Lecture 3

Lecture 3 - LECTURE 3 STATISTICAL INFERENCE Point Estimate The estimator(e.g X applied to the sample gives you a single guess(point estimate about

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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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) 1 , 0 ( N ~ n σ μ X Z = Based on the chosen α , we can determine a value c from the standard normal table such that: Pr(-c Z c) = 1 – α Note: if you want α = .05, you want each tail to have 0.025 For example, if α = .05 then: Pr(-1.96 Z 1.96) = .95 This value c is known as our critical value of the standard normal distribution. From this, we can back out a confidence interval for the true population parameter μ : Pr(-c Z c) = 1 – α Ù Pr ( c n σ μ X c ) = 1 – α Ù Pr ( n σ c μ X n σ c ) = 1 – α Ù Pr ( n σ c X μ n σ c X + ) = 1 – α Ù Pr ( n c X n c X σ μ + ) = 1 – α Thus, given α we found an interval:
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[ n σ c X , n σ c X + ] Such that the probability that μ lies in this interval is (1 – α ). NOTES: Pr ( n c X n c X σ μ + ) = 1 – α Since X is a random variable, the endpoints of this interval are also random variables, and the interval itself is a random interval . For any particular sample, the endpoints take on specific values and the calculated confidence interval either contains or does not contain the unknown parameter μ , but we cannot know whether it does or it does not. The confidence interval is larger the larger is the variance of the population at hand σ , and/or the smaller is the sample sizes n, and/or the greater is the desired confidence level (or equivalently, the smaller α is).
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This note was uploaded on 03/20/2009 for the course ECON 103 taught by Professor Sandrablack during the Winter '07 term at UCLA.

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Lecture 3 - LECTURE 3 STATISTICAL INFERENCE Point Estimate The estimator(e.g X applied to the sample gives you a single guess(point estimate about

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