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LECTURE 3: STATISTICAL INFERENCE Point Estimate:The estimator (e.g. X) applied to the sample gives you a single guess(point estimate) about the unknown parameter (e.g. the population mean μ). Interval Estimate of Confidence Interval: It gives a range of valueslikely to contain the unknown parameter (e.g. μ) at a certain confidence level(i.e. with certain, usually high, probability). What does this mean? Suppose we repeatedly drew samples from a distribution with true population mean μ. For each of these samples, suppose we constructed a 95% confidence interval for μ. In 95% of these samples, the calculated confidence interval around the sample mean would include the true population mean. Next, we calculatetwo-sided confidence intervalsfor a population mean. We can do the same for any other unknown parameter of a distribution, provided that we know the (fixed sample or asymptotic) distribution of an estimator that consistently estimates the unknown parameter of interest. CONFIDENCE INTERALS FOR THE MEAN Case 1: Normal random sample – known variance: Xi~ i.i.d.N(μ, σ2) where μunknown, σ2known. •We specify a confidence level, (1 – α) × 100%, where αis a small number between 0 and 1, e.g. α= 0.01, 0.05, or 0.10, which corresponds to confidence levels of 99%, 95%, and 90%. •αis the significance level.•It is natural to use Xas our estimator of μ. Here we know )nσ,μ(N~X2Standardizing:
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