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Unformatted text preview: STAT 200 Chapter 6 Introduction to Inferences Basic concepts: estimation, estimators and estimates 1. Estimators vs. Estimates: An estimator is a function of random variables X 1 ,X 2 , X n in a random sample of size n ) while an estimate is the realized value of an estimator (i.e., a number) that is obtained when a sample is actually taken. 2. Point Estimation: The process of estimating a single number as an estimate of the population parameter is called point estimation . Parameter Point Estimator Point Estimate Population mean Sample mean X x Population SD Sample SD S s Population proportion p Sample proportion p p (a value) 1 3. Interval Estimation: We might want to specify a range of values within which a population parameter is likely to fall. The process of estimating this range of values is called interval estimation . An interval estimator of a population parameter is a function of the observations from a sample which gives an interval that estimates the population parameter. An interval estimate is the realized range of values (an interval) of its corresponding interval estimator. 4. A desirable property of a point estimator: Unbiasedness If a point estimator has a mean equal to the population parameter it is intended to estimate, the point estimator is said to be an unbiased estimator of the parameter. Otherwise, it is said to be a biased estimator of the parameter. For example, X and p are unbiased estimators of and p , respectively. E ( X ) = X = and E ( p ) = p = p . Interval estimation of population parameters confidence intervals (Section 6.1) Let X 1 ,X 2 , ,X n be a random sample from a population with mean and standard devi ation . Suppose is unknown and is known. We want to obtain an interval estimate of . With X N ( , ), we have X N ( X = , X = n ). If X does not follow the normal distribution, as long as the conditions under the CLT are satisfied, X approx. N ( X = , X = n ) By the 689599.7 rule, P (  2 n < X < + 2 n ) 95% P ( X 2 n < < X + 2 n ) 95% P ( X + 2 n > > X 2 n ) 95% P ( X 2 n < < X + 2 n ) 95% 2 There is an approximately 95% chance that the interval ( X 2 n , X + 2 n ) captures the true value of . We call this interval the 95% confidence interval for . Confidence intervals for : The level C confidence interval for can be constructed using x z * n where x is the sample mean, is the known population standard deviation, and z * is the zscore such that the area (in %) to its right under the zcurve is equal to 100% C 2 ....
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This note was uploaded on 10/21/2010 for the course STATISTICS Stat 200 taught by Professor Eee during the Spring '10 term at The University of British Columbia.
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