ENGR 610 Applied Statistics Fall 2007 - Week 2

ENGR 610Applied StatisticsFall 2007 - Week 2 Marshall University CITE Jack Smith

Overview for Today • Homework problems 1.25, 2.54, 2.55 • Review of Ch 3 • Homework problems 3.27, 3.31 • Probability and Discrete Probability Distributions (Ch 4) • Homework assignment

Homework problems • 1.25 • 2.54 • 2.55

Chapter 3 Review • Measures of… • Central Tendency • Variation • Shape • Skewness • Kurtosis • Box-and-Whisker Plots

Measures ofCentral Tendency • Mean (arithmetic) • Average value: • Median • Middle value - 50th percentile (2nd quartile) • Mode • Most popular (peak) value(s) - can be multi-modal • Midrange • (Max+Min)/2 • Midhinge • (Q3+Q1)/2 - average of 1st and 3rd quartiles

Measures of Variation • Range (max-min) • Inter-Quartile Range (Q3-Q1) • Variance • Sum of squares (SS) of the deviation from mean divided by the degrees of freedom (df) - see pp 113-5 • df = N, for the whole population • df = n-1, for a sample • 2nd moment about the mean (dispersion) (1st moment about the mean is zero!) • Standard Deviation • Square root of variance (same units as variable) • Sample (s2, s, n) vs Population (2, , N)

Quantiles • Equipartitions of ranked array of observations • Percentiles - 100 • Deciles - 10 • Quartiles - 4 (25%, 50%, 75%) • Median - 2 Pn = n(N+1)/100 -th ordered observation Dn = n(N+1)/10 Qn = n(N+1)/4 Median = (N+1)/2 = Q2 = D5 = P50

Measures of Shape • Symmetry • Skewness - extended tail in one direction • 3rd moment about the mean • Kurtosis • Flatness, peakedness • Leptokurtic - highly peaked, long tails • Mesokurtic - “normal”, triangular, short tails • Platykurtic - broad, even • 4th moment about the mean

Box-and-Whisker Plots • Graphical representation of five-number summary • Min, Max (full range) • Q1, Q3 (middle 50%) • Median (50th %-ile) • Shows symmetry (skewness) of distribution

Other Resources • SPSS Tutorial at Statistical Consulting Services • http://www.stats-consult.com/tutorials.html • MathWorld • http://mathworld.wolfram.com • See Probability and Statistics • Wikipedia • http://en.wikipedia.org/wiki/Category:Probability_and_statistics

Homework Problems • 3.27 • 3.31

Chapter 4 Probability • Introduction to Probability • Rules of Probability Discrete Probability Distributions • Probability Distributions • Binomial Distribution • Poisson Distribution • Hypergeometric, Negative Binomial, Geometric Distributions

Introduction to Probability • Probability - numeric value representing the chance, likelihood, or possibility that an event will occur • Classical, theoretical • Empirical • Subjective • Elementary event - a distinct individual outcome • Event - a set of elementary events • Joint event - defined by two or more characteristics

Rules of Probability • A probability P(A) for event A is between 0 (null event) and 1 (certain event) • The complement of P(A) is the probability that A will not occur, and P(not-A) = 1- P(A) • Two events are mutually exclusive if P(A and B) = 0 • If two events are mutually exclusive, then P(A or B) = P(A) + P(B) • If set of events are mutually exclusive and collectively exhaustive, then

Rules of Probability • If two events are not mutually exclusive, thenP(A or B) = P(A) + P(B) - P(A and B), whereP(A and B) is the joint probability of A and B. • The conditional probability of B occurring, given that A has occurred, is given byP(B|A) = P(A and B)/P(A) • If two events are independent, thenP(A and B) = P(A) x P(B) andP(A) = P(A|B) and P(B) = P(B|A) • If two events are not independent, thenP(A and B) = P(A) x P(B|A)

Probability Distributions • A probability distribution for a discrete random variable is complete set of all possible distinct outcomes and their probabilities of occurring, whose sum is 1. • 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.

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 (frequencies) are given by the probability distribution.The standard deviation (X) is then the square root of the variance.

Binomial Distribution • Each elementary event is a Bernoulli event, with one of two mutually exclusive and collectively exhaustive possible outcomes. • 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 • The proportion of trials resulting in xsuccesses, out of n trials, with a constant probability of p, is given by:

Binomial Distribution, cont’d • 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)

Poisson Distribution • Probability for a particular number of discrete events over a continuous interval (area of opportunity) • Assumes a Poisson process (“isolable” event) • Limit case of Binomial distribution for large n • Based only on expectation value ()

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

Other Discrete Probability Distributions • Hypergeometric (pp 159-160) • Bernoulli events, but selected from finite population without replacement • pA/N, where A number ofsuccesses in population N • Approaches binomial for n < 5% of N • Negative Binomial (pp 162-163) • Number of trials (n) until xth success • Binomial with last trial constrained to be a success • Geometric (pp 164-165) • Special case of negative binomial for x = 1 (1st success)

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)

Homework • Ch 4 • Appendix 4.1 • Problems: 4.57,60,61,64 • Read Ch 5 • Continuous Probability Distributions and Sampling Distributions

ENGR 610 Applied Statistics Fall 2007 - Week 2