Basics of Critical Values: What is it and How to Find it?

Critical values define specific regions within the sampling distribution for test statistics. It plays a key role in hypothesis testing and confidence intervals. In hypothesis testing, the critical value is used to decide if a result is statistically significant or not. In confidence intervals, critical values help to calculate the upper and lower limits.

In this article, we will explore critical values, delving into their definition, various types, and the techniques used to compute them. We will solve some examples of critical value to make this concept easier for our readers.

Defining Critical Value

In statistics, the critical value is the value that is compared to test statistics in hypothesis testing to identify whether the null hypothesis should be rejected or not. The null hypothesis will not be rejected if the test statistic value is less than the critical value. Reject the null hypothesis when the value of the test statistic is greater than the critical value.

Graphically, the critical value represents the boundary between the rejection region and the non-rejection region on a probability distribution. The rejection region is the area in the tails of the distribution where the test statistic is unlikely to occur if the null hypothesis is true. Non-rejection region refers to the zone in the middle of the distribution where the test statistic is likely to occur under the null hypothesis.

Formula of Critical Value

The formula for calculating critical values depends on the nature of the statistical test and the distribution involved. We can use either the confidence interval or the significance level to calculate the critical value. In the next selection, we will discuss different formulas for finding critical values.

How to Determine Critical Value?

Here is a general procedure for calculating critical values:

Choose a Significance Level (α): The critical value depends on the significance level that you chose for your test. The common choice is 0.05, but it can vary depending on your analysis.
Determine the Test Distribution: Identify the probability distribution that your test statistic follows. Common distributions include t or chi-squared distribution.
Select the Tail(s): Determine whether you are conducting a one-tailed or two-tailed test. A one-tailed test focuses on one side of the distribution (either the left or the right tail), while a two-tailed test considers both tails.
Consult Statistical Tables: Find out the critical value corresponding to your chosen significance level, distribution, and tails. These tables provide critical values for various α levels and degrees of freedom.
Utilize a Critical Value Calculator: To simplify the process and ensure accuracy, employ a critical value calculator. Input your chosen significance level, test distribution, and tail(s) into the calculator, and it will quickly and exactly calculate the critical value for you.

Types of Critical Value and How to Find Them

There are four main types of critical values:

T - Critical value
Z - Critical value

iii. F - Critical value

Chi-square critical value

Let’s learn how to evaluate these types with easy steps.

T-Critical Value

Determine the alpha level by following the method.
∴ Alpha = 1 – (confidence level / 100)
Alpha will remain the same for the one-tailed test, divide alpha by 2 for the two-tailed test.
Find the degree of freedom (df) by subtracting 1 from sample size.
Degree of freedom (df) = n – 1
Look at the t- t-distribution table.
Find the degree of freedom (df) at the very left side column and alpha at the top row of the t-distribution table. Take the intersection of this row and column, this value will be t- critical value.

Z - Critical value

Find the level of significance (alpha).
Determine the type of test.
For a right-tailed test: Subtract alpha from 0.5.
For a left-tailed test: Subtract the alpha value from the 1.
Subtract alpha/2 from 1 for a two-tailed test.
Look up the resulting area in the Z table.
Sum the values of intersecting row (on the top) and column (most left) to find the z critical value
If you are calculating the left-tailed test, then make sure to use the negative sign for the z-critical value in your calculations.

F - Critical value

Determine the level of significance for your test.
Compute the degrees of freedom for the numerator and denominator by the following method.

o Degree for numerator = n1 – 1

o Degree for denominator = n2 – 1

Look at the F-distribution table.
Locate the degrees of freedom for the numerator (most left column) and denominator (on top row) in the table.
Find the intersection of these degrees of freedom and the alpha value to get the F-critical value.

Chi-square critical value

Find the level of significance (alpha).
Calculate the degrees of freedom (df) by subtracting 1 from the sample size.
Observe the Chi-Square distribution table. Locate the alpha column and the row corresponding to the degrees of freedom (df). The intersection of these values will give you the Chi-square critical value.

Solved Examples of Critical Value

Here are some solved examples of finding critical values for different statistical tests:

Question 1:

Evaluate the t-critical value for a one-tailed test when the level of confidence is 99% and the sample size is 9.

Solution:

Step 1: Find the value of alpha.

Alpha level = 1 – (99 /100)

= 1 – 0.99

= 0.01

Step 2: Find the degree of freedom by subtracting 1 from the sample size.

Degree of freedom (df) = 9 – 1 = 8

Step 3: Observe the t-distribution table for the one-tailed test.

Step 4: Find the degree of freedom at the leftmost column and alpha at the top row of the table. Take the intersection of this row and column, this value will be our t- t-critical value.

The value of alpha and degree of freedom intersect at 2.8965. Therefore, the - Critical value for a one-tailed test is 2.8965

Question 2:

Evaluate the z-critical value for a left-tailed test when the alpha level is 0.452.

Solution:

Step 1: As the test is left-tailed, subtract alpha from 1.

1 – 0.452 = 0.548

Step 2: Find in 0.548 in z-distribution table (left-tailed test)

Step 3: Sum the values of intersecting row (top) and column (most left).

0.1+0.02 = 0.12

It is a left-tailed test, thus multiply it by -1.

Thus, -0.12 is the z-critical value for the left-tailed test.

Conclusion

In this article, we have explained the importance of critical values in hypothesis testing and confidence intervals. We discussed how to calculate them, involving factors like significance levels, test distributions, and test tails. We covered four main types (T, Z, F, and Chi-square) with practical calculation steps and provided examples for clarity.