statsexam3 09c sol

statsexam3 09c sol - Economic Statistics Exam 3 December...

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Economic Statistics Exam 3 December 14, 2009 SOLUTIONS Please sign that you abide by the honor pledge: This exam contains twenty short answer questions (worth 6 points each) and three long answer questions (worth 15 points each). In all problems where you need to make a calculation, simplify your answer as much as possible. Show your work clearly to be eligible for partial credit. Some PDFs: Some probabilities: f ( x ) = 1 2 πσ 2 e ( X μ ) 2 σ 2 P [ x a ] = 1 e λ a f ( x ) = 1 u P [ x ] = x e x ! f ( x ) = e x P [ x ] = n ! x !( n x )! p x (1 p ) n x 1. Suppose that the random variable X is drawn from a normal distribution with a mean of 12 and a variance of 16. What is the probability that X is between 10 and 18? P [10 < X < 18] = P [ 10 12 4 < Z < 18 12 4 ] = P [ 0.5 < Z < 1.5] = 0.4332 + 0.1915 2. A researcher draws a random sample of size N = 9 from a normal distribution. His sample has a mean of 52 and a standard deviation of 6. Construct a 95% confidence interval for the population mean. X ± t α /2, n 1 s x n 52 ± 2.306 6 9 ( )
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3. Suppose that you estimate Y as a quadratic function of X : Y i = β 0 + 1 X i + 2 X i 2 + e i . In Stata, you type: . gen x2 = x^2 . reg y x x2 Source | SS df MS Number of obs = 1000 ---------+------------------------------ F( 2, 997) = 233.17 Model | 48166.9162 2 24083.4581 Prob > F = 0.0000 Residual | 102975.057 997 103.284912 R-squared = 0.3187 ---------+------------------------------ Adj R-squared = 0.3173 Total | 151141.973 999 151.293267 Root MSE = 10.163 -------------------------------------------------------------------------- y | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+---------------------------------------------------------------- x | -1.570586 .3091338 -5.08 0.000 -2.177213 -.9639583 x2 | 4.546693 .2165667 20.99 0.000 4.121714 4.971671 _cons | 1.91043 .397595 4.80 0.000 1.130211 2.690649 -------------------------------------------------------------------------- What is the estimated marginal effect of X on Y (that is, Y X )? For quadratic functions, the marginal effect is Y X = ˆ 1 + 2 ˆ 2 X i . In this case, that is 1.57 + 2 4.54 X i . 4. In the previous question, how would you test the hypothesis that X has no effect at all on Y ? Either write out your null and alternative hypotheses, or write the Stata command(s) you would use to test it. This is a joint test that both 1 and 2 equal zero. You can either state H 0 : 0 = 1 = 0 ; H A : at least one is not zero or give the Stata commands: test x = 0 then test x2 = 0, accum . 5. I have an egg carton with 12 eggs; 4 are fresh, and 8 are rotten. If I select three eggs at random from the carton, what is the chance that all three are rotten? The probability that the first one is rotten is 8 12 . After you’ve removed one rotten egg, the probability that the next egg is rotten is 7 11 . After both of these are gone, the probability of getting a third rotten egg is 6 10 . Therefore, the probability of three rotten eggs in a row is: (8 12 )(7 11 )(6 10 ) .
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6. A dataset contains the following combinations of X and Y : X = 1, 2, 3, 4, 5 Y = 5, 5, 3, 1, 1 Calculate the covariance between the two variables. Cov ( X , Y ) = 1 n 1 ( X i X )( Y i Y ) = 1 5 1 [(1 3)(5 3) + (2 3)(5 3) + (3 3)(3 3) + (4 3)(1 3) + (5 3)(1 3)] = 1 4 [( 4) + ( 2) + 0 + ( 2) + ( 4)] = 10 4 7. Of the customers who walk into Ms. Lentil’s Widget Shop, about ¼ purchase a widget (the other ¾ leave without buying anything). Today, 20 customers enter the shop, but Ms. Lentil has only two widgets in stock. What is the chance that only 2 or fewer of these 20 customers want to purchase widgets?
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statsexam3 09c sol - Economic Statistics Exam 3 December...

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