Econ 281 Fall 2010 Problem Set 3 - Solutions

Econ 281 Fall 2010 Problem Set 3 - Solutions - Econ 281...

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Econ 281 Spring Quarter 2010 Problem Set 3 Solutions 1. Suppose that Y i are independent draws from N(0, 2 Y σ ) for i=1…N. a. Show that 1 ) ( 2 2 = Y i Y E b. Let W= = N i i Y Y 1 2 2 1 .Show that E(W)=N 2. Let Y~ (50, 81). Answer questions b)-d) assuming your sample size=100. a) Why do we have to use the central limit theorem to answer b)-d) b) Calculate P( Y >52.5)= 1-P(Z< 2.777)=1-.9973=.0027 c) Calculate P( Y <51.5)= P(Z<1.66)=.9515 d) Calculate P(48.24< Y <51.76)=P(-1.96<Z<1.96)= P(-1.96)-P(Z<1.96)=.95 e) How big of a sample do you need to get P(49.50< Y <50.50) 99 . 0 N=2149 or bigger 3. Assume that k is an extremely small number. What happens to P(50-k< Y <50+k) as the sample size gets larger and larger? In 1 e) the small number was equal to .5 and it should be clear that increasing N beyond 1652 will bring the probability closer and closer to 1. The same is true in general. No matter how small we choose k, as the sample size increases (towards infinity) the probability P(50-k< Y <50+k) approaches 1. 4. What is the difference between consistency and unbiasedness Consistency is a large sample property of an estimator. It means that the estimator comes arbitrarily close to the true population parameter as the sample size increases. If the expected value of an estimator equals the true population parameter it is unbiased. This property doesn’t depend on the sample size.
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Econ 281 Fall 2010 Problem Set 3 - Solutions - Econ 281...

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