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Unformatted text preview: Chapter 8 Point Estimation and Confidence Interval 8.1 Point estimator The purpose of point estimation is to use a function of the sample data to estimate the unknown parameter. Definition 8.1 A parameter is a constant that describes the population. A statistic is a random variable that can be computed from the sample data without making use of any unknown parameters. The statistic ˆ Θ used to estimate the unknown parameter is called a point estimator of θ . A point estimate is the value of ˆ Θ calculated from the observed sample values. Recall • A statistic ˆ Θ is said to be unbiased if E ( ˆ Θ) = θ . • ¯ X is an unbiased estimator of μ , and S 2 is an unbiased estimator of σ 2 . • Warning : Unbiased estimator is not unique. Definition 8.2 If we consider all possible unbiased estimators of θ , the one with the smallest variance is called the most efficient estimator of θ . 81 8.2 Interval estimation We have proved that the sample mean is an unbiased estimator of the population mean. Suppose a sample of size n is taken from a Poisson distribution and it is found that ¯ x = 3 . 8. Then, 3 . 8 is a point estimate of λ . But, what exactly does this tell us about the true value of λ ? Can we fell reasonably certain , for example, that λ lies somewhere close to ¯ x – say, in the interval from 3 . 7 to 3 . 9. Or, on other hand, is ¯ X so variable that there is a good chance that  ¯ X λ  is fairly large? To address this uncertainty we turn from point estimation to a technique known as interval estimation . Interval estimation is exactly what the name implies. We want to find two statistics, ˆ Θ 1 and ˆ Θ 2 , that can be used to generate an interval of real numbers that we hope contains the true value of the parameter θ being estimated. Definition 8.3 A 100(1 α )% confidence interval for a parameter θ is an interval of the form [ ˆ θ 1 , ˆ θ 2 ] , in which ˆ Θ 1 and ˆ Θ 2 are statistics such that P ( ˆ Θ 1 ≤ θ ≤ ˆ Θ 2 ) = 1 α 8.3 Confidence interval for μ when σ is known Theorem 1 If ¯ x is the value of the sample mean of a random sample of size n from a normal population with the known variance σ 2 , then a 100(1 α )% confidence interval for μ is ¯ x ± z α/ 2 σ √ n = (¯ x z α/ 2 σ √ n , ¯ x + z α/ 2 σ √ n ) z α/ 2 σ √ n is called the margin of error . Example 8.1 What is the average price of statistics books? Below is a random sample of prices: 40 53 39 37 22 35 66 80 95 35 What is a 95% confidence interval for μ ?, assuming σ = 20 Solution : 82 • The interval (37.8, 62.6) is called a 95% confidence interval for μ . We are 95% confident that the unknown μ lies between $37.8, and $62.6 — “We got this interval by a method that gives correct results 95% of the time”....
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 Fall '09
 lisiufeng
 Math, Statistics

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