Critical value is a term used in mathematics to describe a value that separates the region of acceptance and the region of rejection in a hypothesis test. In statistical hypothesis testing, critical values are compared to test statistics to determine whether to reject the null hypothesis or fail to reject it.
These critical values are important because they can provide information about the behavior of the function, such as its local minima or maxima. Critical values are necessary in many areas of mathematics, including calculus, optimization, and differential equations.
Critical values are derived from probability distributions and are dependent on the level of significance chosen for the hypothesis test. They are often used in fields such as economics, social sciences, and engineering, where statistical analysis plays a crucial role in decision-making processes.
In this article, we will discuss the definition of Critical value, importance of Critical value, types of critical value and also an example of the topic will be explained.
Definition of Critical Value
Critical value is a value or a set of values that separate the acceptance region from the rejection region in a statistical hypothesis test. In other words, critical values are the boundary points that are used to determine whether to reject the null hypothesis or fail to reject it.
In mathematics, critical values can also refer to points where the behavior of a function changes or where the derivative of a function equals zero.
Importance of Critical value
Here are some important notes on critical value:
- Critical values are used in hypothesis testing to determine the rejection region for a test statistic. The rejection region is the area of the distribution that corresponds to the chosen level of significance (alpha) for the test.
- The critical value is based on the probability distribution of the test statistic and the level of significance chosen for the test. As such, it varies depending on the type of test statistic being used (e.g., Z-score, t-value) and the specific level of significance chosen (e.g., 0.05, 0.01).
- The critical value is used to determine whether to reject the null hypothesis or fail to reject it. If the statistic test falls in the rejection region reject the null hypothesis is in favor of the another hypothesis. If the test falls outside the rejection region, the null hypothesis cannot be rejected.
- The critical value is important because it helps ensure that hypothesis testing is conducted in a standardized and objective manner. By using a predetermined level of significance and critical value, researchers can avoid subjective judgments and make clear and replicable decisions about whether to reject the null hypothesis.
Overall, the critical value is a key component of hypothesis testing and is essential for making objective and replicable decisions about whether to reject the null hypothesis.
Types of Critical value
The type of critical value used depends on the type of hypothesis test being conducted and the level of significance chosen.
- Z critical value:
The Z critical value is used in hypothesis testing involving the standard normal distribution. It is denoted as Zα/2, where α is the level of significance and Zα/2 is the Z score that corresponds to the upper tail area of α/2.
- T critical value:
The T critical value is used in hypothesis testing involving the t-distribution. It is denoted as tα/2, where α is the level of significance and t α/2 is the t-value that corresponds to the upper tail area of α/2 and n-1 degrees of freedom.
- F critical value:
The F critical value is used in hypothesis testing involving the F-distribution. It is denoted as Fα, df1, df2, where α is the level of significance, df1 and df2 are the degrees of freedom for the numerator and denominator, respectively, and Fα, df1, df2 is the F-value that corresponds to the upper tail area of α.
- Q critical value:
The Q critical value is used in hypothesis testing involving the Q-distribution. It is denoted as Qα, df1, df2, where α is the level of significance, df1 and df2 are the degrees of freedom for the numerator and denominator, respectively, and Qα, df1, df2 is the Q-value that corresponds to the upper tail area of α.
The above types are frequently used to calculate critical value of t, z, and f test by taking the results from the distribution tables
Examples of Critical Value
Here is an example to understand how to calculate critical value.
Example:
Find the critical value for a two-tailed f test conducted on the following samples at a α = 0.05
Variance = 130, Sample size = 31
Variance =100, Sample size = 41
Solution:
To find the critical value for a two-tailed f test, we need to first calculate the F-statistic and degrees of freedom.
Step 1:
The F-statistic is calculated by dividing the larger sample variance by the smaller sample variance. In this case, the larger sample variance is 130 and the smaller sample variance is 100, so:
F = 130/100 = 1.3
The degrees of freedom for the numerator is the sample size of the smaller variance minus one, and the degrees of freedom for the denominator is the sample size of the larger variance minus one.
Step 2:
df numerator = 41 - 1 = 40
df denominator = 31 - 1 = 30
Step 3:
Level of significance of α = 0.05, we can find the critical values for the F-distribution using a statistical table or calculator. For a two-tailed test, we need to find the values that split the α level into two equal parts, one in each tail.
In critical values for the F-distribution the degree of freedom in the numerator is 40 and denominator 30 in the at α = 0.05 are approximately 1.94 and 2.54, respectively.
Therefore, the critical values for the two-tailed F-test with α = 0.05, variance = 130, sample size = 31, and variance = 100, sample size = 41 are 1.94 and 2.54.
If the calculated F-statistic is greater than 2.54 or less than 1.94, null hypothesis reject.
Conclusion
In this article, we have discussed the definition Critical value, importance of critical value, types of critical value, and also with the help of an example topic will be explained. After studying this article everyone can defend this topic.