What is the critical value? Discuss how to evaluate what is the critical value. Discuss how to evaluate it.

A critical value plays a vital role in confidence interval and hypothesis tests and also defines regions in the sampling distribution of a test statistic. In addition to this, the critical value of a particular test can be molded from the distribution of the test statistic and also the significance level.

In this article, we’ll discuss the basic definition of critical value and its types will also be discussed and for better understanding examples will also be provided.

Definition

“Critical value is a range of acceptance regions of the test”. In hypothesis tests, its role is to allow to accept or reject the null hypothesis. It comprises two tests one-tailed hypothesis test and a two-tailed hypothesis test as it is clear from the name that a one-tailed mean it has one critical value similarly two-tailed test has two critical values.

Critical values are basically used to check whether the results are statistically significant or not. Critical values in confidence intervals help to evaluate the lower limits and upper limits.

The formula of critical value

Critical Value = 1 – (α / 2)

and α = 1 – (confidence level / 100)

Types of Critical Value

There are different testing techniques for critical values to check the statistical significance of a particular population or sample. The statistical significance will guide you to where test finding is needed or not in a given situation. Mostly used critical value systems by statisticians to calculate significance are discussed below:

Z Critical Value

Z-score is basically used for identifying how far a data point is deviating from the sample mean. Z-critical values are the standard scores that may be determined from data collection. From this, you easily get to know how many standard deviations your population means are below and above the raw score.

Chi-square Value

Chi-square values come from two different types of chi-square tests: the goodness of fit chi-squares tests and the independence chi-square tests. The work of the goodness of fit chi-square test is to check whether a small collection of sample data is representative of the total population or not while in independence test is to check their link and compare two variables.

F Critical Value

Its role is similar it also uses to check the significance of a given test it can be calculated in a few steps and divided into two mean squares it is commonly used in ANOVA for analysis of variance. it is a value on F-distribution.

Example Section

In this section with the help of examples, the topic will be explained.

Example 1:

Calculate the critical value by a left-tailed z-test where α is 0.1.

Solution:

Step 1:

First, subtract the α value from 0.5.

Thus, 0.5 – 0.1 = 0.4

Step 2:

With the help of the z-distribution table, z = 1.2816

Because of the left-tailed z test then the critical value will be in negative sign, z= -1.2816

Critical value = -1.2816

Example 2:

Evaluate the critical value for a two-tailed f test by using α = 0.5 on the samples given below

Sample size = 31

Sample size = 11

Solution:

Step 1:

N1 = 31,

N2 = 11,

N1 – 1= 30,

N2 – 1 = 10,

Step 2:

For Sample 1

df = 30,

For Sample 2

df = 10

with the help of the F-distribution table using Now using for α = 0.5, the value will be selected where the 30th column and 10th row mean that is F (30, 10) = 0.7403 & 1.5119

Critical Value = 0.7403 & 1.5119

You can also try an online critical value calculator to evaluate the problems of critical value without any effort.

Solved through criticalvaluecalculator.com

Example 3:

Calculate the critical if a one-tailed t-test is under observation on data whose sample size is 8 and α is 0.05.

Solution:

Step 1:

Extract the given data

Sample size = n = 8

Step 2:

To obtain a degree of freedom we use the formula

df = n – 1

df = 8 – 1 = 7

with the help of a one-tailed t distribution table get to know that t (7, 0.05) = 1.8946

Thus, Critical Value = 1.8946

Example 4:

Evaluate the critical value for a two-tailed f test by using α = 0.3 on the samples given below

Sample size = n1 =43

Sample size = n2 =31

Solution:

Step 1:

n1 = 43,

n2 = 31,

n1 – 1= 42,

n2 – 1 = 30,

Step 2:

For Sample 1

df = 42,

For Sample 2

df = 30

with the help of the F-distribution table using Now using for α = 0.3, the value will be selected where the 42nd column and 30th row mean that is F (42, 30) = 0.7090 & 1.4388

Critical Value = 0.7090 & 1.4388

Example 5:

Calculate the critical if a one-tailed t-test is under observation on data whose sample size is 6 and α is 0.01.

Solution:

Step 1:

Extract the given data

Sample size = n = 6

Step 2:

To obtain a degree of freedom we use the formula

df = n – 1

df = 6 – 1 = 5

with the help of a one-tailed t distribution table get to know that t (5, 0.01) = 3.365.

Thus, Critical Value = 3.365.

Conclusion

In this article, with the help of basic definitions and methods of how to calculate it by using different techniques, and also with the help of examples, the topic is explained.