Lecture 18 - 211 Fall 2009

Lecture 18 - 211 Fall 2009 - I Introduction II Descriptive...

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Unformatted text preview: I. Introduction II. Descriptive Statistics III. Probability, Random Variables and Sampling Distributions. A. Probabilty (Chapter 5) 1. Basic Concepts. 2. Rules of Probability. B. Random Variables (Chapter 5) C. Normal Distribution (Chapter 6) D. Sampling Distribution (Chapter 7) B. Random Variables 1. Discrete RVs – What are they? Discrete Part Values obtained by counting • Old stuff – we’ve discussed discrete variables • Numeric values assigned to outcomes of experiment Random Part Random Part • The values occur by chance • There’s a probability for each value. Probability Distribution • Random values sample space. • Each with a probability • Histogram is a probability distribution We know N, Example – 3 Tosses of a coin Tree diagram for all possible outcomes: 3rd Toss 2nd Toss 1st Toss H 0.5 0.5 T 0.5 H T 0.5 H 0.5 T 0.5 H 0.5 T 0.5 H 0.5 T 0.5 (H & H & H) Outcomes: H (H,H,H) (H,H,T) (H,T,H) (H,T,T) (T,H,H) (T,H,T) (T,T,H) (T,T,T) Probabilities: 0.5 T 0.5 H T 0.5 1 1. Discrete Random Variables Example – 3 Tosses of a coin Probability Distribution for h: h 0 1 2 3 P(h) P(h) Or: 0.4 0.3 0.2 0.1 0 0 1 2 3 h 1. Discrete Random Variables Describing the Distribution – What measures? Example: 3 Tosses of a Coin h 0 1 2 3 Totals P(H = h) 0.125 0.375 0.375 0.125 1.000 2 1. Discrete Random Variables Interpretation of Expected Value • E(h) = 1.500. • But, we can never get 1.5 heads on a toss Summary - DRVs X is a DRV, possible values are countable. countable Values Values for X occur randomly – by chance. All All possible values for X and their probabilities constitute the probability distribution of X. probability How do we describe probability distributions? How do we describe probability distributions? Expected Value or Mean: E [ X ] = ∑ x ⋅ P ( x) = μ x Standard Deviation: σX = ∑( x − E [ X ]) ⋅ P( x) 2 Example: Number of Siblings (S) s 0 1 2 3 4 5 6 7 8 9 14 Total Freq. 14 118 95 31 12 4 2 1 1 1 1 280 s 0 1 2 3 4 5 6 7 8 9 14 Total P(S = s) 3 You are completely unprepared for a multiple choice exam of 20 questions. There are 4 possible answers for each question and you guess on each question. Draw a tree diagram to show all possible exam results and their probabilities. Create a probability distribution for the discrete random variable, number of questions answered number correctly. 3. Binomial Distributions Theoretical distribution, a mathematical model. • Discrete Random variables - generated by processes distribution mathematical model • Theoretical distribution – mathematical model of the process. • No experimentation or survey necessary – theoretical distribution gives probabilities. • Processes satisfy binomial conditions. 3. Binomial Distributions Special case of a discrete random variable. Gives probabilities for x successes in n trials Series of n “Bernoulli Trials:” 4 ...
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