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class03_15 - 1.017/1.010 Class 15 Confidence Intervals...

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Unformatted text preview: 1.017/1.010 Class 15 Confidence Intervals Interval Estimates Parameter estimates computed from a random sample x 1 , x 2 ,..., x N can vary around the unknown true value. For any given estimate, we seek a two-sided confidence interval that is likely to include the true value a : U L a a a ≤ ≤ [ a L , a U ] is called an interval estimate . Standardized Statistics Interval estimate is often derived from a standardized statistic . This is a random variable that depends on both the unknown true value a and its estimate . a ˆ An example is the z statistic : ) ˆ ( ˆ ) , ˆ ( a a a z SD a a − = If the estimate is unbiased E [ z ] = 0 and Var [ z ] = 1 , for any x or a probability distribution with finite moments (prove). ˆ Example: Suppose: a = E [ x ] = mean of the x probability distribution = m a ˆ x = sample mean (of random sample outcome x 1 , x 2 ,..., x N ). Then: N x SD E SD E a x x x ) ( ] [ ) ( ] [ ) , ˆ ( x m m x m a z − = − = 1 Deriving Interval Estimates If we know the probability distribution of the standardized statistic z we can derive an interval estimate. Specify the probability 1- α that z falls in the interval [ z L , z U ] for a given value of a (e.g. 0.95): α − = ≤ ≤ 1 ] ) , ˆ ( [ U L z a z...
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class03_15 - 1.017/1.010 Class 15 Confidence Intervals...

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