# One sample t and z tests

One sample t and z tests is part of:

• ### System configuration

• Windows:
• Versions: 9x/Me/NT/2000/XP/Vista/Win 7/Win 8
• Excel: 97 and later
• Processor: 32 or 64 bits
• Hard disk: 150 Mb
• Mac OS X:
• OS: OS X
• Excel: X, 2004 and 2011
• Hard disk: 150Mb.

## Benefits

• Easy and user-friendly
• Data and results shared seamlessly
• Modular
• Didactic
• Affordable
• Accessible - Available in many languages
• Automatable and customizable

### Principle of the 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 not known; 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