IOE366-Ch12-Simple+Regression

IOE366-Ch12-Simple+Regression - Ch 12 Simple Linear...

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Ch 12: Simple Linear Regression and Correlation 12.1 Simple Linear Regression Model 12.2 Estimation of Model Parameters 12.3 Inference about the Slope 12.4 Inference about Future Y and E(Y) 12.5 Correlation 1 Ch 12 - Learning Objectives ± Use SLR for empirical model building nderstand Method of Least Squares ± Understand Method of Least Squares ± Analyze residuals to determine aptness of odels models ± Test hypotheses and construct confidence intervals ± Use SLR models to predict ± Apply correlation models 2 Regression Analysis and Empirical Models Regression Analysis is statistical technique for modeling and ± a statistical technique for modeling and investigating the relationship between two r more variables or more variables. n mpirical Model An Empirical Model is ± a statistical model based on observations ther than theory 3 rather than theory. Scatterplots ± Graphically show empirical relationships etween two variables between two variables ± Example: Torque vs Pressure 4
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5 http://www.enerpac.com/html/products/industrial_tools/Bolting/Cat_pages/HXD_torqpress_325US.pdf Air Pressure Torque i 120.0 psi Nm 50 66.4 52 69.5 4 0 4 100.0 110.0 ue 54 70.4 56 71.9 58 75.4 60 82.8 80.0 90.0 Torq u 62 79.6 64 89.7 66 93.9 60.0 70.0 0 0 0 0 0 68 93.3 70 97.3 72 100.8 4 03 9 40 50 60 70 80 Air Pressure 74 103.9 76 103.4 78 113.6 80 108.8 Y: Torque X: Air Pressure Other variables? What information does one variable (X) provide about another (Y) ? Y is the Dependent Variable The variable one predicts (e.g. Torque) is the Independent Variable X is the Independent Variable Predictor Variable Explanatory Variable Want to estimate the best linear model describing the relationship between Torque and Air Pressure 7 pq 12.1 Simple Linear Regression Model Equation of a Straight Line with Error: 01 Y x β βε =+ + Y = dependent variable, random variable. = independent variable, nothing random about it. , = intercept and slope, unknown constants. 0 1 ε = random error term, ssumption: normally distributed with 8 Assumption: ε is normally distributed with E( ε )=0 and V( ε )= σ 2
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Model has Two Components: ) Deterministic 1) Deterministic 2) Random 2 1 Assume () ~ N (0 ,) y x εβ β σ =− + 9 01 ii i ββ Simple Linear Regression Model the mean response of Y for a given X with inimal error minimal error. rror Terms Observed Value Error Terms + * Y i * Fit Line Y i * 10 * Simple Linear Regression We estimate the line with ˆ Y ’s: ˆˆ ˆ =+ (fitted value) and estimate the errors with ) + ey (residuals) 11 Criterion for Choosing “Best Fit” Line Error Terms: distance from fit line (predicted values) to observed data point (observed values) Criterion: 12 Called “Least Squares” or “Ordinary Least Squares”
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Least Squares Criterion 22 01 Minimize ( ) nn ii y x εβ β = ∑∑ 11 == ince we don't know we need to Since we don t know , ββ estimate them using ˆˆ , 2 13 Minimize ( –– ) ey = Solving 2 Equations … 2 n e 1 0 2( ) 0 ˆ i yx ∂β = = =• = 2 1 1 ) 0 ˆ xy = = = 14 Solution () xx yy −− 1 1 2 ˆ ) = = 1 = 15 = Computational Formula, for 1 ˆ S = 1 ( )( ) / Sx =− ( ) 2 2 / 16
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Example #1 : MINITAB Regression Output Torque (Nm) = -14.2456 + 1.58522 Air Pressure Regression Plot 115 S = 2.68025 R-Sq = 97.1 %
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This note was uploaded on 03/17/2010 for the course IOE 366 taught by Professor Garyherrin during the Winter '10 term at University of Michigan-Dearborn.

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IOE366-Ch12-Simple+Regression - Ch 12 Simple Linear...

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