Hypothesis Testing: Significance Levels Explained

This question relates to hypothesis testing and how the decision changes based on the chosen significance level ($\alpha$).

Understanding Significance Levels

• The significance level ($\alpha$) represents the threshold for rejecting the null hypothesis ($H_0$). It's the probability of making a Type I error (rejecting $H_0$ when it is true).
• Commonly used levels are $\alpha = 0.05$ (5%) and $\alpha = 0.01$ (1%).
• A smaller $\alpha$ (like 0.01) requires stronger evidence to reject $H_0$ compared to a larger $\alpha$ (like 0.05).

Decision Logic

• Given: The null hypothesis could not be rejected at the $\alpha = 0.05$ level. This means the results obtained were not statistically significant enough to discard $H_0$ at the 5% significance level.
• Reasoning: Since the $\alpha = 0.01$ level is more stringent (requires stronger evidence) than the $\alpha = 0.05$ level, if $H_0$ was not rejected at the looser 0.05 level, it certainly will not be rejected at the stricter 0.01 level. The evidence is insufficient for both, and particularly insufficient for the stricter criterion.
• Conclusion: Failing to reject $H_0$ at $\alpha = 0.05$ guarantees that $H_0$ will also not be rejected at $\alpha = 0.01$.

Therefore, the correct conclusion is that the null hypothesis will definitely not be rejected at the .01 level of significance.