# 8oct08 - 45 730 Probability Decision Making Topics for...

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45 730 Probability & Decision Making (10/8/08) Topics for today’s class: Confidence Intervals Population Mean Population Proportion More Simulation Examples

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A Population is the set of all items or individuals of interest Examples: All likely voters in the next election All parts produced today All possible outcomes of iteration of simulation A Sample is a subset of the population Examples: 1000 voters selected at random for interview A few parts selected for destructive testing Outcomes of n runs of the simulation model Populations and Samples
Population vs. Sample a b c d e f g h i j k l m n o p q r s t u v w x y z Population Sample b c g i n o r u y

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Making statements about a population by examining sample results Sample statistics Population parameters (known) Inference (unknown, but can be estimated from sample evidence) Sample Population Inferential Statistics
Sampling Distributions A sampling distribution is a distribution of all of the possible values of a statistic for a given size sample selected from a population

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Chapter 7 Outline Sampling Distributions Sampling Distribution of Sample Mean Sampling Distribution of Sample Proportion Sampling Distribution of Sample Variance
Developing a Sampling Distribution Assume there is a population … Population size N=4 Random variable, X, is age of individuals Values of X: 18, 20, 22, 24 (years) A B C D

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.25 0 18 20 22 24 A B C D Uniform Distribution P(x) x (continued) Summary Measures for the Population Distribution: Developing a Sampling Distribution 21 4 24 22 20 18 N X μ i = + + + = = 2.236 N μ) (X σ 2 i = - =
1 st 2 nd Observation Obs 18 20 22 24 18 18,18 18,20 18,22 18,24 20 20,18 20,20 20,22 20,24 22 22,18 22,20 22,22 22,24 24 24,18 24,20 24,22 24,24 16 possible samples (sampling with replacement) Now consider all possible samples of size n = 2 1st 2nd Observation Obs 18 20 22 24 18 18 19 20 21 20 19 20 21 22 22 20 21 22 23 24 21 22 23 24 (continued) Developing a Sampling Distribution 16 Sample Means

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1st 2nd Observation Obs 18 20 22 24 18 18 19 20 21 20 19 20 21 22 22 20 21 22 23 24 21 22 23 24 Sampling Distribution of All Sample Means 18 19 20 21 22 23 24 0 .1 .2 .3 P(X) X Sample Means Distribution 16 Sample Means _ Developing a Sampling Distribution (continued) (no longer uniform) _
Summary Measures of this Sampling Distribution: Developing a Sampling Distribution (continued) μ 21 16 24 21 19 18 N X ) X E( i = = + + + + = = 1.58 16 21) - (24 21) - (19 21) - (18 N μ) X ( σ 2 2 2 2 i X = + + + = - =

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Comparing the Population with its Sampling Distribution 18 19 20 21 22 23 24 0 .1 .2 .3
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