Decision Rule in Hypothesis Testing

A decision rule is the rule based on which the null hypothesis is rejected or not rejected.

We first state the hypothesis. Then we determine if it is a one-tailed or a two tailed test. We then specify a significance level, and calculate the test statistic. Now we calculate the critical value. If the test statistic follows a normal distribution, we determine critical value from the standard normal distribution, i.e., the z-statistic. Using the test statistic and the critical value, the decision rule is formulated.

For a 5% level of significance, the decision rules look as follows:

1. H0: θ = θ0 versus Ha: θ ≠ θ0

Reject the null hypothesis if test-statistic > 1.96 or if test-statistic < -1.96.

2.  H0: θ ≤ θ0 versus Ha: θ > θ0

Reject the null hypothesis if test-statistic > 1.645

3. H0: θ ≥ θ0 versus Ha: θ < θ0

Reject the null hypothesis if test-statistic < -1.645

In our example, the decision rule will be as follows:

Reject the null hypothesis if test-statistic > 1.96 or if test-statistic < -1.96.

Our value of test-statistic was 4, which is greater than 1.96. Therefore, null hypothesis should be rejected. This was a two-tailed test.

The following chart shows the rejection point at 5% significance level for a one-sided test using z-test.

Power of Test

The power of test is the probability of correctly rejecting the null (rejecting the null when it is false).

The power of test = 1 – P(Type II Error)

The following table illustrates the correct decision, Type I error and Type II error.

	H0 is true	H0 is false
H0 is rejected	Type I ErrorSignificance level, a. Probability of Type I error.