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# Statistics Measurement in Economics ECON 310

UW

GPA 3.6

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This 3 page Class Notes was uploaded by April Jerde on Thursday September 17, 2015. The Class Notes belongs to ECON 310 at University of Wisconsin - Madison taught by Sang Lee in Fall. Since its upload, it has received 10 views. For similar materials see /class/205137/econ-310-university-of-wisconsin-madison in Economcs at University of Wisconsin - Madison.

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Date Created: 09/17/15

ECON 310 TA SECTION WEEK 8 prepared by SANG YOON TIM LEE and IGNACIO MONZON I Normal Distribution Definition The random variable X has a normal distribution with mean y and variance 72 denoted X m N 4472 if its density function f is given by 307 7 27102 Note that 0 Height weight body temperature shoe size the diameter of trees light intensity changes in the log of stock price indices are examples of normal distributions 0 Due to the central limit theorem a binomial distribution with parameters 71 and p is approx imately normal for large n and p not too close to 1 or 0 Polling Multiple choice tests look like normals 0 About 68 of values drawn from a normal distribution are within one standard deviation 7 gt 0 away from the mean y and about 95 are within two standard deviations 11 Shifting and Scaling the Normal Distribution Let a 7E 0 and b be constants If X is normally distributed and Y X b then Y is normally distributed as well That is if X m N y 72 then Y m N W b 202 12 Standard Normal Distribution If Z is a normal random variable with mean 0 and variance 1 then Z has a standard normal distribution Z m N 01 Suppose X m N y 72 then Z E y is a standard normal random variable 13 Sums of Independent Normal Random Variables If X and Y are independent normal random variables then S X Y is also a normal random variable In particular if X m N ma a and Y m N pl712 are independent then S N XWIU ltU 11 Problems 1 Exercise 7 from Chapter 7 Let X1 and X2 be independent random variables each with N a N 04 distribution Determine which of the random variables below are normally dis tributed If the variable is normal specify its distribution a Y 8X1 7 10 b Y 2X 7 c Y 1 7 025X1 d Y X1bX2C e Now assume that X1 and X2 have the same distributions as before except now they are no longer independent In fact assume that they are perfectly negatively correlated C077 X1X2 71 What is the distribution of Y X1 X2 Based on Exercise 8 from Chapter 7 A ight from New York to Chicago is scheduled to take 160 minutes Assume that the actual ight duration is normally distributed with a mean of 150 minutes and a standard deviation of 10 minutes a Determine the probability that a ight s duration is within twenty minutes of its mean duration b A businessman needs to arrive in Chicago less than 180 minutes after takeoff in order to make a meeting What is the probability that he will make it to the meeting on time c An airline wants to advertise the shortest ight duration such that 90 of ights will arrive on time What duration should it announce 3 Exercise 2 from Chapter 9 A chain of brew pubs has established strict guidelines for bar F tenders to help manage costs Each glass should contain 16 ounces of beer however man agement has noted that there is some small variation in volume Specifically the amount of beer poured has a mean volume of 1600 ounces and a standard deviation of 020 ounces The chain of pubs collectively pours 50000 beers per week and beer volumes are iid across pours a What is the expected volume of beer poured per week b What is the variance in the volume poured per week Suppose that a sample of 40 beers are measured for quality control c Determine the expected value of the sample mean d Determine the variance of the sample mean e Describe the probability distribution of the sample mean Exercise 4 from Chapter 9 The amount of time needed to verify a credit card online using a popular security service has a mean of 8 seconds and a variance of 9 seconds The sequence of verifications is iid Determine the standard deviation in the average verification time if a there are 10 customers b there are 100 customers c there are 1000 customers d there are 10000 customers e there are 100000 customers

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