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Two statistical tests that students often get mixed up are the F-Test and the T-Test. This tutorial explains the difference between the two tests.

F-Test: The Basics

An F-test is used to test whether two population variances are equal. The null and alternative hypotheses for the test are as follows:

H₀: σ₁² = σ₂² (the population variances are equal)

H₁: σ₁² ≠ σ₂² (the population variances are not equal)

The F test statistic is calculated as s₁² / s₂².

If the p-value of the test statistic is less than some significance level (common choices are 0.10, 0.05, and 0.01), then the null hypothesis is rejected.

Example: F-Test for Equal Variances

A researcher wants to know if the variance in height between two species of plants is the same. To test this, she collects a random sample of 20 plants from each population and calculates the sample variance for each sample.

The F test statistic turns out to be 4.38712 and the corresponding p-value is 0.0191. Since this p-value is less than .05, she rejects the null hypothesis of the F-Test. This means she has sufficient evidence to say that the variance in height between the two plant species is not equal.

T-Test: The Basics

A two sample t-test is used to test whether or not the means of two populations are equal.

A two-sample t-test always uses the following null hypothesis:

• H₀: μ₁ = μ₂ (the two population means are equal)

The alternative hypothesis can be either two-tailed, left-tailed, or right-tailed:

• H₁ (two-tailed): μ₁ ≠ μ₂ (the two population means are not equal)
• H₁ (left-tailed): μ₁ < μ₂ (population 1 mean is less than population 2 mean)
• H₁ (right-tailed): μ₁> μ₂ (population 1 mean is greater than population 2 mean)

The test statistic is calculated as:

where x₁ and x₂ are the sample means, n₁ and n₂ are the sample sizes, and where s_p is calculated as:

s_p = √ (n₁-1)s₁² +  (n₂-1)s₂² /  (n₁+n₂-2)

where s₁² and s₂² are the sample variances.

If the p-value that corresponds to the test statistic t with (n₁+n₂-1) degrees of freedom is less than your chosen significance level (common choices are 0.10, 0.05, and 0.01) then you can reject the null hypothesis.

Example: Two Sample t-test

A researcher wants to know if the mean height between two species of plants is equal. To test this, she collects a random sample of 20 plants from each population and calculates the sample mean for each sample.

The t test statistic turns out to be 1.251 and the corresponding p-value is 0.2148. Since this p-value is not less than .05, she fails to reject the null hypothesis of the T-Test. This means she does not have sufficient evidence to say that the mean heights between these two plant species is different.

F-Test vs. T-Test: When to Use Each

We typically use an F-test to answer the following questions:

• Do two samples come from populations with equal variances?
• Does a new treatment or process reduce the variability of some current treatment or process?

And we typically use a T-test to answer the following questions:

• Are two population means equal? (We use a two sample t-test to answer this)
• Is one population mean equal to a certain value? (We use a one sample t-test to answer this)

Additional Resources

Introduction to Hypothesis Testing
One Sample t-test Calculator
Two Sample t-test Calculator