A Brief Introduction of Critical Value: Explained with Types and Calculations

In this article, we will learn how we can find critical values. We will also discuss the different types of critical value. After this, we will solve some examples for your better understanding.

Critical value in statistics

The critical value is a line that breaks the curve into two regions, one of which is the acceptance region and the other is the rejection region. If the test statistics value falls in the rejection (critical) region, we reject the null hypothesis and accept the alternative hypothesis.

The critical value tells us whether the null hypothesis is accepted or rejected during hypothesis testing. We have only one critical value in the one-tailed test but two critical values for the two-tailed test. The critical value is used to calculate different hypotheses tests.

How to find critical value?

Let’s learn how to find critical value for a one-tailed and two-tailed test. By using confidence level, we can find critical value easily. Consider the confidence level that is 95%. Follow the following steps to evaluate the critical value,

• To find the alpha (level of significance), subtract the confidence level from 100%. Such that, Alpha= 100% – 95% = 5%
• Change the alpha value into decimal, as 5% equals 0.05.
• For the one-tail test, we use the above method. However, if we want two tail-tests,n, we will divide the alpha level by 2.

Types of Critical Value

There are four different types of critical value.

1. T – Critical value
2. Z – Critical value

• F – Critical value

1. Chi-square critical value

Let’s discuss these types one by one in detail.

     I.        T-Critical value

It is used when,

• the sample size (n) is less than 30, i.e. n < 30
• Population standard deviation is not given

How to Calculate the T – critical value?

• First of all, find the level of significance
• To find the degree of freedom (d_f), subtract 1 from sample size n.
• Now look at the t- distribution table.
• Observe the degree of freedom (d_f) at the left side column and alpha at the top row of the t-table. Then take the corresponding value of this row and column. This value is our T- critical value.

   II.        Z – Critical value

When we use-tests

• If n ≥ 30 (where n is the sample size)
• If the Population standard deviation is given

How to find the Z – critical value?

• Evaluate the level of significance (alphaα).
• For the one-tailed test, alpha is subtracted from 0.5; for the two-tailed test, alpha subtracts from 1.
• Now, look Z table to give the z-critical value.
• After calculating the left-tailed test, we will write a negative sign at the end of the calculations.

  III.        F – Critical value

The f-critical is used to compare the variance of two samples. In the F-critical value, two sample sizes are given. The f-critical value test is also known as ANOVA.

How to find the F – critical value?

• First, calculate the level of significance (alphaα).
• To get 1^st degree of freedom (d_f), 1 subtract from 1^st sample size (n₁) and say x = n₁ – 1
• To get 2^nd degree of freedom (d_f), 1 subtract from 2^nd sample size (n₂) and say y = n₂ – 1
• Now look at the f- distribution table and take the corresponding value of the x column and y row to get the f- critical value.

  IV.        Chi-Square Critical value

In the chi-square value, we compare two variables to determine their relationship.

How to calculate the Chi-Squarecritical value?

• Determine the level of significance (alphaα).
• To get the degree of freedom (d_f), 1 subtract from the sample size, i.e. d_f = n – 1
• To get Chi-square critical value, look at the chi-Square distribution table and take the intersection of the alpha column and degree of freedom row.

Solved Example of critical value

Example 1:

Find a one-tail test when the confidence level is 99%, and the sample size is 7.

Solution:

Here, n = 7, and n < 30, so we will use a t-test.

Step 1: To find alpha, subtract the level of confidence from 100% i.e.

Alpha = 100% – 99% = 1%

Step 2:Covert alpha into decimal

Alpha = α= 1 / 100 = 0.01

Step 3: To get the degree of freedom (d_f), subtract 1 from the sample size.

d_f = 7 – 1 = 6

Step 4:Then Look at t- distribution table

Step 5:Find the degree of freedom (6) at the left side column and alpha (0.01) at the top row of t- distribution table. Take the intersection of this row and column; this value is our T- critical value.

The degree of freedom and alpha intersect at 3.1430, our t-critical value.

Conclusion

In this article, we have discussed the critical value in detail. We learned the method of finding the critical value. Then we discussed different types of critical value. After this, we learned when these types would be used and how to calculate these critical values.

At last, we discussed the Example of critical value with the systematic solution. After reading this article, you can find critical value easily.