CI - 2 Independent Samples - Variance & Standard Deviation

given 2 independent samples:

• 𝑿 = {𝑋₁, …, 𝑋_𝑛} from Normal(𝜇_𝑋, 𝜎_𝑋)
• 𝒀 = {𝑌₁, …, 𝑌_𝑚} from Normal(𝜇_𝑌, 𝜎_𝑌)

the standardized ratio of variances:

• 𝐹 = [(𝑠_𝑋)²/ (𝜎_𝑋)²] / [(𝑠_𝑌)² / (𝜎_𝑌)²]
• 𝐹 = [∑_{1≤𝑖≤𝑛}[𝑋_𝑖 - 𝑋̅] / (𝜎_𝑋)² / (𝑛 - 1)] / [∑_{1≤𝑖≤𝑚}[𝑌_𝑖 - 𝑌̅] / (𝜎_𝑌)² / (𝑚 - 1)]

has F-distribution with with (𝑛 − 1) and (𝑚 − 1) degrees of freedom

how it was computed

Unlike population means and proportions, variances are scale factors, and they are compared through their ratio:

• 𝜃 = (𝜎_𝑋)² / (𝜎_𝑌)²

an estimator of 𝜃 is the ratio of sample variances:

• 𝜃ˆ = (𝑠_𝑋)² / (𝑠_𝑌)²
  • (𝑠_𝑋)² = ∑_{1≤𝑖≤𝑛}[𝑋_𝑖 - 𝑋̅] / (𝑛 - 1)
  • (𝑠_𝑌)² = ∑_{1≤𝑖≤𝑚}[𝑌_𝑖 - 𝑌̅] / (𝑚 - 1)
• 𝜃ˆ = [∑_{1≤𝑖≤𝑛}[𝑋_𝑖 - 𝑋̅] / (𝑛 - 1)] / [∑_{1≤𝑖≤𝑚}[𝑌_𝑖 - 𝑌̅] / (𝑚 - 1)] # 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 1

Its standard form, after we divide each sample variance in 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 1 by the corresponding population variance, is therefore called the Fisher-Snedecor distribution or simply F-distribution with (𝑛 − 1) and (𝑚 − 1) degrees of freedom:

• 𝐹 = [∑_{1≤𝑖≤𝑛}[𝑋_𝑖 - 𝑋̅] / (𝜎_𝑋)² / (𝑛 - 1)] / [∑_{1≤𝑖≤𝑚}[𝑌_𝑖 - 𝑌̅] / (𝜎_𝑌)² / (𝑚 - 1)]

Example F-Distribution

Confidence Interval - Ratio of Population Variances

where:

• 𝐹_𝛼(𝑖, 𝑗) - is the f-distribution with degrees 𝑖 and 𝑗

how confidence interval was constructed

Click here to expand...

start with estimator 𝜃ˆ:

• 𝜃ˆ = (𝑠_𝑋)² / (𝑠_𝑌)²

standardizing it to:

• 𝐹 = [(𝑠_𝑋)²/ (𝜎_𝑋)²] / [(𝑠_𝑌)² / (𝜎_𝑌)²]
• 𝐹 = [(𝑠_𝑋)²/ (𝑠_𝑌)²] / [(𝜎_𝑋)²/ (𝜎_𝑌)²]
• 𝐹 = 𝜃ˆ / 𝜃

we get an F-variable with (𝑛-1) and (𝑚-1) degrees of freedom. Therefore:

Solving the double inequality for the unknown parameter 𝜃, we get:

therefore:

Confidence Interval Without Reciprocal

is a (1 - 𝛼)100% confidence interval for 𝜃 = (𝜎_𝑋)²/ (𝜎_𝑌)²

Reciprocal Property of F-Distribution

𝐹_1-𝛼(𝑣₁, 𝑣₂) = 1 / 𝐹_𝛼(𝑣₂, 𝑣₁)

Confidence Interval With Reciprocal