UASTAT151Ch8 - Ch 8 Statistical Inference Defn Estimation...

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Ch. 8 - Statistical Inference Def’n: Estimation is the assignment of value(s) to a population parameter based on a value of the corresponding sample statistic. An estimator is a rule used to calculate an estimate. An estimate is a specific value of an estimator. Note: in this chapter, always assuming an SRS. - Notation: - Let θ be a generic parameter. - Let θ ˆ be an estimator – a statistic calculated from a random sample - Consequently, θ ˆ is an r.v. with mean E ( θ ˆ ) = θ µ ˆ and std. dev. θ σ ˆ 8.1 Point and Interval Estimates Def’n: A point estimate is a single number that is our “best guess” for the parameter. Æ like a statistic , but more precise towards parameter estimation. An interval estimate is an interval of numbers within which the parameter value is believed to fall. Generic large sample confidence intervals : Def’n: A confidence interval (CI) for a parameter θ is an interval estimate of plausible values for θ . With a chosen degree of confidence, the CI’s construction is such that the value of θ is captured between the statistics L and U , the lower and upper endpoints of the interval, respectively. The confidence level of a CI estimate is the success rate of the method used to construct the interval (as opposed to confidence in any particular interval). The generic notation is 100(1 – α )%. Typical values are 90%, 95%, and 99%. Ex8.1) Using 95% and the upcoming method to construct a CI, the method is “successful” 95% of the time. That is, if this method was used to generate an interval estimate over and over again with different samples, in the long run, 95% of the resulting intervals would capture the true value of θ . Many large-sample CIs have the form: point estimate ± (critical value) × (standard error) where “point estimate” is a statistic θ ˆ used to estimate parameter θ , “standard error” is a statistic θ σ ˆ ˆ used to estimate std. dev. of estimator θ ˆ , “critical value” is a fixed number z defined so that if Z has std. norm. dist’n, then P (- z Z z ) = 1 – α = confidence level
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