Statistical Methods

Unit 8 - F-Distribution

8.4.Testing for the equality of variances

• H₀ : σ₁²= σ₂² (No significant difference between Variances)
• H1: σ₁²≠ σ₂² (Significant difference between Variances)
• Fix α =0.05 Say
• Test statistic: F= s₁²/ s₂² where s₁² is the larger sample variance
• Decision Rule: Reject H₀ (i.e. No significant difference between Variances) if computed test statistic value is more than table F value for (n₁-1); (n₂-1) degrees of freedom otherwise accept H₀.

Example 1
A random sample of sample of 25 mackerels taken gave standard deviation of length 3.8 cm, while a random sample of 35 oil sardine taken gave standard deviation of length 4.5 cm.Does this support equality of length variances of both the species at 5% of significance?
Solution
F test statistic is used to test for equality of variances.
Hence we have to convert the given values standard deviation into variances. Here we have larger variance is for oil sardine. Hence take s₁= 4.5, n₁=35, s₂= 3.8, n₂=25.

• H₀: σ₁²= σ₂² (No significant difference between Variances)
• H₁: σ₁²≠ σ₂² (Significant difference between Variances)
• Fix α =0.05 Say
• Test statistic: F = s₁²/ s₂² where s₁² is the larger sample variance

= 4.5²/3.8²

F_(cal) = 1.40

F_(tab) at (n₁-1, n₂-1)= (34,24)df at 5%=1.79 (approximately)

• Decision Rule: Since F_(cal)=1.40< F_(tab)=1.79, Null hypothesis is accepted.