A test of significance is a procedure by which sample results are used to verify the truth or falsity of a null hypothesis. The key idea behind tests of significance is that of a test statistic (estimator) and the sampling distribution of such a statistic under the null hypothesis. The decision to accept or reject H0 is made on the basis of the value of the test statistic obtained from the data at hand.

If the value of true β2 is specified under the null hypothesis, the t value can readily be computed from the available sample, and therefore it can serve as a test statistic. And since this test statistic follows the t distribution, confidence-interval statements such as the following can be made:

Pr(−tα/2 ≤ˆ (β2 − β∗2)/se ( ˆ β2) ≤ tα/2) = 1 − α where β∗2  is the value of β2 under H0 and where −tα/2 and tα/2 are the values of t (the critical t values) obtained from the t table for (α/2) level of significance and n − 2 df

In the confidence-interval procedure we try to establish a range or an interval that has a certain probability of including the true but unknown β2, whereas in the test-of-significance approach we hypothesize some value for β2 and try to see whether the computed ˆ β2 lies within reasonable (confidence) limits around the hypothesized value.

To test the hypothesis that there is no relationship between the variables x and Y using the model , the null hypothesis is stated as ; H₀: β=0 [no relationship between x and y]. if no prior information about the values of the regression parameters is available , the alternative hypothesis is stated as; H_A: β≠0.

The test statistic is given as  at n-2 degrees of freedom. For a two tailed test (β≠0), the acceptance region is .

The best test is achieved if we take the alternative hypothesis as β<0, where the rejection region is .

If there is prior knowledge about the values of the parameters, .

Alternatively, the F- test can be used to test for a relationship between the variables. .