1 / 25

ENGR 610 Applied Statistics Fall 2007 - Week 3

ENGR 610 Applied Statistics Fall 2007 - Week 3. Marshall University CITE Jack Smith. Overview for Today. Review of Chapter 4 Homework problems (4.57,4.60,4.61,4.64) Chapter 5 Continuous probability distributions Uniform Normal Standard Normal Distribution (Z scores)

debra-spence
Download Presentation

ENGR 610 Applied Statistics Fall 2007 - Week 3

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. ENGR 610Applied StatisticsFall 2007 - Week 3 Marshall University CITE Jack Smith

  2. Overview for Today • Review of Chapter 4 • Homework problems (4.57,4.60,4.61,4.64) • Chapter 5 • Continuous probability distributions • Uniform • Normal • Standard Normal Distribution (Z scores) • Approximation to Binomial, Poisson distributions • Normal probability plot • LogNormal • Exponential • Sampling of the mean, proportion • Central Limit Theorem • Homework assignment

  3. Chapter 4 Review • Discrete probability distributions • Binomial • Poisson • Others • Hypergeometric • Negative Binomial • Geometric • Cumulative probabilities

  4. Probability Distributions • A probability distribution for a discrete random variable is a complete set of all possible distinct outcomes and their probabilities of occurring, where • The expected value of a discrete random variable is its weighted average over all possible values where the weights are given by the probability distribution.

  5. Probability Distributions • The variance of a discrete random variable is the weighted average of the squared difference between each possible outcome and the mean over all possible values where the weights are given by the probability distribution.The standard deviation (X) is then the square root of the variance.

  6. Binomial Distribution • Each elementary event is one of two mutually exclusive and collectively exhaustive possible outcomes (a Bernoulli event). • The probability of “success” (p) is constant from trial to trial, and the probability of “failure” is 1-p. • The outcome for each trial is independent of any other trial

  7. Binomial Distribution • Binomial coefficients follow Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 • Distribution nearly bell-shaped for large n and p=1/2. • Skewed right (positive) for p<1/2, and left (negative) for p>1/2 • Mean () = np • Variance (2) = np(1-p)

  8. Poisson Distribution • Probability for a particular number of discrete events over a continuous interval (area of opportunity) • Assumes a Poisson process (“isolable” event) • Dimensions of interval not relevant • Independent of “population” size • Based only on expectation value ()

  9. Poisson Distribution, cont’d • Mean () = variance (2) =  • Right-skewed, but approaches symmetric bell-shape as  gets large

  10. Other Discrete Probability Distributions • Hypergeometric • Bernoulli events, but selected from finite populationwithout replacement • p now defined by N and A (successes in population N) • Approaches binomial for n < 5% of N • Negative Binomial • Number of trials (n) until xth success • Last selection is constrained to be a success • Geometric • Special case of negative binomial for x = 1 (1st success)

  11. Cumulative probabilities P(X<x) = P(X=1) + P(X=2) +…+ P(X=x-1) P(X>x) = P(X=x+1) + P(X=x+2) +…+ P(X=n)

  12. Continuous Probability Distributions • Differ from discrete distributions, in that • Any value within range can occur • Probability of specific value is zero • Probability obtained by cumulating bounded area under curve of Probability Density Function, f(x) • Discrete sums become integrals

  13. Continuous Probability Distributions (Mean, expected value) (Variance)

  14. Uniform Distribution

  15. Normal Distribution • Why is it important? • Numerous phenomena measured on continuous scales follow or approximate a normal distribution • Can approximate various discrete probability distributions (e.g., binomial, Poisson) • Provides basis for SPC charts (Ch 6,7) • Provides basis for classical statistical inference (Ch 8-11)

  16. Normal Distribution • Properties • Bell-shaped and symmetrical • The mean, median, mode, midrange, and midhinge are all identical • Determined solely by its mean () and standard deviation () • Its associated variable has (in theory) infinite range (- < X < )

  17. Normal Distribution

  18. Standard Normal Distribution where Is the standard normal score (“Z-score”) With and effective mean of zero and a standard deviation of 1

  19. Normal Approximation to Binomial Distribution • For binomial distributionand so • Variance, 2, should be at least 10

  20. Normal Approximation to Poisson Distribution • For Poisson distributionand so • Variance, , should be at least 5

  21. Normal Probability Plot • Use normal probability graph paperto plot ordered cumulative percentages, Pi = (i - 0.5)/n * 100%, as Z-scores- or - • Use Quantile-Quantile plot (see directions in text)- or - • Use software (PHStat)!

  22. Lognormal Distribution

  23. Exponential Distribution Poisson, with continuous rate of change,  Only memoryless random distribution

  24. Sampling Distribution of the Mean • Central Limit Theorem Continuous data (proportion) Attribute data

  25. Homework • Ch 5 • Appendix 5.1 • Problems: 5.66-69 • Skip Ch 6 and Ch 7 • Statistical Process Control (SPC) Charts • Read Ch 8 • Estimation Procedures

More Related