The concept of critical value is fundamental in statistics and hypothesis testing. When conducting statistical tests, researchers aim to make inferences about a population based on a sample of data. The critical value plays a crucial role in determining whether the observed results are statistically significant or simply due to random variation.

The critical value corresponds to a specific point or points on a probability distribution. The choice of distribution depends on factors such as the type of test being conducted (e.g., Z-test, t-test) and the assumptions about the population being tested

In this article we will discuss the critical value definition, Formula, what is critical value, Criteria, and calculation of critical value also with the topic explained with the help of examples.

## What is Critical value?

In statistics, a critical value is a specific value or set of values that separates the critical region from the non-critical region in a hypothesis test. The critical region consists of the values of a test statistic that would lead to the rejection of the null hypothesis, while the non-critical region consists of the values that would not result in the rejection.

The significance level represents the probability of making a **Type I** error, which is the incorrect rejection of the null hypothesis when it is true in most cases, the levels of significance are 0.01 (1%) and 0.05 (5%), however depending on the application, alternative values may be chosen.

The choice of distribution depends on factors such as the sample size, assumptions about the population, and the type of tests like the Z-test, t-test, and chi-square test. Common distributions used include the standard normal distribution (Z-distribution), t-distribution, chi-square distribution, and F-distribution.

## Criteria for Critical Value

The critical value in statistics possesses several key characteristics:

**Threshold**: The critical value serves as a threshold or cutoff point that determines whether the test statistic falls within the critical region or the non-critical region. It divides the probability distribution into these two regions, guiding the decision-making process in hypothesis testing.**Significance Level**: The critical value is determined based on the desired significance level (alpha) for the test.**Distribution-Based**: The critical value is derived from a specific probability distribution that is appropriate for the statistical test being conducted. The choice of distribution depends on factors such as the sample size, assumptions about the population, and the type of test. Common distributions used include the standard normal distribution (Z-distribution), t-distribution, chi-square distribution, and F-distribution.**Tail(s):**Depending on how the hypothesis test is structured, the crucial value can be either one- or two-sided. In a one-sided test, the critical value is located in one tail of the probability distribution, representing the extreme values in only one direction. The critical value for a two-sided test is found in the tails of the distribution, which indicate extreme values in either direction.**Rejection Rule:**The critical value, along with the observed test statistic, helps determine whether to reject or fail to reject the null hypothesis. If the observed test statistic exceeds (or falls below, depending on the test) the critical value, it falls within the critical region, leading to the rejection of the null hypothesis.

## Calculation of Critical value

The calculation of critical values depends on the specific statistical test being conducted and the probability distribution associated with that test. Here are some common examples of how critical values are calculated for different tests:

**Z-Test:**The z critical value a Z-test can be determined by using the standard normal distribution (Z-distribution).

Critical value = Zα/2× σ standard deviation

two-sided test

**T-Test:**The critical value for a t-test is determined using the t-distribution. The formula to calculate the critical value depends on the type of t-test:**One-Sample t-Test:**The critical value is calculated as tα × (s / √n), where tα represents the t-score corresponding to the desired significance level α, s is the sample standard deviation, and n is the sample size.**Independent Samples t-Test:**The critical value is obtained based on the degrees of freedom (df) associated with the test. The critical value is determined by finding the t-score corresponding to the desired significance level α and the appropriate degrees of freedom.**Paired Samples t-Test:**The critical value is calculated based on the degrees of freedom (df) associated with the test. The critical value is determined by finding the t-score corresponding to the desired significance level α and the appropriate degrees of freedom.**Chi-Square Test:**The critical value for a chi-square test is determined using the chi-square distribution. The formula to calculate the critical value is:

Critical value = χ²α, df

Where χ²α, df represents the chi-square value corresponding to the desired significance level α and the degrees of freedom (df) associated with the test.

A critical value calculator is a handy tool to find the critical values of t, z, and chi-square distributions in a fraction of a second.

## Examples of Critical Values

**Example 1:**

Suppose we want to conduct a two-tailed hypothesis test using a Z-test with a significance level of 0.05 (5%).

**Solution:**

**Step 1**:

**Determine the significance level (α/2):**

Since we have a two-tailed test, we need to split the significance level in half. Therefore, α/2 = 0.05/2 = 0.025.

**Step 2:**

** Find the critical value corresponding to the significance level:**

We must use the usual normal distribution table or utilize statistical tools to determine the crucial value. From the standard normal distribution table, the Z-score corresponding to a cumulative probability of 0.975 is approximately 1.96.

**Step 3: **

**Calculate the critical value:**

The negative of the positive critical value is the critical value for a two-tailed test. For this case -1.96 is the critical value.

**Example 2:**

Determine T- critical value if n is 8 and α=0.025.

**Solution:**

To determine the T-critical value with a sample size (n) of 8 and a significance level (α) of 0.025, follow these step-by-step calculations:

**Step 1: calculate (df):**

df = n – 1

df = 8 – 1 = 7

**Step 2: Determine the significance level for each tail:**

Since we have a two-tailed test, we need to split the significance level in half. Therefore, α/2 = 0.025/2 = 0.0125.

**Step 3:** Find the critical t-value that corresponds to the degrees of freedom as well as the significance level.

We must consult the t-distribution table or utilize statistical tools to determine the essential t-value. In each tail, we want the t-value that reflects a cumulative probability of 0.0125.

**Step 4: Evaluate critical t-value:**

The crucial t-value for a two-tailed test with seven degrees of freedom and a significance threshold of 0.0125 can be found using the t-distribution table or software is approximately ±2.998.

## Conclusion

In this article we have discussed the critical value definition, Formula, what is critical value, Criteria, and calculation of critical value also with the topic explained with the help of examples. After studying this article anyone can defend this topic easily.

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