Critical Value: Definition, Steps, Types, & Examples

Critical value refers to a concept commonly used in statistical hypothesis testing. It is a threshold or boundary value that helps determine the acceptance or rejection of a null hypothesis. In hypothesis testing, researchers set up a null hypothesis, which assumes that there is no significant difference or relationship between variables. The possibility of a significant relationship or difference is suggested by the alternate hypothesis.

In this article, we will discuss the critical value definition, critical value steps, and critical value application. In addition, the topic will be explained with the help of examples.

Critical Value

The critical value is a predetermined threshold used to determine the acceptance or rejection of a null hypothesis in statistical hypothesis testing. The critical value varies depending on the significance level chosen for the test, which is typically denoted as α (alpha).

Commonly used significance levels are 0.05 (5%) and 0.01 (1%). These values correspond to the probability of committing a Type I error, which is rejecting the null hypothesis when it is true. The critical value is derived from statistical tables specific to the chosen distribution, such as the t-distribution or the standard normal distribution (z-distribution).

Steps include finding Critical Value

For a confidence interval to find the critical value, we follow the following steps:

Step 1: Analyses the confidence interval’s level.

The confidence level indicates the desired level of certainty or probability related to the confidence interval. Some common confidence levels used to find the critical value are 99%, 95%, and 90%.

Step 2: Analyses the size of the sample and distribution.

The distribution of the test statistic and sample size have an impact on the critical value. The appropriate distribution to use depends on the sample size and the specific test being conducted.

For example, the normal distribution can be utilized if the sample size is big (usually greater than 30) and the population standard deviation is known. The t-distribution is employed when the sample size is limited or when the population standard deviation is unknown.

Step 3: Using distribution find Degrees of freedom.

You must establish the degrees of freedom if you’re utilizing the t-distribution. Here n shows the size of the sample. The degree of freedom depends on the size of the sample. We easily said that df depends on the size of the sample.

Step 4: Find desired critical value in the desired question.

Using a statistical table or a calculator with built-in functions, look up the critical value corresponding to the desired confidence level, degrees of freedom (if using the t-distribution), and the one- or two-tailed test being conducted.

Step 5: Critical value: putt

The critical value represents the distance from the mean or center of the distribution to the boundaries of the confidence interval. It determines how much variability or uncertainty is allowed in the interval.

The critical value is typically denoted as z× for the normal distribution or t× for the t-distribution.

Types of Critical Value:

Critical values are used in hypothesis testing to determine whether to reject or fail to reject a null hypothesis.

Some applications of critical are given below.

• Z-Tests:
  In hypothesis testing with a normal distribution and known population parameters. We utilize critical value if the sample mean significantly differs from a given value. The null hypothesis is rejected when test statics fall within the region of critical. The critical value is based on the intended level of significance (e.g., 0.05).
• T-Tests:
  When the population parameters are unknown and the sample size is small, t-tests are used. Critical values from the t-distribution are used to determine if the sample mean differs significantly from a specified value. The critical value is determined by the desired degree of significance, much like z-tests.
• Chi-Square Tests:
  Critical values from the chi-square distribution are used in tests of independence or goodness of fit. These tests assess whether observed frequencies differ significantly from expected frequencies. The critical value helps determine if the observed data falls within an acceptable range or deviates significantly from the expected distribution.
• F-Tests:
  F-tests are used in regression analysis or analysis of variance (ANOVA) to compare variances or check the significance of regression models. To assess whether the observed variation is statistically significant or whether the model adequately accounts for the data’s fluctuation, critical values from the F-distribution are used.

Confidence Intervals:

Critical values are also used in constructing confidence intervals. For example, when estimating a population mean or proportion, the critical value is used to determine the margin of error around the point estimate.
Example section
Example 1:
Find the critical value if a one-tailed t-test is being conducted on data with a sample size of 15 at a value α is 0.075.
Solution:
Given data
Sample size= 15
Α=0.075
Do we find the critical value=?
Step 1:
sample size (n)= 15
α (signific interval) = 0.075
Degree of freedom = 15 – 1 = 14
Step 2:
To use a statical table we find the critical value of given interval and degree.
Using a one-tailed table (0.075, 14) = 1.5236.

Therefore, the critical value for the given one-tailed t-test with a sample size of 15, df 14, and α = 0.075 is approximately 1.5236.
You can also try a t value calculator to find the t critical value in a fraction of a second.

Example 2:
If value α = 0.025
Variance = 120, Sample size = 31
Variance = 80, Sample size = 11
Critical value=?
Solution:
n1 = 31,
n2 = 11,
n1 – 1= 30,
n2 – 1 = 10,
Sample 1 df = 30
Sample 2 df = 10
To find the critical value, we use the F distribution table for α = 0.025
the value at the intersection of the 30th column and 10th row is
F (30, 10) = 3.3110
Answer: Critical Value = 3.3110

Conclusion

In this article, we have discussed the critical value definition, critical value steps, and critical value application. In addition, the topic will be explained with the help of examples. After completely understanding this article, anyone can defend this topic easily.