realMT2008solns - EC 420, Spring 2008 Midterm Show all...

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EC 420, Spring 2008 Midterm Show all necessary work. You will have the entire class period to complete this exam. This exam is “open book” but “closed note”; you may use the course text (Wooldridge), but notes, old homework questions and answers, etc., are strictly forbidden. Part One (short answer and proof) : 1. What is a sampling distribution? What does the Central Limit Theorem state, and why is it so awesome (important for us statisticians)? (3 points) A sampling distribution describes how a sample statistic (such as the sample mean, the sample regression slope, the sample regression intercept, etc) varies from one sample to the next. The Central Limit Theorem is indeed awesome – it says that the sampling distribution of the sample mean and sample regression “betas” are all approximately normally distributed, as long as the sample size is “large enough” (n>20 is a typical rule of thumb). It is important because it means that we can compute the probability of observing a particular sample statistics if we knew the population parameter – this is fundamental for computing p-values and doing hypothesis tests. 2. What are the similarities between the slope of a simple regression line, the correlation between two variables, and the r-squared of a simple regression model? What are the differences? (3 points) If the r-squared equals zero, so does the correlation and the slope coefficient. The correlation and the slope coefficient always are the same sign, and the r-squared just equals the square of the correlation between x and y . The differences are that, while the slope tells us how steep a regression line is, it does not really tell us how well the line fits the data…the correlation and the r-squared do that. 3. Consider the following data for ( x , y ): (1,2), (4,5), (1,4), (3,5) and (4,7). I run a regression of y on x , and I find that 1 ˆ β = 1 and 0 ˆ = 2. (3 points) a. Graph the underlying data. b. Graph the predicted values for each observation. c. Graph the regression line. d. Indicate the residuals for each observation.
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4. Provide proofs for the following. Provide an explicit description of each of the steps you take. (1 point each) a. Show that 0 ) ( 1 = - = X X n i i . 0 ) ( 1 1 1 = - = - = - = = = X n X n X X X X n i n i i n i i Where the third equals sign comes from the definition of X and the fact that the sum of a constant (like X ) is just n times that constant. b. Using the result from part a, show that 0 ) ( 1 = - = X X X n i i OK, this one is really easy:
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0 0 ) ( ) ( 1 1 = = - = - = = X X X X X X X n i i n i i
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Part Two : Many people think that the quality of college that one attends will affect one’s yearly salary later in life. You wish to investigate this idea. You have access to a data set that includes information from 1000 college graduates on their annual salary at age 40 (“ salary ”) and a measure of the quality of the college they attended (“ quality ”). This measure is the college’s score from the U.S. News “America’s Best Colleges” survey,
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This note was uploaded on 03/30/2008 for the course ECON 420 taught by Professor Toddelder during the Spring '08 term at Michigan State University.

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realMT2008solns - EC 420, Spring 2008 Midterm Show all...

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