Suppose that independent samples of sizes n1 and n2 are taken from two normally

Chapter 8, Problem 8.125

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Suppose that independent samples of sizes n1 and n2 are taken from two normally distributed populations with variances 2 1 and 2 2 , respectively. If S2 1 and S2 2 denote the respective sample variances, Theorem 7.3 implies that (n1 1)S2 1 /2 1 and (n2 1)S2 2 /2 2 have 2 distributions with n1 1 and n2 1 df, respectively. Further, these 2-distributed random variables are independent because the samples were independently taken. a Use these quantities to construct a random variable that has an F distribution with n1 1 numerator degrees of freedom and n2 1 denominator degrees of freedom. b Use the F-distributed quantity from part (a) as a pivotal quantity, and derive a formula for a 100(1 )% confidence interval for 2 2 /2 1 .