Econ140Final_Soln_Fall2008 - Econ 140 Fall 2008 FINAL EXAM...

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Econ 140 Department of Economics Fall 2008 Prof. Armando Levy FINAL EXAM You have 180 minutes to complete the exam. There are 6 questions worth 135 total points. The problems are not ranked by difficulty. Please write your answers on the exam itself and show all work. Good luck. 1. Suppose you are playing Craps in Las Vegas, and have just rolled an 8 on your first roll (thereby setting the point). To determine your “passline” bet you will continue to roll the pair of dice until you either duplicate your roll on 8 (and you win) or you roll a 7 (and lose). Let X be the random variable that is the number of rolls you will make after this first roll of 8 to determine the outcome. Assume the dice are fair and rolls are independent of each other. a. (5 points) What is the probability that X=1 ? 5/36 + 6/36 = 11/36 b. (3 points) What is the probability that X>1 ? 1-P(X=1) =25/36 c. (2 points) What is the probability that X=2 conditional on X>1 ? 11/36 d. (5 points) Sketch the cdf of X for X <= 4 . Page 1 of 13
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Econ 140 Department of Economics Fall 2008 Prof. Armando Levy 2. More on dice: Suppose we want to test whether a particular die is “loaded”—i.e. not equally likely to show any of the six outcomes (you may remember this was an issue in “Oceans 13”). We roll the die n times and record the number of times we rolled 1, 2,. .,6 denoted n 1 ,…, n 6 . a. (5 points) One thing we could do is calculate the sample expectation of rolling k as n k/ n and test whether it’s significantly different than 1/6 at the 5% significance level. What is the test statistic you would construct? ) ) ( / )]( 6 / 1 ( ) / [( ) ˆ ( / ) 6 / 1 ˆ ( / ) )( / ( / ) ˆ 1 ( ˆ ) ˆ ( , / ˆ 5 . 1 2 k k k k k k k k k k k k n n n n n n p se p so n n n n n n p p p V n n p - - = - - = - = = b. (5 points) What is the distribution under the Null Hypothesis? For large n, standard normal, you can also specify the exact binomial distribution(for all n). For small n, it would not be a bad idea to use the t with n-1 degrees of freedom. Page 2 of 13
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Econ 140 Department of Economics Fall 2008 Prof. Armando Levy c. (5 points) Explain why it would be invalid to perform six of these tests, one for each of the possible outcomes from our experiment? The tests are not independent of each other. Given p1,. ., p5 then p6=1-p1-…-p5. That is given the first 5 tests, the sixth is not a random variable, it is deterministic . d. (5 points) Describe a simple regression equation that you could specify to test whether the die shows 3 dots 1/6 th of the time as it should. t t y ε α + = where y t =1 if the t th roll is a 3 and zero otherwise. Here α is the expectation of y t which is 1/6. 3. In this question, we will use regression models to try to identify and estimate sibling peer effects. Our data set consists of observations of schooling outcomes and individual characteristics for i = 1,…,N households in Indonesia. Each of our households has two siblings. Y1 i denotes the test score of the oldest sibling, and
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This note was uploaded on 02/17/2010 for the course ECON 141 taught by Professor Staff during the Fall '08 term at University of California, Berkeley.

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Econ140Final_Soln_Fall2008 - Econ 140 Fall 2008 FINAL EXAM...

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