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Statistical Inference ch2

# Statistical Inference ch2 - Topic Two Inferences about a...

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MSOR221 1 / 8 Topic Two Inferences about a Population: Interval Estimation Point and Interval Estimation A point estimation refers to the procedure that uses the information in the sample to arrive at a single number that is intended to be close to the true value of the target parameter. An interval estimation refers to the procedure that uses the sample information to arrive at the lower limit and the upper limit that are intended to enclose the parameter of interest. In either case the actual estimation is accomplished by using an estimator for the target population parameter. An estimator θ ˆ is a random variable that is used to estimate an unknown parameter θ in the population. It is a mathematical function that tells how to calculate the value of an estimate based on the random variables, X 1 , X 2 , …, X n , contained in the sample. An actual numerical value obtained from an estimator is called an estimate . Thus, when a single value, or point, is given as an estimate of an unknown parameter, it is called a point estimate; when a range (or an interval) of values is provided, it is called an interval estimate. For instance, X may be used as an estimator of μ , in which case x is an estimate of this parameter. Also, S 2 may be used as an estimator of σ 2 , in which case s 2 is an estimate of this parameter. Similarly, P ˆ may be used as an estimator of p , in which case p ˆ is an estimate of this parameter. Since estimators are random variables, consequently, estimators have their probability distributions or, more properly, sampling distributions. That is, by repeated sampling from population, a probability distribution of the values of the estimates can be constructed. Various statistical properties can thus be used to decide which estimator is most appropriate in a given situation, which will give us the most information at the lowest cost, which will expose us to the smallest risk, and so forth. The following properties are used to evaluate the goodness of estimators. Unbiaedness When we are estimating an unknown population parameter, it is highly desirable for the sampling distribution of the estimator to cluster about the target parameter. In other words, the mean (or expected value) of the sampling distribution of the estimator would equal the parameter estimated; that is, θ θ = ) ˆ ( E . An estimator θ ˆ is an unbiased estimator if it satisfies this property; otherwise θ ˆ is said to be biased . When θ ˆ is a biased estimator of θ , it may be of interest to know the extent of the bias , given by θ θ θ - = ) ˆ ( ) ˆ ( E B . If the bias tends to be smaller when sample size n grows larger, i.e. 0 ) ˆ ( lim = θ B n , the estimator is asymptotically unbiased .

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MSOR221 2 / 8 Efficiency In addition to preferring unbiasedness, the variance of the sampling distribution of the estimator is also preferred to be as small as possible. Given several unbiased estimators of a given parameter and all other things being equal, the estimator with the smallest variance is usually selected.
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Statistical Inference ch2 - Topic Two Inferences about a...

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