Critical value is a cut-off value used to mark the beginning of a region where the test statistic obtained in the theoretical test is unlikely to fail. Compared to the obtained test statistic to determine the critical value at hypothesis testing, Null hypothesis is rejected or not. Graphically, the critical value divides the graph into an accepted and rejected region for hypothesis testing. It helps to check the statistical significance of the test statistics. So, critical values are simply the function’s output at these critical points.

In this article, we will learn more about the critical value, its formula, types, and how to calculate its value.

What is Critical Value?

Critical values are essential components in hypothesis testing. They are calculated to help determine the significance of test statistics in relation to a specific hypothesis. The distribution of these test statistics guides the identification of critical values. In a one-tailed hypothesis test, there is one critical value, while in a two-tailed test, there are two critical values, each corresponding to a specific level of significance.

Critical Value Definition

Critical values are often defined as specific points on a scale used in statistical tests. These points help determine whether the results of a test are statistically significant or not. They serve as thresholds for making decisions about hypotheses being tested.

Critical Value Formula

There are different formulas for calculating the critical value, depending on the distributional nature of the test statistic. Confidence intervals or significance levels can be used to determine a critical value.

Critical Value Confidence Interval

Critical values play a crucial role in hypothesis testing, and they’re closely linked to confidence intervals. Let’s say we’ve set a 95% confidence interval for our test. To find the critical value:

• Step 1: Subtract the confidence level from 100% (100% – 95% = 5%.

• Step 2: Convert this to a decimal to get α (α = 5%).

• Step 3: If it’s a one-tailed test, α remains the same as in step 2. For a two-tailed test, divide α by 2.

• Step 4: Depending on the type of test, find the critical value in the distribution table using the α value.

T-Critical Value

T-test is used when the population trend is not observed and the sample size is less than 30. The t-test is conducted when the population data follow the Student t distribution. The t critical value can be calculated as follows.

Specify Alpha Level

• First, we set a level of confidence, which we call alpha (α). This is usually 0.05 or 0.01, but it can be different depending on the study.

• Next, we figure out the degrees of freedom (df). It’s just one less than the sample size. Degrees of freedom tell us how many values in the final calculation are free to vary.

• Now, we look at a table called the t-distribution table. If we’re doing a one-tailed test (meaning we’re only interested in one direction, like if something is greater or less than), we use a one-tailed t-distribution table. If it’s a two-tailed test (we’re interested in both directions), we use a two-tailed t-distribution table.

• In this table, we find the row that matches our degrees of freedom and the column that matches our alpha level. The number where they intersect is our t-critical value.

Test Statistic for One sample t-test: [Tex]t = \frac{\overline{x} – \mu}{s/\sqrt{n}}[/Tex]. [Tex]\overline{\rm x}[/Tex] is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the size of the sample.

Test Statistic for Two samples t-test: t = [Tex] t = \frac{(\overline{x}_1 – \overline{x}_2) – (\mu_1 – \mu_2)}{\sqrt{\frac{s^2_1}{n_1} + \frac{s^2_2}{n_2}}}[/Tex]

Decision criteria:

• Reject the null hypothesis if test statistic > critical t value (right-tailed hypothesis test).

• Reject the null hypothesis if test statistic < t critical value (left-tailed hypothesis test).

• Reject the null hypothesis if test statistic is not in the region of acceptance (two-tailed hypothesis test).

Z-Critical Value

A ‘Z test’ is performed on a normal distribution when the population mean is known and the sample size is greater than or equal to 30. The critical value of Z can be calculated as follows.

To Find the alpha value

• Subtract the alpha level from 1 for a two-tailed test. For a one-tailed test, subtract the alpha level from 0.5

• The area from the Z distribution table to obtain the z critical value. For a left-tailed test, a negative sign needs to be added to the critical value at the end of the calculation.

The Test statistic for Two sample z-test: [Tex]z = \frac{(\overline{x}_1 – \overline{x}_2) – (\mu_1 – \mu_2)}{\sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}}[/Tex]

Test statistic for One sample z-test: [Tex]z = \frac{\overline{x} – \mu}{\sigma/\sqrt{n}}[/Tex]. [Tex]\sigma[/Tex] is the population standard deviation.

F-Critical Value

The F test is commonly used to compare differences between two samples. The test statistic thus obtained is also used for regression analysis. The critical value of f is given as:

Find the alpha level

• Subtract 1 from the size of the original sample. This gives them the first freedom. Say, x.

• Similarly, subtract 1 from the second sample size to obtain the second df. Say, y.

• f Using the distribution x and row y, the distribution table will produce a critical value of f.

Test Statistic for large samples: f = [Tex]\sigma[/Tex]₁² / [Tex]\sigma[/Tex]₂². [Tex]\sigma[/Tex] ₁² variance of the first sample and [Tex]\sigma[/Tex] ₂² variance of the second sample.

Test Statistic for small samples: f = s₁² / s₂². S₁¹ variance of the first sample and S₂² variance of the second sample.

Chi-Square Critical Value

The chi-square test is used to check whether the sample data are consistent with the population data. It can also be used to compare two variables to see if they are correlated. The critical chi-square value is given as:

Select the alpha level

• Subtract 1 from the sample size to find the degree of freedom(df).

• Using the chi-square distribution table, the critical chi-square value is obtained by intersecting the df value in the row and column of the alpha value.

Chi-Square test statistic: [Tex]\Chi[/Tex]² = [Tex]\Sigma[/Tex] (O_i – E_i)² / E_i .

Also, Check:

• Variance and Standard Deviation
• Difference Between Variance and Standard Deviation
• Mean, Variance and Standard Deviation

Solved Questions on Critical Value

Question 1: Find the critical value for a right-tailed t-test with a sample size of 15 and α = 0.025.

Solution:

Given: n = 15, α = 0.025

Degrees of freedom (df) = n – 1 = 15 – 1 = 14

Using the t-distribution table for α = 0.025 and df = 14, the critical value is found to be t(14, 0.025) ≈ 2.145.

Critical Value = 2.145

Question 2: Calculate the critical value for a left-tailed z-test with α = 0.05.

Solution:

Given: α = 0.05

First, subtract α from 0.5: 0.5 – 0.05 = 0.45.

Using the z-distribution table, the value corresponding to the cumulative probability of 0.45 is approximately -1.645.

Critical Value = -1.645

Question 3: Determine the critical value for a two-tailed t-test with a sample size of 25 and α = 0.01.

Solution:

Given: n = 25, α = 0.01

Degrees of freedom (df) = n – 1 = 25 – 1 = 24

Since this is a two-tailed test, we need to split α in half: 0.01/2 = 0.005.

Using the t-distribution table for α = 0.005 and df = 24, the critical value is found to be t(24, 0.005) ≈ ±2.797.

Critical Values = ±2.797

Practice Problems on Critical Value

P1. A study examines if a new pain medication provides faster pain relief compared to a placebo. They use a one-tailed test with [Tex]\alpha[/Tex] = 0.05. Assuming a large enough sample size for normality, what is the critical value from the z-distribution?

P2. A social scientist wants to see if there’s a difference (either positive or negative) in life satisfaction scores between urban and rural residents. They conduct a survey and use a t-test with [Tex]\alpha[/Tex] = 0.1 and a combined sample size of 100 (50 urban, 50 rural). What are the degrees of freedom (df) and the critical t-value for a two-tailed test?

P3. Find the critical value for a chi-square test with df = 10 and [Tex]\alpha[/Tex] = 0.05

P4. Calculate the critical value for an F-test with df₁ = 3, df₂ = 20, and [Tex]\alpha[/Tex] = 0.10

Conclusion

The significance value is an important aspect of statistical hypothesis testing and is a requirement that the hypothesis based on sample data may not be rejected. When informed decisions are made, appropriate logic and the use of critical criteria are necessary to control Type I errors, ensure the validity of statistical inferences, facilitate comparison across studies, and increase both the rigor and reliability of statistical analysis in the evaluation and decision-making process.

Related Articles:

• Probability and Statistics
• Data Handling
• Chi-Square Test
• Hypothesis Testing Formula

FAQs on Critical Value

What is the most important advantage of hypothesis testing?

A critical value is a threshold used in statistical hypothesis tests to determine whether the null hypothesis should be rejected on the sample data. to decide on the null hypothesis, it is compared with the test statistic.

How is the importance of the goal related to its importance?

The critical value is determined by the chosen level of significance ([Tex]\alpha[/Tex]), which represents an acceptable probability of Type I error(false positive). Lower significance levels lead to more stringent critical values.

What are the differences between one-tailed and two-tailed critical values?

A single significant tree is used in trials in which the alternative hypothesis specifies a direction(greater than or less than), while a double significant tree is used when the alternative is scalar(not equal to).

How do degrees of freedom affect important values?

Significant differences in the t-distributions depend on the degree of freedom(sample size minus one).

What happens if the p-value is less than the mean?

A low p-value(less than alpha) indicates that the test statistic falls in the rejection region, resulting in the rejection hypothesis.

Why is it important to understand important values in hypothesis testing?

Important logic is important because it sets decision criteria for hypothesis testing, controls for type I errors, verifies statistical inferences, and guides decisions based on sample data.

Can important values be negative?

Depending on the classification and direction of the test(horizontal, vertical, two-tailed), the critical values can be negative or positive.