midterm_th_sol - MS&E 226 Small Data Solutions for...

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MS&E 226 Solutions for Take-Home Midterm Examination “Small” Data PROBLEM 1. A medical researcher wants to investigate the efficacy of a nutritional supplement designed to help athletes gain muscle. The researcher recruits athletes i = 1 , .., n (where n is even), none of whom are taking the supplement at the beginning of the experiment. Once the experiment starts, some athletes start taking the supplement, while others do not. If athlete i takes the supplement, this is denoted by X i = 1 ; and if athlete i does not take the supplement, this is denoted by X i = 0 . For each athlete i , the researcher measures the change in the lean body mass of the athlete from the beginning of the experiment to 3 months later; this difference is denoted by Y i . The researcher also knows that muscle gain depends on the the number of hours the athlete exercises per week, which we denote by Z i for athlete i . Indeed, in the population, given X and Z , the corresponding Y is distributed as: Y = β 0 + β 1 X + β 2 Z + ε, (1) where ε is normal (0 , 1) , and independent across athletes. Unfortunately, the Z i ’s are not available to the researcher. Her primary goal is to establish whether an athlete will gain more muscle on average if she uses the supplement, even if she does not change her exercise routine. a) Interpret the coefficient β 1 , and explain why this is the quantity of interest to the researcher. Since the researcher does not have the Z i ’s, she decides to go ahead with a simple regression of Y on X , with an intercept term; i.e., she estimates the linear model: Y i ˆ β 0 + ˆ β 1 X i , using ordinary least squares (OLS). Our goal is to develop some understanding of whether this is a good idea. Before you begin the next part in R, set the seed of the random number generator with the following code: set.seed(1) b) Assume that β 0 = 0 , β 1 = 1 , β 2 = 1 . First, assume the Z i are i.i.d. normal (5 , 1) random variables; and also assume that whenever Z i > 5 , then X i = 1 ; and when Z i < 5 , then X i = 0 . Repeat the following 1000 times: Draw n = 500 random samples of Z , X , and Y according to the preceding description. Run OLS of Y against X , and record ˆ β 1 .
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Make a histogram of the resulting 1000 values of ˆ β 1 . Now, again assume that the Z i are i.i.d. normal (5 , 1) random variables; but instead assume that the X i are i.i.d. Bernoulli random variables with P ( X i = 1) = P ( X i = 0) = 1 / 2 . ( Hint : The rbinom function can be used to generate Bernoulli random variables in R; in particular rbinom(n,1,p) generates a vector of n Bernoulli random variables with probability p of success on each.) Repeat the following 1000 times: Draw n = 500 random samples of Z , X , and Y according to the preceding description. Run OLS of Y against X , and record ˆ β 1 .
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