ChE253K Lecture 15 -- Central Limit Theorem

ChE253K Lecture 15 -- Central Limit Theorem - Class...

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1 ChE 253K Lecture 15 Class Business Welcome Back from Spring Break HW06 – Probability Due Today HW07 – Normal Dist’n Due Friday 5pm @ CPE 1.420
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2 ChE 253K Lecture 15 The Normal Distribution & The Central Limit Theorem Lecture 15 -- Inferential Statistics
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3 ChE 253K Lecture 15 Outline Of This Lecture Heuristic Justification for CLT of the Mean Bernoulli Trials and Galton Board The Central Limit Theorem of the Mean Sampling Probabilities Standard Normal Dist’n Calc’s Student’s t Dist’n Calcs Confidence Intervals & Hypothesis Testing Preview
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4 ChE 253K Lecture 15 Readings Re: This Lecture Populations & Samples Chapter 6, Sect. 6.1 Sampling Dist’n of Mean (large sample; σ known) Chapter 6, Sect. 6.2 Sampling Dist’n of Mean (small smpl; σ unknown) Chapter 6, Sect. 6.3
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5 ChE 253K Lecture 15 Random Sample Pop’n Param’s 1) Draw a small random sample (n items) from a large population of N items 2) Calculate mean and std dev’n of the sample 3) Infer the mean and std dev of the population Random: Each item sampled Random: Sample of n items Random: Sample mean & std dev
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6 ChE 253K Lecture 15 Variability Of Sample Mean Samples Vary Sample Means Vary Daily SO X emissions from a plant (Lecture 04) Sample x(1-20) x(21-40) x(41-60) x(61-80) Pop’n Mean 17.7 19.2 19.4 19.3 18.91 Std Dev 6.5 4.2 5.0 6.9 13.82 Sources: Lecture 04
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7 ChE 253K Lecture 15 Random Sampling Normal Dist’n CL Theorem 1) Bernoulli Trials Binomial Normal Dist’n with p = ½ and n 1) Data influenced by many small and unrelated random errors/effects are normally distributed. 1) The sampling dist’n of the mean is normal (Central Limit Theorem) Heuristic Justification
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8 ChE 253K Lecture 15 Binomial Normal Distribution Binomial Bell Bell Normal Dist’n n = 128 32 48 64 80 96 ( 29 ( 29 ( 29 2 2 2 ( , , ) 1 2 ( , , 1 ) x f x e f x np np p μ σ μ σ σ π - - = = - ( 29 ( , , ) ! (1 ) ! ! x n x b x n p n p p x n x - = - -
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9 ChE 253K Lecture 15 Random Sample Binomial Dist’n Random Sample Bernoulli Trials Each random item is an exp’t of a Trial An item above the pop’n mean is “success.” The result is the number above μ . Bernoulli Trials Binomial Dist’n Equal number above and below the center. Distribution of results is Normal.
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10 ChE 253K Lecture 15 Randomized Data Normal Dist’n Results influenced by many small random errors/effects are normally distributed. Galton Board: The ball’s final bin is the result of a series of random left or right bounces.
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11 ChE 253K Lecture 15 Galton Board Demonstrations http://www.mathsisfun.com/ probability/quincunx.html http://www.jcu.edu/math/isep/ Quincunx/Quincunx.html
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12 ChE 253K Lecture 15 Random Sample Galton Dist’n Random Sample
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