StatlectThe Digital Textbook

# Test statistic

The concept of test statistic is found in the theory of hypothesis testing. In a test of hypothesis, the test statistic is a function of the sample data and its value is used to decide whether or not to reject the null hypothesis. If the test statistic falls within a critical region, fixed ex-ante, then the null hypothesis is rejected.

## Example

Suppose you observe independent draws from a normal distribution having unknown mean and unknown variance .

Suppose you want to test the null hypothesis that the mean is equal to a specific value :

A test statistic, called t-statistic, can be constructed using the sample data:where is the sample meanand is the adjusted sample variance

After selecting a critical regionthe null hypothesis that is rejected if the test statistic falls within this critical region, that is, if

As discussed in the lecture entitled Hypothesis testing about the mean, this testing procedure leads to simple analytical formulae for the probability of committing Type I and Type II errors.

## More details

Go to the lecture entitled Hypothesis testing for a rigorous definition of the concept of test statistic.