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Paul Cartledge

Ch. 19 - 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. ˆ Let - θ be an estimator – a statistic calculated from a random sample Cons-equently, θ is an r.v. with mean E( θˆ) = µ θ and std. dev. σ θ 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%. Ex19.1) 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 θˆ used to estimate parameter θ, “standard error” is a statisticσˆ used to estimate std. dev. of estimator θˆ, θ “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 The product of the “standard error” and “critical value” is the margin of error. Note: critical value z often denoted by z α/2here the notation reflects P(Z > z) = α/2. Ex19.2) if the confidence level is 95%, then α/2 = 0.025 and z 0.025= 1.96. (diagram drawn in class) Table 19X0 – Critical values for usual confidence levels 100(1 – α)% α α/2 zα/2 90% 0.10 0.050 1.645 95% 0.05 0.025 1.96 99% 0.01 0.005 2.58 ˆ ˆ The estimator θ and its standard error σθˆare defined so that, when the sample size n is sufficiently large, the sampling distribution of ˆ θθ− ▯ N(0,1) σˆˆ θ Thus, ⎛ θ −θ ⎞ P − z ≤ ≤ z ≈1−α ⎝ σθˆ ⎠ Algebraic manipulation yields Pθ − zσ ≤ˆ ≤θ + zσ ˆ ˆ)≈1−α θ θ Large Sample CI for Population Proportion Recall the 3 rules regarding
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