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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 n independent draws [eq1] from a normal distribution having unknown mean mu and unknown variance sigma^2.

Suppose you want to test the null hypothesis that the mean is equal to a specific value $mu _{0}$:[eq2]

A test statistic, called t-statistic, can be constructed using the sample data:[eq3]where Xbar_n is the sample mean[eq4]and $s_{n}^{2}$ is the adjusted sample variance[eq5]

After selecting a critical region[eq6]the null hypothesis that $\mu =\mu _{0}$ is rejected if the test statistic falls within this critical region, that is, if[eq7]

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.

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