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Unformatted text preview: Econ 319, Fall 2008 TA: Simon Kwok Exam 3 solutions Part 1 1. Bias of an estimator ^ & is the di/erence between the expectation of ^ & and the true characteristic & . i.e. Bias ( ^ & ) = E ( ^ & ) & & . 2. The variance of an estimator ^ & measures the variation of the estimator ^ & about its mean E ( ^ & ) . i.e. V ar ( ^ & ) = E [( ^ & & E ( ^ & )) 2 ] . 3. The signi&cance level of a test is the highest probability of rejecting the null hypothesis if it is in fact true. It is usually set by the statistician before conducting hypothesis testing. 4. The power of a test is the probability of rejecting the null hypothesis if the alternative hy pothesis is in fact true. It is equal to one minus the probability of type II error. 5. Type I error is rejecting the null hypothesis if it is in fact true. Its probability is equal to the signi&cance level of the test. Type II error is accepting the null hypothesis if the alternative hypothesis is in fact true. Its probability is equal to one minus the power of the test. Part 2 1. (a) & X = 1 320 320 X i =1 X i , & Y = 1 206 206 X i =1 Y i . (b) E & X = 1 320 320 X i =1 E ( X i ) = 1 320 320 X i =1 ¡ X = ¡ X , so bias ( & X ) = 0 . Similarly, bias ( & Y ) = 0 . V ar ( & X ) = 1 320 2 320 X i =1 V ar ( X i ) = 1 320 2 320 X i =1 ¢ 2 X = & 2 X 320 ....
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