In the field of statistics, the concept of critical value keeps an exceptional position. It plays a key role in several analytical and scientific fields confidence intervals, hypothesis testing as well as in decision-making. The critical value is integral and helps in dealing with data analysis.

A test statistic is measured against a critical value while conducting a hypothesis test. We reject the null hypothesis if the test statistic is greater than the critical threshold. Critical value is a specific value that comes from a test statistic’s sample distribution, such as the t-distribution or the normal distribution, and it is employed to observe whether it is to accept or reject the null hypothesis (H₀).

In this article, we have elaborated on the important term critical value. We will present its definition, formula, and importance. We will also give some examples to grasp the concept of the critical value.

Definition of Critical Value:

In statistical hypothesis testing, the critical value is a threshold or boundary that assists in determining whether to accept or reject the null hypothesis. In a statistical test, the value distinguishes between the zone of acceptance and the region of rejection.

The desired significance level (α), which denotes the likelihood of making a Type I mistake (wrongly rejecting a valid null hypothesis), is what determines it. The significance level commonly indicated as (alpha), which represents the likelihood of committing a Type I mistake (erroneously rejecting a valid null hypothesis), is used to calculate critical values.

Formula for Critical Value:

The formula for calculating critical values depends on the statistical distribution and the desired significance level (α). Here are a few common distributions and their associated critical value formulas

Z-Distribution (Normal Distribution):

For a standard normal distribution, you can find critical values using the z-table or a calculator. The critical value is denoted as zα/2, where α/2 is half of the significance level. For example, if α = 0.05 (a common choice), then the critical value would be z0.025.

T-Distribution:

When working with small sample sizes or when the population standard deviation is unknown, the t-distribution is used. The significance level and degrees of freedom (df) are used to calculate the critical value. Utilizing statistical tables or software, you could find the t-critical values.

Chi-Square Distribution:

The number of degrees of freedom and the level of significance are what decide the chi-square test’s extremely important value. The formula is written asχ²α that stands for the significance level.

F-Distribution:

Critical values are determined using the F-distribution and are dependent on the degrees of freedom for the numerator and denominator as well as the significance level for ANOVA and other F-tests.

Role of Critical Values:

Critical values serve as decision points in hypothesis testing. When conducting a hypothesis test, you compare the test statistic (calculated from your sample data) to the critical value.

The critical value helps determine whether to reject the null hypothesis or fail to reject it. If the test statistic falls in the rejection region (beyond the critical value), you reject the null hypothesis; otherwise, you do not reject it.

Importance of Critical Values:

Statistical Significance: Critical values help establish statistical significance. They provide a clear threshold for determining whether the observed data is unlikely to occur by chance alone.

• Objective Decision-Making: Critical values offer an objective basis for making decisions in hypothesis testing. They help prevent subjective judgments and biases.
• Standardization: Critical values provide a standardized approach to hypothesis testing, making it possible for researchers and statisticians worldwide to use the same principles and criteria.
• Quality Control: In fields such as manufacturing and quality control, critical values are used to set tolerance limits, ensuring products meet certain standards.
• Scientific Research: Critical values play a crucial role in scientific research, allowing researchers to draw conclusions and make inferences based on data.

Examples:

Example 1:

Consider that a one-tailed t-test is conducted on data having:

Sample size = 15

α = 0.025, Find the critical value.

Solution:

Step 1: Given data

n = 15

α = 0.025

Degree of freedom (df) = n – 1

df = 15 – 1

df = 14

Step 2: Using t-distribution table,

So,

T(0.025, 140) = 2.145

Therefore, the critical value for the given one-tailed t-distribution is 2.145

A t table calculator is an online resource to find the t critical value of the given values according to the distribution table in couple of seconds.

Example 2:

Determine the critical value for a two-tailed f-test conducted on the following samples at a significance level (α) of 95%.

Variance = 120, sample size = 25

Variance = 80, sample size = 17

Solution:

Step 1: Given data

n₁ = 25, n₂ = 17

α = 0.05 (95% confidence level)

Step 2:

df₁ = n₁ – 1

df₁ = 25 – 1

df₁ = 24

df₂ = n₂ – 2

df₂ = 17 – 1

df₂ = 16

Step 3: We will observe the f distribution table for α = 0.05.

The value at the intersection of the 24^th column and the 16^th row.

So,

F(24, 16) = 2.24

Therefore,

Critical value = 2.24

Wrap Up:

We can summarize the whole article as the critical values are essential for statistical analysis and offer a distinct framework for evaluating hypotheses and making decisions. In this article, we have addressed an important term critical value, its definition, and its importance.