oh3 - LECTURE 3: STATISTICAL INFERENCE Point Estimate: The...

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LECTURE 3: STATISTICAL INFERENCE Point Estimate : The estimator (e.g. X ) applied to the sample gives you a single guess (point estimate) about the unknown parameter (e.g. the population mean ) : Interval Estimate or : It gives a range of values likely to contain the unknown parameter (e.g. ) at a certain (i.e. with certain, usually high, probability). What does this mean? Suppose we repeatedly drew samples from a distribution with true population mean For each of these samples, . around the sample mean would include the true population mean. Next, we calculate two-sided con&dence intervals for a pop- ulation mean . We can do the same for any other unknown parameter of a distribution sample or asymptotic) distribution of an estimator that con- sistently estimates the unknown parameter of interest. 1
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CONFIDENCE INTERVALS FOR THE MEAN CASE 1: X i i:i:d:N ( ± 2 ) where unknown, ± 2 known. ± We specify a con±dence level, (1 ² ² ) ³ 100% ; where ² is a small number between 0 and 1, e.g. ² = 0.01, 0.05, 99% , 95%, and 90%. ± ² is the signi±cance level. ± It is natural to use X as our estimator of Here we know X N ( ± 2 n ) Standardizing: Z = X ² ± p n N (0 ; 1) ± Based on the chosen ² , we can determine a value c from the standard normal table such that Pr ( ² c ´ Z ´ c ) = 1 ² ² ± Note: if you want ² = : 05 , you want each tail to have 0.025 2
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For example, if = : 05 then Pr( ± 1 : 96 ² Z ² 1 : 96) = : 95 3
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This value c , is known as our critical value of the stan- dard normal distribution true population parameter . Pr ( ± c ² Z ² c ) = 1 ± ± , Pr ± c ² X ± ² p n ² c ! = 1 ± ± , Pr ± c ² p n ² X ± ² c ² p n 0 ± = 1 ± ± , Pr ± X ± c ² p n ² ± ² ± X + c ² p n ± = 1 ± ± , Pr X ± c ² p n ² ² X + c ² p n ± = 1 ± ± Thus, given ± we found an interval ² X ± c ² p n ; X + c ² p n ³ such that the probability that lies in this interval is (1 ± ± ) . 4
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NOTES: Pr X c p n ± ± ± X + c p n ± = 1 ² ² Since X is a random variable, the endpoints of this in- terval are also random variables, and the interval itself is a random interval . For any particular sample, the endpoints take on speci&c values and the calculated con- unknown parameter ± , but we cannot know whether it does or it does not. ² The con&dence interval is larger, the larger is the vari-
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This note was uploaded on 04/02/2008 for the course ECON 103 taught by Professor Sandrablack during the Winter '07 term at UCLA.

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oh3 - LECTURE 3: STATISTICAL INFERENCE Point Estimate: The...

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