In statistics, a critical value is a threshold or boundary point used in hypothesis testing and constructing confidence intervals. It is a specific value derived from the sampling distribution of a test statistic, such as the **t-distribution** or the **normal distribution**, and is used to determine whether to reject or fail to reject the null hypothesis.

In this article, we will discuss the definition of critical value, the Formula, and the importance of critical value. Also, with the help of examples,the topic will be explained easily. After completely understanding this article, anyone can defend this article.

## Critical Value in Statistics

The critical value in statistics is those value when we compared to a test statistic in hypothesis testing to find out the null hypothesis are rejected are not. The null hypothesis is a statement that assumes there is no significant difference or relationship between variables, while the alternative hypothesis contradicts the null hypothesis by asserting that there is a significant difference or relationship.

Hypothesis testing involves collecting sample data and comparing it to a critical value to make an inference about the population.

Critical value critical value in sctatistics formula depends on the test static distribution so, there are different kind of formula which is used to find the critical value. One of the most common critical formulas is to be used which is an interval of confidence or level of significance that can be used to find the critical value.

## Importance of Critical Value

**Hypothesis Testing:**

Critical values play a central role in hypothesis testing. They help determine whether the observed data provide sufficient evidence to favor the alternative hypothesis inrejecting the null hypothesis.

By comparing the computed test statistic to the critical value, statisticians can make informed decisions about the validity of their hypotheses and draw meaningful conclusions from their analyses.

**Confidence Intervals:**

Critical values are used to construct confidence intervals, which provide a range of plausible values for a population parameter. The critical value determines the width of the interval and reflects the desired level of confidence.

Confidence intervals are valuable in statistical inference as they help quantify the uncertainty associated with sample estimates and provide a measure of the precision of the estimated parameter.

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**Significance Level:**

The critical value is chosen based on the desired significance level, and it establishes the threshold for determining statistical significance. A lower significance level corresponds to a more stringent test, requiring stronger evidence to reject the null hypothesis.

**Decision-making:**

Critical values provide a clear and objective criterion for decision-making in statistical analysis. By comparing the test statistic to the critical value, statisticians can determine whether to reject or fail to reject the null hypothesis.

This helps researchers and decision-makers make informed choices based on the evidence available from the data.

**Standardization:**

Critical value in statistics are often used to standardize test statistics, transforming them into a common scale that facilitates comparison and inference across different datasets and distributions.

This standardization enables researchers to draw conclusions based on a set of predefined critical values rather than relying on specific parameter values or sample characteristics.

**Reproducibility and Consistency:**

Critical values provide a standardized and consistent approach to statistical analysis. By adhering to predefined critical values and significance levels, researchers can ensure that their findings are replicable and comparable across studies. This consistency promotes transparency and allows for the objective evaluation of results by other researchers.

## How to find critical value in statistics?

**Example 1:**

Let’s say we have a sample of 80 observations, and we want to determine whether the sample mean differs from the population mean by a significant amount at a 90% level of confidence.

**Solution:**

Step 1:**Set up hypotheses:**

H_{0}: μ = μ_{0} (population mean)

H_{a}: μ ≠ μ_{0} (sample mean is different from population mean)

Step 2:**Significance level:**

Significance level = 1 – 0.90 = 0.10

because the confidence level is 90%.

The two-tailed significance level is 0.10 divided by 2 for each tail of the distribution, resulting in α/2 = 0.05 for each tail.

Step 3: **Critical value:**

To find the critical value, we need to determine the z-value corresponding to the significance level and the two tails.

Since the sample size is relatively large (n = 80) and assuming the population standard deviation is unknown, we can use the standard normal distribution (z-distribution).

Using statistical software or a standard normal distribution table,we find that the critical z-value for a significance level of 0.05 (two-tailed) is approximately **±1.645.**

Step 4:**Make a decision:**

In this step, we compare the calculated test statistic (z-value) with the critical value to make a decision.

If the calculated test statistic falls outside the range of **-1.645 to +1.645**, we reject the null hypothesis and conclude that the sample mean is significantly different from the population mean at a 90% confidence level.

**Example 2:**

Suppose a one-tailed t-test is being conducted on data with a sample size of 12 at

α = 0.025. Critical value?

**Solution:**

To find the critical value for a one-tailed t-test here, with a sample size of 12 and α = 0.025, we need to consult the statistical software or t-distribution table.

Step 1:

sample size (n)= 12

α (signific interval) = 0.025

Degree of freedom= 12 – 1 = 11

Step 2:

To solve this, we use t distribution table of statistical software

Using a one-tailed table (0.025, 11) = 2.718.

Therefore, the critical value for the given one-tailed t-test with a sample size of 12 and α = 0.025 is approximately **2.718.**

## Conclusion:

In this article, we have discussed the definition of critical value, the Formula, and the importance of critical value. Also, with the help of examples, the topic will be explained easily. After studying this article anyone can defend this topic easily.