# Why does the critical value varies inversely with the level of significance in hypothesis testing?

The level of significance $\alpha$ is supposed to the probability at which the null hypothesis $H_0$ can be rejected. But when checking the statistical tables for $t_\alpha$, $z_\alpha$, and $\chi^2_\alpha$, the values increase with decrease in $\alpha$ leading to the conception that the acceptance region increases as the strictness increases.

Eg., in goodness of fit test

$\chi^2_{0.05} = 21.026 \qquad\chi^2_{0.01} = 26.217 \qquad \text{where } \nu=12$

$\alpha=0.01$ means lesser probability of error than $\alpha=0.05$, hence more stricter. But gives a greater range for the acceptance region in the test where $H_0$ is rejected when $\chi^2 > \chi^2_\alpha$.

But this cannot be so. How could one explain this?

• The $P$ value is the proportion of the distribution which is greater than the critical value, not less than it. For an increasing $\chi^2$ value, you have a smaller acceptance region, assuming the same df. – Chris C Nov 8 '15 at 15:00
• We are becoming stricter when we require harder evidence against $\text{H}_0$, which is to say, we expand the region of insufficient evidence (non-rejection of $\text{H}_0$). – Richard Hardy Nov 8 '15 at 19:41
• @ChrisC, I am not sure if I understand your last sentence. I would say, the acceptance versus the rejection regions are separated by the critical value, and the critical value depends on d.f. Meanwhile, the regions do not depend on the actual realization of the test statistic, $\chi^2$. – Richard Hardy Nov 8 '15 at 19:45
• @RichardHardy Sorry, I must have minced my words. By acceptance region, I meant the region where you reject your null. I should have been more accurate with my terminology. While keeping the same df, if you decrease $\alpha$ / increase $\chi^2_{crit}$ then you are decreasing the area under the probability curve where you would reject $H_0$. I thought it would illustrate what I was saying but unfortunately not. – Chris C Nov 8 '15 at 19:54

The level of significance $α$ is supposed to the probability at which the null hypothesis $H_0$ can be rejected.

Almost. Let's be careful here. Keeping in mind that a type I error is the case where you reject when $H_0$ is true, $\alpha$ is the probability of a type I error under the rejection rule "reject when the statistic is at least as large as the critical value".

But when checking the statistical tables for $t_α$, $z_α$, and $χ^2_α$, the values increase with decrease in $α$

... exactly as they should.

leading to the conception that the acceptance region increases as the strictness increases.

Correct. To have smaller type I error rate (smaller probability of rejection when H0 is true), the rejection region must get smaller ... so the part that isn't the rejection region must get bigger.

$α=0.01$ means lesser probability of error than $α=0.05$

Less probability of a Type I error. The distinction matters here.

But gives a greater range for the acceptance region in the test

Correct -- less chance of a rejection in the case when you shouldn't reject.

• Thanks, @Glen_b. I know I can always count on you when it comes to hypothesis testing. Regarding your answer, correct me if I'm wrong: greater $\alpha$ means more probability of making type I error, and hence less level of confidence that your prediction right. Thus the stricter limits for greater $\alpha$. – Ébe Isaac Nov 8 '15 at 17:26
• "greater α means more probability of making type I error" yes (but this only happens when the null is true) ... "hence less level of confidence that your prediction right" -- not sure what you mean by this. – Glen_b Nov 8 '15 at 17:29
• lesser level of confidence that your prediction is right: by prediction, I mean accepting the null hypothesis to be correct. – Ébe Isaac Nov 8 '15 at 17:59