Critical Value

The rejection region of a hypothesis test is defined by a number known as the critical value. This value changes according to the type of hypothesis test and the data. A critical value is used to compare the hypothesis testing with a test statistic. Based on the critical value, we can decide whether to reject the null hypothesis.

We do not reject the null hypothesis if the test statistic is less extreme than the critical value. If the test statistic goes beyond the critical value, we reject the null hypothesis and support the alternative hypothesis. A critical value divides a distribution graph in two regions; acceptance and rejection regions. The null hypothesis becomes invalid if the test statistic falls in the rejection region. Otherwise, it becomes valid, if the statistics fall in the acceptance region.

The formula for the critical value depends on the test statistic distribution type. If the test is one-tailed, it has only one critical value. A two-tailed test has two critical values. To find the critical value, we use the confidence interval or significance level, with different formulas for different distributions. 

Critical value confidence interval: By using the confidence interval, we can calculate the critical value of one-tailed and two-tailed tests. For example, if the confidence interval of a hypothesis test is 95%. It means that we are 95% sure that the value will lie within the range. Also, there is a 5% chance of error. Now the steps to find the critical value are:

Step 1: Subtract the confidence interval from 100%. Here, 95% is the confidence level. When we subtract the confidence interval from 100%, we will get a significance level, denoted as α, alpha. 100% - 95% = 5%

Step 2: Convert the percentages to decimal form. Here, we get 5% and convert it into decimal form. 
5% = 5 /100 = 0.05. 
So, α = 0.05

Step 3: In the one-tailed test, α stays the same. In the two-tailed test, the alpha level (α) is divided by 2. 

Step 4: Find the critical value. Depending on the test type, we can use the alpha value (α) to look it up in a corresponding distribution table.
The different test types are:

• T Critical value 
• Z Critical value
• F Critical value
• Chi-Square Critical value

T Critical value 

When the population standard deviation is unknown, a t-test is used. Also, the size of the sample is less than 30. If the population data follows a student’s t-distribution, a t-test is applicable. To calculate the t critical value, we have to follow certain steps. They are:

Step 1: Find out the alpha level.

Step 2: To figure out the degrees of freedom (df), subtract 1 from the sample size. 

Step 3: For the one-tailed test, we can use the one-tailed t-distribution table. Likewise, for the two-tailed test, use the two-tailed t-distribution table. 

Step 4: From the left side of the table, identify the df value and alpha value from the top row. The number at the intersection of the row and column is the critical value. 

The test statistic for one sample t-test: 
t = x̄ - μs / √n 
Here, x̄ is the sample mean
Μ is the population mean
s is the sample standard deviation
n is the size of the sample

The test statistic for two samples t-test: 
t = (x̄1 - x̄2) - (μ1 - μ2) / (√s21 / n1) + (s22 / n2)

Here, it is the test statistic 
x̄1  and x̄2 are the sample means of Group 1 and Group 2
μ1 and μ2 are the population means 
S21 and S22 are the sample variances 
n1 and n2 are the sample sizes 

Z Critical value

When the sample size is greater than or equal to 30 and if the population standard deviation is known, then we can implement the z test. It is used for a normal distribution. In the calculation process of z critical value, first, we need to figure out the alpha level. 

Next, for a two-tailed test, subtract the alpha level from 1. For a one-tailed test, we should subtract the alpha level from 0.5.

Based on the alpha value, use the z distribution table to find the area that corresponds to z critical value. For a left-tailed test, we add a negative sign to the critical value. 

The test statistic for one sample z-test: 

z = (x̄ - μ) / (σ / √n)

Here,  σ is the population standard deviation. 

The test statistic for two samples, z-test:

z = (x̄1 - x̄2) - (μ1 - μ2) / (√σ21 / n1) + (σ22 / n2)

F Critical value

To compare the variances of two samples, we use the F test. The test statistic is also useful in regression analysis. To find the F critical value, we need to follow these steps:
Figure out the alpha level.
To get the degree of freedom (df1) of the first sample, subtract 1 from the size of the first sample.
To get the degree of freedom (df2) of the second sample, subtract 1 from the size of the second sample.
Find the value where the df1 column and the df2 row intersect. That will be the F critical value. 

The test statistic for large samples: 

f = σ21 / σ22

Here, σ21 and σ22 are the variances of two samples.

The test statistic for small samples: 

f = s21 / s22

Here, s21 and s22 are the variances of the two samples. 

Chi-Square Critical Value

We use the chi-square test to check whether a sample represents the population data accurately. It compares two connected variables and is used to determine test results. The chi-square value can be calculated by following these steps:

Find the alpha level.
To decide the degree of freedom(df) subtract 1 from the sample size.
The intersection point of the df row and the column of the alpha value gives the chi-square critical value. 

The test statistic for chi-squared test statistic: 

x2 = Σ (O - E)2 / E
Here, O is the observed frequency
E is the expected frequency. 

The concept of critical value is useful in various real-life scenarios, where decisions are made up on facts rather than guesses. Critical value plays a vital role in situations where need to make data-based choices. 

• In the field of medicine and research, professionals can make decisions according to the critical value. If the test statistic is greater than the critical value, then the result is good to go. Otherwise, the research result is not good enough. 

• The weather forecasters can use the critical value in their job field. It helps them to warn the public if the weather pattern crosses the critical value. 

• The concept of critical value is helpful for the business and marketing industries. They can evaluate the sales, profit, or performance of their products by the critical value. If the product’s data is greater than the critical value, it makes a profit to the company. 

• Educational institutions and academies can assess their student’s performance by calculating the critical value. If the scores of the students are beyond the expected value, they are performing very well.