Hypothesis testing is a statistical technique that helps in making decisions about a population by analyzing sample data. Hypothesis testing involves formulating a null and alternative hypothesis, collecting sample data, calculating a test statistic, and then comparing the test statistics to a critical value.
The critical value is a crucial component in hypothesis testing, as it helps to determine whether the null hypothesis should be rejected or not. In this article, we are going examine the critical value with its types. We will learn how it is calculated and how it is used in hypothesis testing.
Critical value: Definition
A critical value is a specific value used in statistical hypothesis testing that helps determine whether to reject or accept a null hypothesis. It is calculated based on the significance level (alpha level) and the degree of freedom. The comparison of the critical value with the calculated test statistics derived from the sample data is a crucial step in statistical analysis.
In statistical hypothesis testing, when the value of the test statistic exceeds the critical value, it leads to the rejection of the null hypothesis and the acceptance of the alternative hypothesis. If the calculated value of the test statistics is equal to or smaller than the critical value, the null hypothesis is accepted.
How is the Critical Value calculated?
To find the critical value in statistical hypothesis testing, follow the following steps:
- Determine the significance level (alpha level) by subtracting the confidence level from 100%.
- Change the significance level to a decimal point.
- Calculate the degree of freedom (df) for your test. This value depends on the sample size and the number of estimated parameters.
- Look up the critical value in the statistical table, such as a t-distribution table or chi-square distribution table. The table will provide the critical value that corresponds to your significance level and degrees of freedom.
Types of critical value
Different types of critical values depend on the statistical test and level of significance (alpha). The most common types are:
- Z- Critical value
- T- Critical value
- F- Critical value
- Chi-square critical value
Let’s discuss the above types of critical values one by one.
Z- Critical value
The z-critical value is used in hypothesis testing when the sample size (n) is greater or equal to 30, and the population standard deviation is known
Method to find Z-critical value:
Determine the significance level (alpha). For a one-tailed test, subtract alpha from 0.5; for a two-tailed test, subtract alpha from 1. Next, use a z-table to find the critical value. In the case of a left-tailed test, write a negative sign at the end of the calculation.
However, we can use a z-critical value by using the following formula:
T- Critical value
The T-critical value is used in hypothesis testing when the sample size (n) is less than 30, and the population standard deviation is unknown.
Method to find T-critical value:
To calculate the T-critical value, the first step is determining the significance level (α). Next, subtract 1 from the sample size (n) to obtain the degree of freedom. Then, refer to the t-distribution table and find the corresponding value by locating the degree of freedom on the left side column and the level of significance on the top row. The intersection of the row and column will give the T-critical value.
Alternatively, a t value calculator can be a helpful resource to find the t critical value according to the t distribution tables.
F- Critical value
The F-critical value is used in hypothesis testing when comparing the variance of two samples. It is the ratio of the variances of two samples and is used to determine whether they are significantly different.
Method to find F-critical value:
To find the f-critical value, start by determining the level of significance. Then, subtract 1 from the first sample size to get the first degree of freedom and label it as x. i.e. x = n1 -1. Subtract 1 from the second sample size to get the second degree of freedom and label it as y. i.e. y = n2 -1. Next, please look at the f-distribution table and locate the corresponding value where the x in the column and y in the row intersect. This value will be f-critical value.
Chi-square critical value
The chi-square critical value is used in hypothesis testing to determine the goodness of fit or independence between two categorical variables. It measures the difference between observed and expected values.
Method to find Chi-square critical value:
Determine the level of significance (alpha). To determine the degree of freedom, subtract one from the sample size, represented as df = n – 1. Refer to the chi-square distribution table and locate the point where the alpha in the column and degree of freedom in the row intersect, and this point is our chi-square critical value.
Solved example of critical value
Example:
Determine the t-critical value for a one-tailed test with a confidence level of 99% and the sample size is 5.
Solution:
Step 1: Determine the level of significance (alpha) by subtracting the level of confidence from 100% i.e.
Alpha = 100% – 99% = 1%
Step 2: Covert it into a decimal point.
Alpha = α = 1 / 100 = 0.01
Step 3: Calculate the degree of freedom (df) by subtracting 1 from the sample size (n)
df = 5 – 1 = 4
Step 4: Refer to the t-distribution table and find the corresponding value by locating the degree of freedom (that is 4 in the given example) on the left side column and the level of significance (0.01) on the top row. The intersection of the row and column will give the T-critical value.
| df | α = 0.10 | α = 0.05 | α = 0.025 | α = 0.01 |
| 1 | 3.0780 | 6.3140 | 12.7100 | 31.8200 |
| 2 | 1.8860 | 2.9200 | 4.3030 | 6.9650 |
| 3 | 1.6380 | 2.3530 | 3.1820 | 4.5410 |
| 4 | 1.5330 | 2.1320 | 2.7760 | 3.7470 |
| 5 | 1.4760 | 2.0150 | 2.5710 | 3.3650 |
| 6 | 1.4400 | 1.9430 | 2.4470 | 3.1430 |
| 7 | 1.4150 | 1.8950 | 2.3650 | 2.9980 |
| 8 | 1.3970 | 1.8600 | 2.3060 | 2.8960 |
Therefore, the degree of freedom (df) and alpha (α) intersect at 3.7470, so 3.7470 is the t-critical value.
Conclusion
In this article, we have provided a comprehensive explanation of critical values. We have covered the methods for determining critical values, the various types of critical values, and their appropriate usage. Lastly, we have provided a solved example of critical value with a step-by-step solution. By understanding this article, you will be able to determine any critical value easily.
