Critical Value

A critical value is a number used in hypothesis testing to decide whether to accept or reject a result. It is found by looking at the distribution of the test statistic and the chosen significance level. If the test is one-tailed, it has one critical value, but if it is two-tailed, it has two critical values. These critical values help define the boundaries where results are considered unusual or unlikely under the null hypothesis. So, depending on the test type and significance level, the critical value(s) separate expected results from surprising ones.

A critical value is a number used in hypothesis testing to help decide whether to reject the null hypothesis. It is compared with the test statistic obtained from the data. If the test statistic is less extreme than the critical value, we keep the null hypothesis. But if the test statistic is more extreme than the critical value, we reject the null hypothesis and accept the alternative hypothesis.

The critical value splits the distribution into two parts: the acceptance region and the rejection region. If the test statistic falls inside the rejection region, the null hypothesis is rejected. Otherwise, it is not rejected. This helps us understand if the results are unusual enough to doubt the initial assumption.

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. 


\(5\% = \frac{5}{100} = 0.05\)


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

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: 

\(t = \frac{(x̄ - μ)}{(s / √n)}\)

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: 
 

\(t = \frac{[(\bar{X}_1 - \bar{X}_2)-(μ_1 - μ_2)]}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}\)
 

Here, it is the test statistic 

\(x̄_1\) and \(x̄_2\) are the sample means of group 1 and group 2.

\(μ_1\) and \(μ_2\) are the population means.

\((S_1)^2\) and \((S_2)^2\) are the sample variances.

\(n_1\) and \(n_2\) are the sample sizes.

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: 

\(z = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}} \)

Here,  σ is the population standard deviation. 

The test statistic for two samples, z-test:

\(z = \frac{ \left( \bar{X}_1 - \bar{X}_2 \right) - \left( \mu_1 - \mu_2 \right) } { \sqrt{ \frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2} } } \)

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 (\(df_1\)) of the first sample, subtract 1 from the size of the first sample.

To get the degree of freedom (\(df_2\)) of the second sample, subtract 1 from the size of the second sample.

Find the value where the \(df1\) column and the \(df_2\) row intersect. That will be the F critical value. 

The test statistic for large samples: 

\(f = \frac{(σ_1)^2} {(σ_2)^2}\)

Here, \((σ_1)^2\) and \((σ_2)^2\) are the variances of two samples.

The test statistic for small samples: 

\(f =\frac{ (s_1)^2 }{ (s_2)^2}\)

Here, \((s_1)^2\) and \((s_2)^2\) are the variances of the two samples. 

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: 

\(x^2 = \frac{Σ (O - E)^2}{E}\)

Here, O is the observed frequency

E is the expected frequency.

For a right-tailed z test, we have to calculate the critical value for a 0.0079 alpha level. First, we need to subtract the alpha level from the given value. For example, let us take 0.5.

Therefore, we get 

Now, let us use the z distribution table to find the area closest to 0.4921.

The nearest area in the table is 0.4922.

The intersection of this value is 2.4 and 0.2, and thus the critical value is 2.42.

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Here is an example of the table:

While working on a topic like critical value, some tips and tricks can be used to enhance the problem-solving process. In this section, we will learn about such tips and tricks.
 

• Understand that the critical value marks the boundary in hypothesis testing where you decide whether to reject or accept the null hypothesis.
   
• Learn the common significance levels (like 0.05 or 0.01) and how they relate to confidence levels, such as 95% or 99%.
   
• Practice using Z-tables and t-tables to find critical values, depending on whether the population standard deviation is known or not.
   
• Visualize the concept using a normal distribution curve, this helps you see where the critical region lies.
   
• Work on plenty of examples involving both one-tailed and two-tailed tests to gain confidence and accuracy.
   

• Teachers can start teaching about the meaning before teaching the formula. Learning the meaning before the formula helps them in understanding whether something is unusual or likely.
   

• Children can easily understand new concepts with the help of some image explanations. Teachers can use the blackboard to draw a bell curve to highlight the rejected region by shading it. We can then ask the students to mark the critical value with a bold line.
   

• Parents can make use of some of the real-life situations to make their children understand the critical value. We can ask them to measure the jump length variation with their friend to understand who is exceptionally fair, or toss coins to see who is fair.
   

• Children can also understand this by doing activities such as rolling dice to predict what counts as too many sixes and flipping coins 50 times to check if results fall inside or outside a chosen critical range.

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. 
   
• For sports analytics, coaches use statistical tests with critical values to check if training methods truly improve performance.