501_Lecture_06 - Introduction to Inference Section 6.1...

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Introduction to Inference Section 6.1 Estimating with Confidence
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Introduction Distinguish chance variations from permanent features of a phenomenon: Give SAT test to a SRS of 500 California seniors sample mean = 461 What does it say about the mean SAT score of 420,000 HS seniors in CA? Is 12/20 vs. 8/20 improvements in treatment vs. control group strong enough evidence in favor of a drug?
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Methods of formal inference rely on the assumption that the data come from properly randomized experiment (e.g. SRS) Statistics gives methods that give correct results a prescribed (high) percentage of times (if repeated many times) Most prominent: confidence intervals and tests of significance
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Estimating with confidence Example 1 : Observe 15 plots of corn with yields (in bushels): 138, 139.1, 113, 132.5, 140.7, 109.7, 118.9, 134.8, 109.6, 127.3,115.6, 130.4, 130.2, 111.7, 105.5 Sample Mean = 123.8 What can be said about the (population) mean yield of this variety of corn?
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Statistical confidence: Assume that yield is N( µ , σ) with unknown µ and σ=10 (just assume σ is known) Then 68-95-99.7% rule: 95% of time sample mean is within 2 standard deviations of population mean 2×2.58 = 5.16 from µ Thus, 95% of time: 10 , , ( ,2.58) 15 σ μ = = : X N N N n 5.16 5.16 - < < + X
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Put differently: 95% of time The random interval covers the unknown (but nonrandom) population parameter µ 95% of time. Our confidence is 95%. We need to be extremely careful when observing this result. 5.16 5.16 μ - < < + X X
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Example 1 (cont) : This particular confidence interval may contain µ or not… However, such a systematic method gives intervals covering the population mean µ in 95% of cases. is between 5.16 123.8 5.16 (118.64, 128.96) μ = = X
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Confidence Intervals (CIs): Typically: estimate ± margin of error Always use an interval of the form (a, b) with endpoints calculated from data Confidence level (C) gives the probability that such interval(s) will cover the true value of the parameter. It does not give us the probability that our parameter is inside the interval. (In Example 1: C = 0.95)
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Confidence interval for population mean µ : Data: SRS of n observations Assumption: population distribution is N(µ,σ) with known σ and unknown µ Notation: z * percentiles from standard normal distribution * Confidence interval for : σ μ X z n
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How to find z * :
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Example: C= 95%. Find z * from table A. See also last row in Table D.
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Example 2 : Tim Kelley weighs himself once a week for several years. Last month he weighed himself 4 times with an average of 190.5. Examination of Tim’s past data reveals that over relatively short periods of time, his weight measurements are approximately normal with a standard deviation of about 3. Find a 90% confidence interval for his mean weight for last month. Then, find a 99% confidence interval.
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Width of CI increases with confidence level: More confidence wider interval Less confidence narrower interval
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This note was uploaded on 02/20/2012 for the course STAT 501 taught by Professor Staff during the Spring '08 term at Purdue University.

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501_Lecture_06 - Introduction to Inference Section 6.1...

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