Comparing a test statistic and a critical value is the classical approach to hypothesis testing. This is best utilized for distributions that give areas and require you to look-up the critical value (such as the Student's t distribution) instead of distributions that have you look-up a test statistic to find out an area (similar to the normal distribution).
The Classical Approach as well has three distinct decision rules, based on whether it is a left tail, right tail or a two tail test.
One of the problems with the Classical Approach is that when a distinct level of significance is desired, a different critical value should be read from the table.
The P-Value Approach, brief for Probability Value, approaches hypothesis testing from a different way. Rather than comparing z-scores or t-scores in the classical approach, you are comparing the probabilities or areas.
The level of significance (that is, alpha) is the region in critical region. That is, the area in tails to the right or left of critical values.
The p-value is the region to right or left of the test statistic. When it is a two tail test, then look-up the probability in one tail and doubles it.
When the test statistic is in critical region, then p-value will be less than the level of significance. It doesn’t matter whether it is a left tail, right tail or a two tail test. This rule for all time holds.
Refuse the null hypothesis when the p-value is less than the level of significance.
We will fail to refuse the null hypothesis when the p-value is more than or equal to the level of significance.
The p-value approach is fine suited for the normal distribution whenever doing computations by hand. Though, most of the statistical packages will provide the p-value however not the critical value. This is as it is simple for a computer or calculator to determine the probability than it is to determine the critical value.
The other benefit of p-value is that the statistician instantly knows at what level the testing becomes important. That is, the p-value of 0.06 would be refused at a 0.10 level of significance; however it would fail to refuse at a 0.05 level of significance.
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