In statistics, a critical value is a value that separates a region of rejection from a region of non-rejection in a statistical hypothesis test. Critical value is a value that separates the acceptance and rejection regions in a hypothesis test, based on a given level of significance (alpha).

It is a boundary or threshold that determines whether a statistical test will reject the null hypothesis or fail to reject it. The critical value is determined by the distribution of the test statistic and the level of significance chosen for the test. By using a table or using statistical software we can find easily critical value.

The critical value is dependent on the significance level, sample size, and the type of test being performed. They play a crucial role in hypothesis testing and help determine the validity of statistical inferences.

In this article, we will discuss the definition of critical value, the Meaning of critical value, Approaches of Critical value, types of critical value, and also discuss the example of critical value.

Table of Contents

#### Definition of Critical Values

A critical value is a threshold value used in hypothesis testing that separates the acceptance and rejection regions based on a given level of significance. In statistical hypothesis testing, a critical value is a threshold value that is used to determine whether a test statistic is significant enough to null hypothesis reject.

It is based on the level of significance chosen for the test and is determined by the distribution of the test statistic. The critical value separates the acceptance and rejection regions, and if the test statistic falls in the rejection region, the null hypothesis is rejected.

Critical values play a crucial role in hypothesis testing as they help to determine the validity of statistical inferences. The critical value is based on the level of significance (alpha) chosen for the test and is determined by the distribution of the test statistic.

It is used to define the region of rejection, which consists of the extreme sample statistics that are unlikely to occur if the null hypothesis is true. Critical values can be obtained from tables or calculated using statistical software and are essential in determining the validity of statistical inferences.

#### Critical Value Approach | Steps of Hypothesis Testing

The approach of critical value involves several steps in statistical hypothesis testing:

- Formulate the null hypothesis and alternative hypothesis.
- Choose the level of significance (alpha) for the test.
- Determine the appropriate test statistic to use for the hypothesis test.
- Determine the test statistic distribution under the null hypothesis.
- Calculate the test statistic value using the sample data.
- Determine the critical value from the distribution of the test statistic based on the level of significance.
- Compare the critical value with the test statistic value.
- If the test statistic value is greater than or equal to the critical value, reject the null hypothesis in favor of the alternative hypothesis; if not, fail to reject the null hypothesis.
- Calculate the p-value to determine the strength of evidence against the null hypothesis, if desired.

The critical value approach is widely used in hypothesis testing to determine the validity of statistical inferences. It involves determining a threshold value that separates the acceptance and rejection regions based on a given level of significance, which helps to determine whether the test statistic is significant enough to reject the null hypothesis.

The approach of critical value is used in various statistical tests such as t-tests, F-tests, and chi-square tests. The critical value approach is widely used in statistical hypothesis testing as it provides a clear and objective method to determine the validity of statistical inferences.

#### Different Types of Critical Value

There are four different kinds of crucial values, depending on the statistical test that is run on

the statistical data. A list of critical value types is given below:

- F-critical value
- T- critical value
- Z- critical value
- Chi-square-critical value

**F- Critical value**

When testing a hypothesis involving F-distribution then we use the F-critical value. It is denoted by the following notation Fα, df2, df1, here α is the level of significance and df2, df1 denotes the degree of freedom for the denominator and nominator, respectively, and also Fα, df2, df1 is the F critical value that corresponds upper tail area of α.

The F-critical value is used to determine whether to reject or fail to reject the null hypothesis in a hypothesis test involving variances. If the calculated F-statistic is greater than or equal to the F critical value, the null hypothesis is rejected, indicating that there is a significant difference in variances between the groups being compared.

**T- Critical value**

When testing a hypothesis involving T-distribution then we use the T-critical value. It is denoted by the following notation tα/2, here α is the level of significance and tα/2 is the t-critical value that corresponds to the upper tail area of α/2 and also n-1 is the degree of freedom.

A t critical value calculator is the best way to find the t value of your required input to avoid table searches and possible mistakes.

**Z- Critical value**

In hypothesis when involving standard normal distribution used the Z critical value. It is denoted by the following notation Zα/2, here α is the level of significance and Zα/2 is the Z score critical value that corresponds to the upper tail area of α/2.

**Chi-square- Critical value**

The Chi-Square critical value is a value used in statistical hypothesis testing to determine the

significance of the Chi-Square statistic. It is based on the level of significance (alpha) chosen for

the test and the degrees of freedom associated with the Chi-Square distribution.

#### Critical Value Example | Formula

Find the critical value for a two-tailed f test conducted on the following samples at an α = 0.05

Variance = 120, Sample size = 61

Variance = 80, Sample size = 51

**Solution:**

Step 1:

n1 = 61

n2 = 51

Step 2:

Sample df1 = n1 – 1= 60

Sample df2= n2 – 1 = 40

Step 3:

For α = 0.05, using the F distribution table, the value at the intersection of the 60th column and 40th row is

F (60, 40) = 1.637

Critical Value = 1.637

#### FAQs Questions and Answers

**1. How is the critical value calculated?**

– The critical value is calculated based on the distribution of the test statistic and the desired significance level. In most cases, the critical value is determined from tables or statistical software. For example, in a t-test with a sample size of 10 and a significance level of 0.05, the critical value can be looked up from a t-distribution table with 9 degrees of freedom.

**2. Can critical values be negative?**

– Yes, critical values can be negative, depending on the distribution of the test statistic.

**3. What happens if the test statistic exceeds the critical value?**

– If the test statistic exceeds the critical value, the null hypothesis is rejected, indicating that there is evidence to support an alternative hypothesis. This means that the observed difference between two groups or variables is unlikely to have occurred by chance.

#### Summary

In this article, we have discussed the definition of critical value, the Meaning of critical value, Approaches of Critical value, and types of critical value, and also with the help of examples, the topic will be explained. After studying this article anyone can defend this topic easily.

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