ChE253K Lecture 15 -- Central Limit Theorem

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

This preview shows pages 1–13. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 Dist’n Calc’s Student’s t Dist’n Calcs Confidence Intervals & Hypothesis Testing Preview

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 - = - -
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
11 ChE 253K Lecture 15 Galton Board Demonstrations http://www.mathsisfun.com/ probability/quincunx.html http://www.jcu.edu/math/isep/ Quincunx/Quincunx.html

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
12 ChE 253K Lecture 15 Random Sample Galton Dist’n Random Sample
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern