ChE253K Lecture 15 -- Central Limit Theorem

ChE253K Lecture 15 -- Central Limit Theorem - 1 ChE 253K...

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Unformatted text preview: 1 ChE 253K Lecture 15 Class Business Welcome Back from Spring Break HW06 Probability Due Today HW07 Normal Distn Due Friday 5pm @ CPE 1.420 2 ChE 253K Lecture 15 The Normal Distribution & The Central Limit Theorem Lecture 15 -- Inferential Statistics 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 Distn Calcs Students t Distn Calcs Confidence Intervals & Hypothesis Testing Preview 4 ChE 253K Lecture 15 Readings Re: This Lecture Populations & Samples Chapter 6, Sect. 6.1 Sampling Distn of Mean (large sample; known) Chapter 6, Sect. 6.2 Sampling Distn of Mean (small smpl; unknown) Chapter 6, Sect. 6.3 5 ChE 253K Lecture 15 Random Sample Popn Params 1) Draw a small random sample (n items) from a large population of N items 2) Calculate mean and std devn 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 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) Popn 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 7 ChE 253K Lecture 15 Random Sampling Normal Distn CL Theorem 1) Bernoulli Trials Binomial Normal Distn with p = and n 1) Data influenced by many small and unrelated random errors/effects are normally distributed. 1) The sampling distn of the mean is normal (Central Limit Theorem) Heuristic Justification 8 ChE 253K Lecture 15 Binomial Normal Distribution Binomial Bell Bell Normal Distn 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 -- = =- ( 29 ( , , ) ! (1 ) ! ! x n x b x n p n p p x n x- =-- 9 ChE 253K Lecture 15 Random Sample Binomial Distn Random Sample Bernoulli Trials Each random item is an expt of a Trial An item above the popn mean is success. The result is the number above . Bernoulli Trials Binomial Distn Equal number above and below the center. Distribution of results is Normal. 10 ChE 253K Lecture 15 Randomized Data Normal Distn Results influenced by many small random errors/effects are normally distributed. Galton Board: The balls final bin is the result of a series of random left or right bounces. 11 ChE 253K Lecture 15 Galton Board Demonstrations http://www.mathsisfun.com/ probability/quincunx.html http://www.jcu.edu/math/isep/ Quincunx/Quincunx.html 12 ChE 253K Lecture 15 Random Sample...
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ChE253K Lecture 15 -- Central Limit Theorem - 1 ChE 253K...

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