Obtaining A Critical Region And p-Value

In this article, we will discuss how important obtaining a critical region is in the analytical and data science problem-solving. The critical region is the set of all values of the test statistic that causes us to reject the null hypothesis and accept the alternative hypothesis. It varies from one case to another due to different probability values being taken up. It also depends on the fact whether the test is one-tailed or two-tailed. In Figure 1, the yellow part is termed as the critical region or rejection region, while the region inside the yellow portion is termed as the acceptance region. Now, if the p-value lies in the yellow region we have to reject our null hypothesis (H0). The points X and -X vary from one probability level to another (Greater the percentage value, greater will be the value of X and -X.)

Two-Tailed Test

In the case of the two-tailed test, there would be two critical regions (As shown in the above graph). In the cases when we are interested to find whether the values are different or not equal, we use the two-tailed test. If percentage level is assumed to be (1-α) level, then both the critical regions would be of size α/2.  The following hypothesis is an example of a two-tailed test.

H0: µ = µ0
H1: µ ≠ µ0

One-Tailed Test

One-tailed tests are used when we are interested only in the extreme values that are greater than or less than a comparative value (say µ0). In the case of one-tailed tests, there is only one critical region.

One-tailed tests are of two types-


H0: µ = µ
µ < µ0


H0: µ = µ0
H1: µ > µ0

In the case of one-tailed test, the critical region is of the value α (Unlike α/2 in the case of two-tailed).

Now, in order to obtain the critical value, we must know the type of hypothesis, the distribution the test follows, the percentage level at which we are working and lastly whether the test is two-tailed or one (right or left tailed). We’ve discussed all the above terms above, so now obtaining the value beyond which the critical region lies would be easy to find.

Step 1: Check the null and the alternative hypothesis.
Step 2: Take note of the distribution the test follows.
Step 3: Calculate the degrees of freedom, if any.
Step 4: Open the tables and look up for the distribution.
Step 5: If it is a two-tailed test at suppose 95% level of a Normal distribution, then look up for the value of 2.5% (α/2). And if it is one tailed test then look up for the value of 5% and then put a negative sign depending on the fact whether it is left or right tailed.

In the case of normal distribution, we do not require to calculate the degree of freedom, but in the cases of other distribution like t-distribution or chi-square distribution, we need to calculate the degree of freedom. On the other hand, both Normal, as well as the t distribution, are symmetrical so we need to just check one value and just replace signs, but in the case of non-symmetrical distributions, we need to check the individual values.

For example, if we are working on the chi-square distribution at 95%, we need to first find the degree of freedom of that chi-square and then check the value of both 97.5% as well as 2.5%. To begin, just follow the steps and practice with the Normal distribution. Once you’ve mastered it, go for the calculation of the degree of freedom and then the other distributions.

Critical regions are as critical as their name suggests and hence should be calculated carefully, or else we might end up in a wrong conclusion (Type 1 or Type 2 error).

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