Chapter 8

# Chapter 8 - IENG 213 Probability and Statistics for...

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IENG 213: Probability and Statistics for Engineers Instructor: Steven E. Guffey, PhD, CIH © 2002-2009

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2 Random Sampling Determine sample mean and sample variance as estimators of true mean and variance. Populations and sampling – Population: totality of observations with which we are concerned – Size: number of observations. Finite, but could be very large – Each observation in a population is a value of the random variable, X, having probability distribution f(x). – Value could be anything: binary, count, measurement Sample: subset of a population – Sample should be representative of the larger population – Selecting so that some elements are under or over represented produces biased estimate of mean and variance Most robust way to avoid bias is to sample randomly
3 Definition 8.3 Random Sample Let X 1 , X 2 , X 3 , … X n be n independent variables, each having the same probability distribution f(x). They are a random sample of size n from the population f(x) if joint probability distribution is: f(x 1 , x 2 , x 3 ,.…, x n ) = f(x 1 ) f(x 2 ) f(x 3 ) . ...f(x n ) I.e., values are independent

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4 8.2 Some Important Statistics Use random samples to estimate unknown population parameter values. Statistic: any function of the random variables constituting a random variable Central tendency: center of the data Sample Mean, median and mode Definition 8.5: Sample mean of a random sample, X 1 , X 2 , X 3 , … X n of sample size n : n X X n i i = = 1
5 Sample Variance Variance of the sample, s 2 , is an estimate of the variance of the population, σ 2 . 1 1 2 2 - - = = n X X s n i i     2 2 1 1 1 1 n n i i i i X X n n N-1: Because it is a sample, not population     2 2 1 1 1 n n i i i i n X X n n

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6 Sample Variance Variance of the sample, s 2 , is an estimate of the variance of the population, σ 2 .   2 2 1 n i i X X n Sample ( 29 1 1 2 2 - - = = n X X s n i i Population
7 Example 8.1 Compute Mean and Variance Too simple to take class time

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8 Example 8.2 Compute Sample Std Dev Too easy to take class time
9 HW p. 234-235, prob 1, 3, 5, 7, 13

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10 8.3 Data Displays and Graphical Methods Box and Whisker Plot (Boxplot) Shows range from 25% to 75% as box Median 75% 95% outliers 25% Note: in this case the median is not the center of the box Diagnostic tool Look for symmetry “Outliers”: sometimes defined as 1.5 times the 25-75% range Whiskers show extreme observations Be able to interpret and label
11 Quantile Plot of Cumulative Distribution Know what tells you about normality Definition 8.8: A quantile of a sample, q(f), is a value for which the specified fraction, f, of the data values is less than or equal to q(f) q(0.5) = median q(0.25)= 25% Quantile plot: data values on vertical axis; empirical assessment of the fraction of the observations exceeded by the data value on the horizontal axis.

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## This note was uploaded on 04/03/2009 for the course IENG 213 taught by Professor Staff during the Spring '08 term at WVU.

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Chapter 8 - IENG 213 Probability and Statistics for...

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