My nephew was born last summer. He has 19 cousins on his father's side (it's a...

  

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My nephew was born last summer. He has 19 cousins on his father's side (it's a big family). I wish to know the mean, μ, of the distribution of the ages of my nephew's cousins (which is the variable X). I take a sample of 4, with ages X₁, X₂, X3, and X₁. These are all drawn from the same underlying population. Instead of calculating the sample mean of these four, I do the following calculation to create an estimator of μ, which I call X*. X* = 0.1(X₁) + 0.2(X₂) + 0.3(X3) +0.4(X4) 3a. 3b. 3c. 3d. 3e. Explain what it means to say that X* is unbiased? Show that X* is unbiased. If the variance of X is 15, what is the variance of X ? If the variance of X is 15, what is the variance of the X? Which estimator is more efficient, the sample mean X or X*? Explain. My nephew was born last summer. He has 19 cousins on his father's side (it's a big family). I wish to know the mean, μ, of the distribution of the ages of my nephew's cousins (which is the variable X). I take a sample of 4, with ages X₁, X₂, X3, and X₁. These are all drawn from the same underlying population. Instead of calculating the sample mean of these four, I do the following calculation to create an estimator of μ, which I call X*. X* = 0.1(X₁) + 0.2(X₂) + 0.3(X3) +0.4(X4) 3a. 3b. 3c. 3d. 3e. Explain what it means to say that X* is unbiased? Show that X* is unbiased. If the variance of X is 15, what is the variance of X ? If the variance of X is 15, what is the variance of the X? Which estimator is more efficient, the sample mean X or X*? Explain.