their use requires the assumption that the samples are distributed normally. Moreover, it also assumed that the observations are independent and identically distributed.

Two-tailed or one-tailed test

Three types of test are possible depending on the alternative hypothesis chosen:

  • For the two-tailed test, the null H0 and alternative Ha hypotheses are as follows: H0 : µ = µ0 Ha : µ ≠ µ0
  • In the left one-tailed test, the following hypotheses are used: H0 : µ = µ0 Ha : µ < µ0
  • In the right one-tailed test, the following hypotheses are used: H0 : µ = µ0 Ha : µ > µ0