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statsexam2f06sol

# statsexam2f06sol - Economic Statistics Exam 2 November 2...

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Economic Statistics Exam 2 Last name: November 2, 2006 First name: PPID: Circle class time: 9:30 - 11:00 12:30 - 2:00 Please sign that you abide by the honor pledge: This exam contains fifteen short answer questions ( worth 4 points each ) and four long answer questions ( worth 15 points each ) . Please answer in the space provided; if you need more room, indicate clearly that the answer is continued on the back of the page. 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 !" 2 e # ( X # μ ) 2 " 2 P [ x ! a ] = e " # a f ( x ) = 1 u ! ! P [ X ] = μ x e ! μ X ! f ( x ) = ! e " ! x P [ X ] = N ! X !( N ! X )! p X (1 ! p ) N ! X Short Questions ( 4 points apiece ) 1. X is a random variable with a continuous distribution. Its probability density function is some function, f ( x ) . How would we find the probability that X is between two values ( for example, between 2.13 and 4.56 ) , using the PDF? P [2.13 ! X ! 4.56] = f ( x ) dx 2.13 4.56 " 2. The sum of many small random variables tends to have a normal distribution, regardless of the distribution of the individual random variables. What is the analogous statement for the log - normal distribution? The product of many small random variables tends to have a lognormal distribution.

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Econ 400 Midterm 2, page 2 of 8. 3. What is the difference between an estimate and an estimator? The estimator is a rule or formula used to guess the value of some statistic; an estimate is the value you obtain with a specific sample. 4. After a long night of trick - or - treating, the number of candies in my bag is a random variable, distributed normally with mean of 300 and variance of 400. What is the chance that I have fewer than 250 candies in my bag? The z - score is 250 ! 300 400 = ! 50 20 = ! 2.5 . P [ X < 250] = F z ( ! 2.5) = 0.0062. 5. The lifespan of a male born in the U.S. in 1999 is 74 years. What is the chance that a person lives to be this age or older? ( You must decide the most appropriate distribution to use in this case. ) Durations are usually modeled using the exponential distribution. Given the exponential distribution, the probability of living to be 74 or older is P [ x ! 74] = e " # 74 , where ! = 1/ μ = 1/ 74 . Thus, P [ x ! 74] = e " 74/74 = e " 1 . 6. State the central limit theorem. As the sample size ( N ) gets large, the sample mean ( X ) has an approximately normal distribution with mean μ X = μ X and variance ! X 2 = ! X 2 / N . 7. I want to know the average number of hours that people watch television in a month. When I take a random sample of size 100, there is a 31.74 % chance that my sample mean differs from the true mean by 5 hours or more ( in either direction ) .
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