3. (28p) You are given data on a simple random sample of two opposite-sex couples. That...

   

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3. (28p) You are given data on a simple random sample of two opposite-sex couples. That is, we have randomly sampled two couples from the population of opposite-sex couples. Let the heights of the two men be M, M2; and let the heights of the two women be W, W2. The population distribution of men's heights has mean and variance . The population distribution of women's heights has mean w and variance w. You want to estimate the average height of people in opposite-sex couples. (a) (6p) What is the population mean height () of people in opposite-sex couples? (b) (6p) Consider the sample average F = 1- (M + M + W + W2). Show that this estimator is unbiased. 4 (c) (8p) Suppose that couples form assortatively on height so that within couples, the covariance between the heights of the man and the woman, cov(M, W), is 0.5. What is the sample variance of H? (Assume that the cross-moments between non-paired men and women are zero, that is, the covariances between any two individuals not in the same couple are zero) (d) (8p) [requires calculus] Suppose instead that couples form randomly so that the covariance between the heights of the man and the woman is zero. Suppose also that you know o and . You may now want to consider a sample average that puts different weights on the men and the women in the sample as an estimator. What is the sample weighted average estimator that has the smallest sample variance? [NB: A weighted average is a weighted sum of the observations such that the sum of the weights is equal to 1, so your estimator is = YMM+YM2 + YwW + YwW2 where 2M+2Yw = 1] 3. (28p) You are given data on a simple random sample of two opposite-sex couples. That is, we have randomly sampled two couples from the population of opposite-sex couples. Let the heights of the two men be M, M2; and let the heights of the two women be W, W2. The population distribution of men's heights has mean and variance . The population distribution of women's heights has mean w and variance w. You want to estimate the average height of people in opposite-sex couples. (a) (6p) What is the population mean height () of people in opposite-sex couples? (b) (6p) Consider the sample average F = 1- (M + M + W + W2). Show that this estimator is unbiased. 4 (c) (8p) Suppose that couples form assortatively on height so that within couples, the covariance between the heights of the man and the woman, cov(M, W), is 0.5. What is the sample variance of H? (Assume that the cross-moments between non-paired men and women are zero, that is, the covariances between any two individuals not in the same couple are zero) (d) (8p) [requires calculus] Suppose instead that couples form randomly so that the covariance between the heights of the man and the woman is zero. Suppose also that you know o and . You may now want to consider a sample average that puts different weights on the men and the women in the sample as an estimator. What is the sample weighted average estimator that has the smallest sample variance? [NB: A weighted average is a weighted sum of the observations such that the sum of the weights is equal to 1, so your estimator is = YMM+YM2 + YwW + YwW2 where 2M+2Yw = 1]