There are two ways we can conduct a hypothesis test. I will illustrate with an example from the normal distribution assuming a two tailed test.

Using a Test Statistic

We have a null distributionand a single observationW e findSuppose we are conducting a 5% test. We find a test statistic and corresponding to 5% – assuming a two tailed test - by calculatingand looking up the corresponding value ofin the normal distribution table, we findIf the value of z we found usingis bigger than this we reject the null hypothesis.

Using a Probability

We have a null distributionand a single observationW e findSuppose we are conducting a 5% test. Everything so far is as it was above. But now, instead of finding the test statistic corresponding to 5% - assuming a two tailed test – we use the calculated value of z to find a probability. If the probability we find is less than 2.5% -since we are conducting a two tailed test - we reject the null hypothesis.

Things to Remember

Big test statistic implies reject null hypothesis and small test statistic implies do not reject null hypothesis.

Small probability implies reject null hypothesis and big probability implies do not reject null hypothesis.

The test statistic is related to the probability of the null hypothesis being true – if the test statistic is large the probability of the null hypothesis being true is small, so reject the null hypothesis - if the test statistic is small the probability of the null hypothesis being true is large so do not reject the null hypothesis.