The critical value is used to explain the limit between the taking and exclusion of the null hypothesis. This is an identical value that is used to determine the result of the significance level of the hypothesis test. In this portion, we talk about the explanation of the critical value, its types, and its actual applications.

In the examples, we will discuss the method of calculating the critical values.

What is the critical value?

In statistics, a critical value is a value that is used to find the rejection region in a hypothesis test. Hypothesis testing is a fundamental concept in inferential statistics, where we use sample data to make inferences about a population.

The critical value is chosen based on the significance level (α) of the hypothesis test, which represents the maximum probability of making a Type I error (incorrectly rejecting a true null hypothesis). Common significance levels are 0.05 (5%) and 0.01 (1%).

Types of critical value

Below are a few types of critical value.

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

1.   T-critical value

We use the t value when we don’t know the population standard deviation and when the sample size is fewer than 30. The t value depends on the value of α and the degree of freedom. The value of T-critical can be measured as,

• The significance level α must be expressed
• Test statistic t = (x̄ – µ)/(s/√n) with µ=(n-1) degrees of freedom
• We use a one-tailed test when a hypothesis is one-tailed
• Two-tailed test is applicable only when the hypothesis is two-tailed

Real-life Application of T critical value

The real use of one-tailed tests in real life in manufacturing and medicine, while the two-tailed test is used in engineering and weight loss. Various examples in real life here cover only a few.

2.   Z-critical value:

z-score is useable only when we identify the population standard deviation. The value of the zee-score is founded on the average spreading. Z-score is applied when the sample size is superior to 30. Z -score can be measured as.

• Label the level of significance α, i.e., H.  (α=0.05)
• for a two-tailed test subtract the significance level α from one. for a one-tailed test subtract the significance level α from 0.05
• Test statistics for one-tailed Z= (x̄ – µ)/(σ/√n) σ is the population standard deviation
• Test statistics for two-tailed Z= (x₁– x₂) /( √σ₁²/n₁+ σ₂²/n₂)
• Assign the rejection region or critical region

Real-life Application of Z-value

The z -value is used in the biology field for the size of new species such as compared to the mean population of other species.  Z-score is used for shoe size as compared to another mean population size. Z values are used for checking the blood pressure of one person to the average blood pressure of another population

3.   Chi-square critical value

We use the chi-square value for the partition of acceptance and rejection region for the chi-square distribution.  The value of the chi-square depends on the level of significance alpha and also on the degree of freedom. (Represented by α). The following steps for chi-square.

• State the unworkable and additional hypotheses.
• Pick the level of significance (α)
• Test statistics to be used

χ² = ∑^r ∑^c (O_i – E_i)²/E                                                         

µ=(r-1) (c-1)

r=number of rows

c=number of columns

• Compute the expected frequencies and calculate the value of the chi-square
• Fix the critical region
• Draw the conclusion and reject the null hypothesis if the calculated Chi-square falls in the electrical region otherwise accept.

Real-life use of Chi-square

We use the chi-square value mostly used in sociology phycology and marketing for the significance of differences between observed and excepted frequencies.

4.   F-value:

We use f -the test for the comparison of two variances and his result is always greater than 1. We calculated the F -test by this. The f critical values can be painstaking (measured) as,

• Variety of worth (significance) level
• For the first degree of freedom subtract 1 from the first sample and say “X”
• Subtract 1 from the size of the second sample we got 2^nd degree of freedom and say “y”

F = S₁² / S₂²

• S₁ = First variation
• S₂ = Second variation

Real-life use of f-test

F-tests mostly used in the biological field.

How to find critical value?

Example 1:

Find the critical value for one-tailed exam z where α=0.5 when the sample is 10

Solution:

Given data is

Step 1:

α = 0. 5, n = 10

Step 2:

Now we find the degree of freedom

Degree of freedom (df) = n-1, 10-1=9

Step 3:

Using the one-tailed test distribution table

The critical value is (9, 0.5) = 0

Example 2

Find the critical value for the two-tailed f-test showed on the following samples at α = 0.275, variance = 54, variance = 56 n₁ = 42, n₂ = 35

Solution:

Step 1:

Given data

α = 0.275

S₁ = 54

S₂ = 56

Step 2

Now we find the degree of freedom for n₁ and n₂

n₁ – 1 = 42 – 1 = 41

n₂ – 1 = 35 – 1 = 34

Step 3

By using the f distribution table and critical value calculator

F = (0.4357 & 2.5931)

A critical value calculator is an alternate way to solve the problems of finding critical values to get the results in a fraction of seconds.

Example 3

Find the critical value for left-tailed z test α=0.02

Solution:

Step 1:

First, we subtract α from 0.5

Thus

0.02 – 0.5 = 0.48

Step 2:

By using the z distribution table Z = (2.053)

So, for left tailed z test Z = (-2.053)

= 2.053

Wrap Up

In this article, we have discussed the intro of critical value, types of critical value T-value, Z-value value, F-value, and chi-square value, and read the use of these critical values in real life. In the example section, we have covered the method of finding the critical values.