The Classical Approach to hypothesis testing is to compare a test statistic and a critical value. It is best used for distributions which give areas and require you to look up the critical value (like the Student's t distribution) rather than distributions which have you look up a test statistic to find an area (like the normal distribution).
The Classical Approach also has three different decision rules, depending on whether it is a left tail, right tail, or two tail test.
One problem with the Classical Approach is that if a different level of significance is desired, a different critical value must be read from the table.
The P-Value Approach, short for Probability Value, approaches hypothesis testing from a different manner. Instead of comparing z-scores or t-scores as in the classical approach, you're comparing probabilities, or areas.
The level of significance (alpha) is the area in the critical region. That is, the area in the tails to the right or left of the critical values.
The p-value is the area to the right or left of the test statistic. If it is a two tail test, then look up the probability in one tail and double it.
If the test statistic is in the critical region, then the p-value will be less than the level of significance. It does not matter whether it is a left tail, right tail, or two tail test. This rule always holds.
You will fail to reject the null hypothesis if the p-value is greater than or equal to the level of significance.
The p-value approach is best suited for the normal distribution when doing calculations by hand. However, many statistical packages will give the p-value but not the critical value. This is because it is easier for a computer or calculator to find the probability than it is to find the critical value.
Another benefit of the p-value is that the statistician immediately knows at what level the testing becomes significant. That is, a p-value of 0.06 would be rejected at an 0.10 level of significance, but it would fail to reject at an 0.05 level of significance. Warning: Do not decide on the level of significance after calculating the test statistic and finding the p-value.
Here is a proportion to help you keep the order straight. Any proportion equivalent to the following statement is correct.
|The test statistic is to the p-value as the critical value is to the level of significance.|