In hypothesis tests, critical values play a crucial role in making decisions about the null hypothesis. A critical value serves as a threshold for comparison with the computed test statistic. This comparison helps statisticians decide whether to reject or accept the null hypothesis.
The purpose of this article is to provide an overview of critical value definitions and their significance in hypothesis testing. We will learn how to calculate the critical value with the help of examples.
Definition of Critical Value
Critical values are thresholds or cutoff points that are used in hypothesis testing to decide whether to reject or fail to reject the null hypothesis. These values are compared with the calculated test statistic to make a decision.
Null hypotheses should not be rejected if the test statistic is less extreme than the critical value. The null hypothesis will be rejected when the test statistic value is more extreme than the critical value. Critical values depend on the significance level and the type of hypothesis test being conducted.
How to Calculate Critical Value?
You can use the confidence interval to find the critical value for a hypothesis test. Follow these steps to calculate the critical value:
- Calculate alpha by subtracting the confidence level from 100%.
- Convert the obtained value of alpha to a decimal.
- If you are conducting a one-tailed test, the alpha value remains the same. You would divide the alpha by 2 for a two-tailed test.
- Depending on whether it’s a one-tailed or two-tailed test, you can now look up the critical value in the appropriate distribution table using the α value you calculated in Step 3.
The upcoming sections will provide a detailed explanation of the precise steps required to locate and determine the critical value from the distribution table. You can also use a critical value calculator to get rid of manual calculations and the use of distribution tables, which can be time-consuming and prone to human error.
T-Critical Value
The t-critical value is a specific value taken from the t-distribution. It is used when the sample size is less than 30 and the population standard deviation is not given. The steps to find the t-critical value are as follows:
- Identify the significance level for your test.
- Calculate the degrees of freedom by subtracting one from the sample size.
Degree of freedom = df = n – 1
- Look up the degrees of freedom in the left column of the t-distribution table and the significance level in the top row.
- The critical value is located at the intersection of this particular row and column.
Formula to calculate test statistic for a one-sample t-test:
t = (x̄ – μ) / (s / √n)
Where:
- x̄ is the sample mean,
- μ is the population mean,
- s is the sample standard deviation,
- n is the sample size.
Test Statistics for a two-sample t-test:
t = (x̄1 – x̄2) – (μ1 – μ2) / √ ((s1² / n1) + (s2² / n2))
Decision Criteria:
You make decisions about the null hypothesis based on the calculated test statistic and the critical value from the t-distribution.
- For the right-tailed hypothesis test: Reject the null hypothesis if t-critical value < test statistics
- Left-tailed hypothesis test: Reject the null hypothesis if t-critical value > test statistics
- For Two-tailed hypothesis Tests: Reject the null hypothesis if the test statistic lies in the non-rejection region.
This criterion is used for all types of tests. Just the test statistic and Critical value will change.
Z-Critical Value
This value is used in statistics to determine the threshold beyond which an observation is considered unusual or statistically significant in a normal distribution. It can be obtained by following steps:
- Determine the significance level (α) for your test.
- Adjust the significance level based on the test type:
Adjusted significance level for two-Tailed Test = 1 – (α/2)
The adjusted significance level for the one-tailed Test = 1 – α
- Use a standard normal distribution table to find the cumulative probability (area) corresponding to the Z-critical value associated with the adjusted significance level.
- For a left-tailed test, add a negative sign to the Z-critical value before calculating the area.
For a one-sample z-test:
z = (x̄ – μ) / (σ / √n)
For a two-sample z-test:
z = ((x̄1 – x̄2) – (μ1 – μ2)) / √ ((σ1² / n1) + (σ2² / n2))
F-Critical Value
The F-critical value is a threshold value derived from the F-distribution that helps determine whether the variances being compared are significantly different or if the tested model has overall significance. The steps to find the F-critical value are as follows:
- Find out the level of significance.
- Determine the df for both the numerator and the denominator.
Degree of freedom for numerator = n1 – 1
Df for denominator = n2 – 1
- Locate the row that corresponds to the numerator degrees of freedom and the column that corresponds to the denominator degrees of freedom in the F-distribution table.
- You will find the F-critical value at the intersection of these rows and columns.
For large samples:
Test statistic = σ₁² / σ₂²
Where:
- σ₁² is the variance of the first sample
- σ₂² is the variance of the second sample
For small samples:
Test statistic = s₁² / s₂²
Where:
- s₁² is the squared variance of the first sample
- s₂² is the squared variance of the second sample
Solved Example to Find Critical Value Using Table
Example:
Find the one-tailed t-test with a significance level of 0.025 and a sample size of 11.
Solution:
Given:
α = 0.025
Sample size = 11
Degree of freedom = df = 11 – 1 = 10
Look up 0.025 at the top row and 10 in the very left column of the one-tailed t-test.
Alpha (0.025) and degree of freedom (10) intersect at 2.228. Thus 2.228 is our critical value for the one-tailed test.
Final Words
In this article, we explored the significance of critical values in hypothesis testing. We outlined the calculation of critical values using confidence intervals and focused on t-critical values for smaller sample sizes. We also explained z-critical values in normal distributions and F-critical values for variance comparisons, providing the formulas for each.