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Understanding Critical Values in Statistics: Significance, Steps, and Calculation

A critical value in statistics is a specific point on a statistical distribution that is used as a reference to determine the acceptance or rejection of a null hypothesis during hypothesis testing. It serves as a cutoff point beyond which a statistical test will lead to rejecting the null hypothesis in favor of an alternative hypothesis.

Critical values are determined based on the chosen level of significance (mostly signifies as alpha, α) for a given statistical test. They are typically obtained from probability distributions like the t-distribution, chi-square distribution, or F-distribution.

This article will explain the following fundamental terms of the critical value:

  • Critical value: What is?
  • Steps to calculate the critical value
  • Critical Values in F-Distribution
  • Critical Values in Chi-Square Distribution
  • Solved Problems of Critical Value.

Let’s explore all the terms and gain more insight into the critical value.

Critical Value

The critical value in statistics refers to a specific point on a statistical distribution that sets the boundary for decision-making during hypothesis testing. It is a determined threshold used to determine whether to accept or reject the null hypothesis.

If the test statistic value lands inside the rejection region, the null hypothesis gets rejected; if not, it remains unelected. Critical values can relate to either a one-tailed (left or right tail) or a two-tailed test. For a two-tailed test, two critical values are needed, while a one-tailed test requires just one critical value.

These values are derived from probability distributions like the t-distribution, chi-square distribution, or F-distribution and depend upon the chosen level of significance (often denoted as alpha, α).

Steps to Calculate Critical Value:

Determining critical value depends on the distribution of the test statistic. It may involve using the confidence interval or the significance level (alpha) as part of the calculation process.

Step 1: Determine the significance level by converting the confidence interval into alpha (α): α = 1 – (confidence level divided / 100).

Step 2: Differentiate between a one-tailed and a two-tailed test as critical values and distribution tables vary for each type. In a one-tailed test, the alpha value remains unchanged. However, for a two-tailed test, the alpha value is divided by 2.

Step 3: Critical values are specific to individual tests. Reference a distribution table to find the critical value corresponding to a test according to its alpha value.

Further details regarding the Step 3 process will be provided in the following section.

Critical Values in F-Distribution:

The F-distribution is used when comparing variances of two independent sample populations. Critical values in the F-distribution correspond to different levels of significance and degrees of freedom for the numerator and denominator in the F-test. This can be calculated by the following process:

Step 1: Determine the α

Step 2: Compute the 1st degree of freedom with the subtraction of one from the initial sample size. Say A

Step 3: Compute the 2nd df by deducting one from the 2nd sample size. Say B

Step 4: Consulting the F-distribution table involves finding the critical value by locating the point where the A column and B row intersect.

Decision criteria:

  • If test Statistics (Right-tailed) > Reject F critical value.
  • If test Statistics (Left tailed) < Reject F-critical value.
  • Reject the null hypothesis in a Two-Tailed Test if the test statistic exceeds the upper or lower critical values.

Critical values in Chi-Square distribution:

The chi-square test serves to evaluate whether sample data aligns with population data and can also evaluate the relationship between two variables. The critical value for chi-square is provided as follows:

  1. Determine the α.
  2. Compute the Df by deducting 1 from the sample size n.
  3. Consulting the chi-square distribution table involves finding the chi-square critical value by locating the point where the row corresponding to df intersects with the column representing the chosen alpha value.

Key Points Regarding Critical Values:

  • A critical value is used as a benchmark to evaluate the validity of rejecting the null hypothesis based on a comparison with the test statistics.
  • It marks the boundary distinguishing between the acceptance and rejection regions within the distribution graph.
  • Critical values come in four types: z, f, chi-square, and t.

Problems of Critical Values:

These examples demonstrate how we calculate the critical value.

Problem 1:
Compute the critical F-value for this scenario:

α = 0.01

Sample 1:

  • Variance = σ1 = 20
  • Sample size = 21

Sample 2:

  • Variance = σ2 = 10
  • Sample size = 16

Solution:

Let’s compute the F-critical value:

Step 1: Degrees of Freedom Calculation:
For Sample 1:

Sample size−1 = 21−1 = 20

For Sample 2:
df2 ​= Sample size − 1= 16 – 1 = 15

Step 2: Critical F-value Determination:

Using the F distribution table for α = 0.01, the value at the intersection of the 20th column and 15th row is 3.37.

Step 3: Decision making

f = σ12 / σ22 = 4

The calculated test statistic f = 4 exceeds the critical F-value of 3.373, we reject the null hypothesis. This suggests that there is a significant difference in the variances between Sample 1 and Sample 2 at a significance level of α=0.01.

Wrap up:

This article explored critical values in statistics, essential in hypothesis testing decisions. Derived from distributions like t, chi-square, or F-distributions, these values help accept or reject null hypotheses. By determining significance levels, we differentiate between one-tailed and two-tailed tests, using critical values as thresholds.

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