chapter7 - CHAPTER 7: CONFIDENCE INTERVALS 7.1 Point...

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1 CHAPTER 7: CONFIDENCE INTERVALS 7.1 Point Estimation Given a random sample x 1 , x 2 , . .., x n from a population, estimate the unknown population parameter θ . Examples of population parameters: θ =μ (mean), or θ = σ (standard deviation), or θ =p (population proportion). Estimator: A statistic ˆ θ that uses the sample values x 1 , x 2 ,…, x n to estimate an unknown parameter of the population. Examples of estimators: 1. 12 ... n x xx x n ++ + = is an estimator of µ . 2. 2 2 1 () 1 n i i x x s n = = is an estimator of σ 2 . Population Sample
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2 Point estimate of a population parameter θ is the value of an estimator ˆ θ for a particular sample. Unbiased Estimators An estimator ˆ of a population parameter θ is unbiased if its sampling distribution is centered at the parameter. Example : Estimator ˆ = x . Population with mean µ All possible samples of size n •••••••••••••••• µ sample averages For each sample the sample mean x is determined The sampling distribution of the sample mean is centered at μ . x is an unbiased estimator of µ
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3 ••••• ••••• ••••••• θ θ Standard Error Standard error of an estimator ˆ θ is the standard deviation of the sampling distribution of the estimator. ••••• ••••• θ θ As before, standard error will be denoted by SE. 1 ˆ 2 ˆ Unbiased estimator of θ Biased estimator of θ 1 ˆ 2 ˆ Standard error of 1 ˆ is smaller than standard error of 2 ˆ , the estimates produced by 1 ˆ tend to more accurate estimates of the parameter than those produced by 2 ˆ .
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4 Example 1 : Estimator ˆ θ = x . Standard deviation of x is . SE n σ = In case σ is unknown, standard error can be estimated as` follows: . s SE n where s is the sample standard deviation. Notice that the standard error of x decreases as the sample size n increases. The magnitude of standard error shows how trustworthy x is as an estimate of the mean µ . Example 2 : Estimator ˆ = ˆ p (sample proportion) Standard deviation of ˆ p is (1 ) . p p SE n = As p is unknown, standard error can be estimated as` follows: ˆˆ ) . p p SE n Note that also in this case SE decreases, as n increases.
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5 7.2 Confidence Intervals about μ when σ is known Given a random sample of size n from a population with unknown mean µ and known standard deviation σ , find two numbers L and U such that L µ U. Assume that the population is normal or n 30. Then the sampling distribution of x is approximately normal (CLT). Define L xm =− and , Ux m =+ where m is a small positive number. L U x μ m m ••••••••• ••••••••• μ x Sample mean values Notice that m would have to be very large (very wide interval) to make the interval capture μ . However, wide interval contains no useful information about μ !
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6 •••••••••
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This note was uploaded on 04/10/2008 for the course STAT 151 taught by Professor Henrykkolacz during the Fall '07 term at University of Alberta.

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chapter7 - CHAPTER 7: CONFIDENCE INTERVALS 7.1 Point...

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