mtermw04_soln - EC120B Winter 2004 Midterm Solution Prof....

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Unformatted text preview: EC120B Winter 2004 Midterm Solution Prof. Berman (20) 1. A. Given a random sample of 40 observations of variables Y and X from some population, provide a formula to estimate 1, the slope of the population linear regression line. ^ Estimate for 1: 1 = Sxy/Sxx= (Xi- X )(Yi- Y )/ (Xi- X )2 B. Write a formula to estimate a 95% confidence interval for 1. ^ ^ 95% confidence interval for 1: 1 1.96 *se( 1) C. Now assume that 140 of your classmates sample from that population and estimate a confidence interval as in part A. Let z be the number of confidence intervals (between 0 and 140) that do not contain 1. Estimate the expected value of z, E(z)? Using the CLT to approximate, prob(interval does not contain 1) 0.05 for each classmate. E(z) 140(0.05)=7 (10 Bonus) D. What's the variance of z, V(z)? The variance of a binary variable is p(1-p), where p is the probability of a "1." The variance of the sum of n independent binary variables is just n times the variance of each. V(z) np(1-p)=140(0.05)(1-0.05)=6.65 1 (40) 2. For each statement below, indicate if it is true or false and write a sentence justifying your answer. A. If X1, ... XN are a random sample from a population with E(X) = : then (Xi- )=0 True [ ] / False [X] Explain: (Xi- X )=0 But X in a finite sample So (Xi- ) 0 B. In random sampling from a population with expectation E(Y) = :, (Yi- Y )2 is greater than (Yi- )2 since the mean of Y is an imprecise estimate of the population mean :. True [ ] / False [X ]. Explain: Y is the least squares estimator. It minimizes (Yi- Y )2 as you showed in problem set #1. Thus, (Yi- Y )2 is always less than or equal to (Yi- )2 C. Assume that E(Y|X) = " + X. If X is randomly assigned we can interpret as the effect of X on Y. True [X ] / False [ ]. Explain: There is no omitted variable bias if X is randomly assigned, so is the effect of X on Y. D. Assume that E(Y|X) = " + X. If X is not randomly assigned there is no useful interpretation of . True [ ] / False [X]. Explain: The regression coefficients are still useful at least for prediction even if they don't inform us about the effect of X on Y. This is the case for most economic data, as we don't have randomly assigned X variables. For example, you could guess a child's age by observing their height or shoe size, even though neither of those actually cause age. 2 (10) 3. Assume that E(Y|X) = 0 + 1X. We draw a random sample of size N, construct least squares estimators b0 and b1 , correctly construct a 95% confidence interval for 1 and correctly test the hypothesis that 1 =0. Choose the correct answer. a) Confidence intervals for 1 will tend to shrink as N grows larger. b) Confidence intervals for 1 will tend to be more accurate as N grows larger. c) The power of the hypothesis test 1 =0 with a 5% chance of rejection if true will tend to increase as N grows larger. d) The level of a hypothesis test 1 =0 with a 5% chance of rejection if true will remain constant as N grows larger. e) All of the above. f) None of the above. ^ a is correct: Notice that the formula for se( 1) has the sample size in the denominator, so the standard error shrinks with N, which implies narrower confidence intervals with higher N. b is correct because CLT is more relevant as N grows, making the normal distribution a better approximation of the exact sampling distribution at any particular sample size. c is correct because power is the probability of rejecting the null hypothesis it is false. As N grows, the confidence interval shrinks so the rejection regions grow. Thus the probability of rejecting a false hypothesis increases. d is correct because the level of the test is 5% and it remains constant by definition as N grows. e is the correct answer. Nobody chose f! (30) 4. What does the Central Limit Theorem say? The CLT says that under general conditions the distribution of Y is well-approximated by a normal distribution when the sample size is large and that approximation improves as the sample size grows. How have we used it so far in this course? We have used the CLT to build confidence intervals for , o and 1. ^ ^ ^ For example, 1 N(1,V(1)) for large samples so 1 N( 1,V( 1)), which we have used to construct a confidence interval for 1. The same argument applies to confidence intervals for other regression coefficients, and for the population mean, . 3 - Survey on Sleep, Study and Caffeine Lisa Ton How many hours did you sleep in the last 24 hours? How many hours do you sleep in a typical night? How many hours did you study for this class in the last 24 hours? How many hours work did you do for this class in the last week? How many of the 9 classes so far have you attended? Did you study for this exam alone / mostly alone / mostly in a group ? ___7___ hours __7.5__ hours __4____ hours __19___ hours ___9___classes (choose one). How many milligrams of caffeine have you had in the last 24 hours? __30___ mg. (A can of soda ~ 70 milligrams) (1 cup of coffee ~ 100 milligrams) How many milligrams of caffeine do you consume in a typical day? How well do you think you did on this exam? (Circle one) 1 2 Very poorly 3 4 5 6 7 8 9 10 Very well __20___ mg. How tired were you during the exam? (Circle one) 1 Very tired 2 3 4 5 6 7 8 9 N/A 10 Very awake What is your current G.P.A.? Is there anything else you would like to share about your study habits? (Please comment below) I prefer studying with friends and food. This survey is completely anonymous and confidential. Professor Berman will not see any of your responses to the survey and your name will remain anonymous throughout the process of the study. Thank you for your participation! Survey number _______ 4 Results: . summ mt30 Variable | Obs Mean Std. Dev -----------+-----------------------------------mt30 | 133 74.76692 15.11616 Midterm Grade 120 Predicted Midterm Grade 100 80 60 40 5 10 15 Problem Set #1 Grade 20 . gen ps1sq=ps110^2 (13 missing values generated) . reg mt ps110 ps1sq , robust Regression with robust standard errors Number of obs = 122 F( 2, 119) = 7.83 Prob > F = 0.0006 R-squared = 0.0774 Root MSE = 14.508 -----------------------------------------------------------------------------| Robust mt30 | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------ps110 | 6.729824 2.143841 3.14 0.002 2.484806 10.97484 ps1sq | -.2133915 .0769797 -2.77 0.006 -.365819 -.060964 _cons | 25.85432 13.98274 1.85 0.067 -1.832903 53.54155 ------------------------------------------------------------------------------ 5 ...
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This note was uploaded on 08/09/2009 for the course ECON 120B taught by Professor Jeon during the Spring '08 term at UCSD.

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