Homework Assignment #1

1. Let Y ₁ Y ₂ Y ₃ and Y ₄ be independent, identically distributed random variables from a population with mean µ and variance ∂ ² . Let Ybar = ¼( Y ₁ + Y ₂ +Y ₃ + Y ₄ ) denote the average of these four random variables.
   1. What are the expected value and variance of Ybar in terms of mean and variance?
   2. Now, consider a different estimator of µ: W = 1/8 Y ₁ + 1/8 Y ₂ +1/4 Y ₃ +1/2 Y ₄ This is an example of a weighted average of the Y _i . Show that W is also an unbiased estimator of µ. Find the variance of W.
   3. Based on your answers to parts a and b, which estimator of µ do you prefer, Ybar or W?
2. The data set BWGHT.RAW contains data on births to women in the United States. Two variables of interest are the dependant variable, infant birth weight in ounces ( bwght ) and an explanatory variable, average number of cigarettes the mother smoked per day during pregnancy (cigs). The following simple regression was estimated using data on n = 1388 births: bwght hat = 119.77 - .514 cigs
   1. What is the predicted birth weight when cigs = 0? What about when cigs = 20 (one pack per day)? Comment on the difference.
   2. Does this simple regression necessarily capture a causal relationship between the child’s birth weight and the mother’s smoking habits? Explain.
   3. To predict a birth weight of 125 ounces, what would cigs have to be? Comment.
   4. What would the coefficients be if birth weight is measured in grams (hint: 1 ounce=28.35 gram).

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