Lecture 3 - LECTURE 3 STATISTICAL INFERENCE Point Estimate...

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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 values likely 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 calculate two-sided confidence intervals for 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: X i ~ i.i.d.N( μ , σ 2 ) where μ unknown, σ 2 known. We specify a confidence level , (1 – α ) × 100%, where α is 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%. α is the significance level. It is natural to use X as our estimator of μ . Here we know ) n σ , μ ( N ~ X 2 Standardizing:
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