# hw3 - Homework 3 Solutions 7.4 a X Y is an unbiased...

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Homework 3 Solutions 7.4. a. X - Y is an unbiased estimator for μ 1 - μ 2 since E ( X - Y ) = μ 1 - μ 2 . To see this, E ( X - Y ) = E ( X ) - E ( Y ) = 1 m m X i =1 E ( X i ) - 1 n n X i =1 E ( Y i ) = μ 1 - μ 2 For the observed data, the estimate of μ 1 - μ 2 is x - y = 121 . 44 - 113 . 80 = 7 . 64. b. By a proposition for linear combinations of independent random variables, we have V ( X - Y ) = V ( X ) + V ( Y ) = 1 m 2 m X i =1 V ( X i ) + 1 n 2 n X i =1 V ( Y i ) = σ 2 1 m + σ 2 2 n (1) The standard error is just the square-root of the variance: p σ 2 1 /m + σ 2 2 /n . To obtain an estimator, plug in the estimators S 2 1 and S 2 2 for σ 2 1 and σ 2 2 , respectively. The estimate is r s 2 1 m + s 2 2 n = r 12 . 29 2 18 + 11 . 28 2 15 = 4 . 11 c. Although they are biased, the usual estimators of σ 1 and σ 2 are S 1 and S 2 , respectively. From this, an estimator of σ 1 2 is S 1 /S 2 . The estimate (from the data) is s 1 s 2 = 12 . 29 11 . 28 = 1 . 09 d. If only one male and one female are sampled, then result (1) above with

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hw3 - Homework 3 Solutions 7.4 a X Y is an unbiased...

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