Hypothesis Testing Significance Level Relationship

In hypothesis testing, the significance level, denoted by $\alpha$, represents the probability of rejecting the null hypothesis (H₀) when it is true. Common values include 0.01 and 0.05.

Decision Rule in Two-Tailed Tests

For a two-tailed test, the critical region (where H₀ is rejected) is split between the two tails of the distribution. The total area in the critical region equals $\alpha$.

• A smaller $\alpha$ (e.g., 0.01) corresponds to a smaller critical region, making it harder to reject H₀.
• A larger $\alpha$ (e.g., 0.05) corresponds to a larger critical region, making it easier to reject H₀.

Impact of Changing Significance Level

If H₀ could not be rejected at the 0.01 level of significance ($\alpha = 0.01$), it means the calculated test statistic did not fall into the critical region defined by $\alpha = 0.01$.

Now, consider the 0.05 level of significance ($\alpha = 0.05$). Since $0.05 > 0.01$, the critical region for $\alpha = 0.05$ is larger than that for $\alpha = 0.01$. The critical region for $\alpha = 0.05$ includes the entire critical region for $\alpha = 0.01$ plus additional areas in the tails.

Therefore, if the test statistic was not in the rejection region for $\alpha = 0.01$, it is guaranteed to be in the non-rejection region for $\alpha = 0.05$ *only if* the test statistic falls within the original non-rejection region. However, the test statistic could potentially fall into the *newly added* parts of the rejection region specific to $\alpha = 0.05$.

Consequently, we cannot definitively say H₀ will be rejected or not rejected. The decision depends on the exact value of the test statistic relative to the critical values for $\alpha = 0.05$.

Conclusion

The null hypothesis may or may not be rejected at the 0.05 level of significance.