The Mechanics Of Carrying Out Tests Of Hypotheses

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A statistic used in formulating the decision rule is called a test statistic. The set of values of the test statistic that lead to the rejection of the null hypothesis is called the rejection region or the critical region.

The set of values of the test statistic that are not in the critical region constitutes the acceptance region. A value of the test statistic that separates the acceptance and rejection regions is called a critical value.

A test of statistical hypothesis is completely determined by providing the relevant test statistics and the rejection region.

Significance Probability (P-Value) The approach we have adopted so far involves picking a specific level of significance a. at the outset and then, on the basis of the value of the test statistic, rejecting Ho or not rejecting it at that level.

An alternate way which is widely used in journals and technical reports is to provide the probability of obtaining a value of the test statistic as extreme as, or even more extreme than, the one computed from the data, if indeed Ho is true.

This probability ...
... is just the area in the appropriate tail(s) of the distribution of the test statistic when Ho is true and is called the F-value or the significance probability of the observed value.

When calculating a f-value, it is important to keep the alternative hypothesis HA in perspective, as the nature of HA (less than, greater than, not equal to) determines the tail(s) in which the values of the test statistic are significant. The smaller the P-value, the stronger the justification for rejection of Ho.

There is an advantage to this method of reporting results by providing the P-values. We are free to pick our own level of significance in arriving at a decision.

Thus, if the calculated P-value is less than or equal to the stipulated level of significance of a, then we reject Ho. On the other hand, if the calculated P-value is greater than the stipulated a, then we do not reject Ho.

To sum up, the P-value (or significance probability) associated with the test of a hypothesis is the smallest a for which the observed data would call for rejection of the null Hypothesis in favor of the alternative. If a is the stipulated level of significance of the test, hen: (1) reject Ho if P-valued a and (2) do not reject Ho if P-value a.

Steps to Be Followed for Testing a Hypothesis Step 1. State the alternative hypothesis HA. (This is the research hypothesis in the investigation and specifies a range of possible values for the parameter that is being tested. It is important in deciding whether the test is one-tail or two-tail. Rejection of Ho leads to the acceptance of HA. )

Step 2. State the null hypothesis. Under Ho give a specific value of the parameter.

Step 3. Pick an appropriate test statistic.

Step 4. Stipulate the level of significance a, the probability of rejecting Ho wrongly. The critical point (s) are determined by the value of a. Together with Step 1, formulate the decision rule, that is, determine the values of the test statistic that will lead to the rejection of H0 (the critical region) . If the value of a is not stipulated, provide the P- value in Step 6 below.

Step 5. Take a random sample and compute the value of the test statistic.

Step 6. The final step consists of making the decision in light of the decision rule formulated in Step 4. It is important to translate the conclusions into no statistical language for the benefit of the uninitiated.

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