# 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

Included in