# 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 Con°dence Interval : It gives a range of values likely to contain the unknown parameter (e.g. ° ) at a certain con°dence level (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, suppose we constructed a 95% con°dence interval for ° . In 95% of these samples, the calculated con°dence interval 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 , provided that we know the (°xed 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: Normal random sample ±known variance: 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, or 0.10, which correspond to con°dence levels of 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
± For example, if ² = : 05 then Pr( ² 1 : 96 ´ Z ´ 1 : 96) = : 95 3

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± This value c , is known as our critical value of the stan- dard normal distribution ± From this, we can back out a con°dence interval for the true population parameter ° . Pr ( ² c ´ Z ´ c ) = 1 ² ² , Pr ² c ´ X ² ° ± p n ´ c ! = 1 ² ² , Pr ° ² c ± p n ´ X ² ° ´ c ± p n 0 ± = 1 ² ² , Pr ° ² X ² c ± p n ´ ² ° ´ ² X + c ± p n ± = 1 ² ² , Pr ° X ² c ± p n ´ ° ´ X + c ± p n ± = 1 ² ² ± Thus, given ² we found an interval ² X ² c ± p n ; X + c ± p n ³ such that the probability that ° lies in this interval is (1 ² ² ) . 4
NOTES: Pr ° X ² 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- °dence interval either contains or does not contain the unknown parameter ° , but we cannot know whether it does or it does not. ± The con°dence interval is larger, the larger is the vari- ance of the population at hand ±; and/or the smaller is the sample sizes n; and/or the greater is the desired con°dence level (or equivalently, the smaller ² is).

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