A critical value in statistics is a specific point on a statistical distribution that is used as a reference to determine the acceptance or rejection of a null hypothesis during hypothesis testing. It serves as a cutoff point beyond which a statistical test will lead to rejecting the null hypothesis in favor of an alternative hypothesis.

__Critical values__ are determined based on the chosen level of significance (mostly signifies as alpha, α) for a given statistical test. They are typically obtained from probability distributions like the t-distribution, chi-square distribution, or F-distribution.

This article will explain the following fundamental terms of the critical value:

- Critical value: What is?
- Steps to calculate the critical value
- Critical Values in F-Distribution
- Critical Values in Chi-Square Distribution
- Solved Problems of Critical Value.

Let’s explore all the terms and gain more insight into the critical value.

## Critical Value

The critical value in statistics refers to a specific point on a statistical distribution that sets the boundary for decision-making during __hypothesis testing__. It is a determined threshold used to determine whether to accept or reject the null hypothesis.

If the test statistic value lands inside the rejection region, the null hypothesis gets rejected; if not, it remains unelected. Critical values can relate to either a one-tailed (left or right tail) or a two-tailed test. For a two-tailed test, two critical values are needed, while a one-tailed test requires just one critical value.

These values are derived from probability distributions like the t-distribution, chi-square distribution, or F-distribution and depend upon the chosen level of significance (often denoted as alpha, α).

## Steps to Calculate Critical Value:

Determining critical value depends on the distribution of the test statistic. It may involve using the confidence interval or the significance level (alpha) as part of the calculation process.

**Step 1:** Determine the significance level by converting the confidence interval into alpha (α): α = 1 – (confidence level divided / 100).

**Step 2:** Differentiate between a one-tailed and a two-tailed test as critical values and distribution tables vary for each type. In a one-tailed test, the alpha value remains unchanged. However, for a two-tailed test, the alpha value is divided by 2.

**Step 3:** Critical values are specific to individual tests. Reference a distribution table to find the critical value corresponding to a test according to its alpha value.

Further details regarding the **Step 3** process will be provided in the following section.

## Critical Values in F-Distribution:

The F-distribution is used when comparing variances of two independent sample populations. Critical values in the F-distribution correspond to different levels of significance and degrees of freedom for the numerator and denominator in the F-test. This can be calculated by the following process:

**Step 1:** Determine the α

**Step 2:** Compute the 1^{st} degree of freedom with the subtraction of one from the initial sample size. Say A

**Step 3:** Compute the 2^{nd} df by deducting one from the 2^{nd} sample size. Say B

**Step 4:** Consulting the F-distribution table involves finding the critical value by locating the point where the **A** column and **B** row intersect.

**Decision criteria:**

- If test Statistics (Right-tailed) > Reject F critical value.
- If test Statistics (Left tailed) < Reject F-critical value.
- Reject the null hypothesis in a Two-Tailed Test if the test statistic exceeds the upper or lower critical values.

## Critical values in Chi-Square distribution:

The __chi-square test__ serves to evaluate whether sample data aligns with population data and can also evaluate the relationship between two variables. The critical value for chi-square is provided as follows:

- Determine the α.
- Compute the Df by deducting 1 from the sample size n.
- Consulting the chi-square distribution table involves finding the chi-square critical value by locating the point where the row corresponding to df intersects with the column representing the chosen alpha value.

## Key Points Regarding Critical Values:

- A critical value is used as a benchmark to evaluate the validity of rejecting the null hypothesis based on a comparison with the test statistics.
- It marks the boundary distinguishing between the acceptance and rejection regions within the distribution graph.
- Critical values come in four types: z, f, chi-square, and t.

## Problems of Critical Values:

These examples demonstrate how we calculate the critical value.

**Problem 1:**

Compute the critical F-value for this scenario:

α = 0.01

Sample 1:

- Variance = σ
_{1}= 20 - Sample size = 21

Sample 2:

- Variance = σ
_{2}= 10 - Sample size = 16

**Solution:**

Let’s compute the F-critical value:

**Step 1: ****Degrees of Freedom Calculation:**

For Sample 1:

Sample size−1 = 21−1 = 20

For Sample 2:

df_{2} = Sample size − 1= 16 – 1 = 15

**Step 2:** **Critical F-value Determination:**

Using the F distribution table for α = 0.01, the value at the intersection of the 20^{th} column and 15^{th} row is 3.37.

**Step 3: **Decision making

f = σ_{1}^{2}_{ / }σ_{2}^{2} = 4

The calculated test statistic **f = 4** exceeds the critical F-value of 3.373, we reject the null hypothesis. This suggests that there is a significant difference in the variances between Sample 1 and Sample 2 at a significance level of α=0.01.

## Wrap up:

This article explored critical values in statistics, essential in hypothesis testing decisions. Derived from distributions like t, chi-square, or F-distributions, these values help accept or reject null hypotheses. By determining significance levels, we differentiate between one-tailed and two-tailed tests, using critical values as thresholds.