Let 𝑆^2_X and 𝑆^2_Y be the respective variances of two independent random samples of sizes 𝑛 and 𝑚 from 𝑁(𝜇_X, 𝜎^2_X) and 𝑁(𝜇 , 𝜎2). Use the fact that 𝐹 = [𝑆^2_X/𝜎^2_X]/[𝑆^2_Y/𝜎^2_Y] has an 𝐹 distribution, with parameters 𝑟_1 = 𝑚 − 1 and 𝑟_2 = 𝑛 − 1, we have 𝑃(𝑐 ≤ 𝐹 ≤ 𝑑) = 1 − 𝛼, where 𝑐 = 𝐹_(1-𝛼/2) (𝑟_1, 𝑟_2) and 𝑑 = 𝐹_(𝛼/2) (𝑟_1, 𝑟_2)

Derive the formula of the 100(1 − 𝛼)% two-sided confidence interval for 𝜎^2_X/𝜎^2_Y. Your answer should have exactly 4 paragraphs. Make sure that the first paragraph starts with the word "steps" and that each paragraph is separated by two new lines.
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