🙅🏼‍♂️ F Test

• In agricultural experiments the performance of a treatment is assessed not only by its mean but also by its variability. Hence, it is of interest to us to compare the variability of two populations.
• In testing of hypothesis, the equality of variances, the greater variance is always placed in the numerator and smaller variance is placed in the denominator.
• F-test is used to test the equality of two population variances, equality of several regression coefficients, ANOVA.
• ‘F’ test can be used for testing the significance of several differences.
• F-test was discovered by G.W. Snedecor.
• The range of F is 0 to ∞.

Process

• Let x₁, x₂, ……, xn₁ and y₁, y₂,……. , yn₂ be the two independent random samples of sizes n₁ and n₁ drawn from two normal populations N(μ₁, σ₁²) and N(μ₂, σ₂²) respectively. S₁² and S₂² are the sample variances of the two samples.
• Null hypothesis H₀: σ₁² = σ₂²
• Under H₀, the test statistic becomes

  

  

• Where, S₁² > S₂² (remember V shape)
• Which follows F-distribution with (n₁ - 1, n₂- 1) d.f.
• If calculated value of F < table value of F with (n₂-1, n₁-1) d.f. at specified level of significance, then the null hypothesis is accepted and hence we conclude that the variances of the populations are homogeneous otherwise heterogeneous.

Example

• The heights in meters of red gram plants with two types of irrigation in two fields are as follows:

• Test whether the variances of the two system of irrigation are homogeneous.

Solution:

• H₀: The variances of the two systems of irrigation are homogeneous.
• i.e. σ₁² = σ₂²
• Under H₀, the test statistic becomes

  

• Where, S₁² > S₂²

  

  

  

• F calculated value = 1.78
• Table value of F_0.05 for (n₁ - 1, n₂ - 1) d.f. = 3.23
• Calculated value of F < Table value of at 5% level of significance, H₀ is accepted and hence we conclude that the variances of the two systems of irrigation are homogeneous.