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**Simple Linear Regression**

Chapter 11 Simple Linear Regression Slides for Optional Sections No optional sections

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**Probabilistic Models General form of Probabilistic Models**

Y = Deterministic Component + Random Error where E(y) = Deterministic Component

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Probabilistic Models First Order (Straight-Line) Probabilistic Model

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**Probabilistic Models 5 steps of Simple Linear Regression**

Hypothesize the deterministic component Use sample data to estimate unknown model parameters Specify probability distribution of , estimate standard deviation of the distribution Statistically evaluate model usefulness Use for prediction, estimatation, once model is useful

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**Fitting the Model: The Least Squares Approach**

Reaction Time versus Drug Percentage Subject Amount of Drug x (%) Reaction Time y (seconds) 1 2 3 4 5

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**Fitting the Model: The Least Squares Approach**

Least Squares Line has: Sum of errors (SE) = 0 Sum of Squared errors (SSE) is smallest of all straight line models Formulas: Slope: y-intercept

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**Fitting the Model: The Least Squares Approach**

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**Model Assumptions Mean of the probability distribution of ε is 0**

Variance of the probability distribution of ε is constant for all values of x Probability distribution of ε is normal Values of ε are independent of each other

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An Estimator of 2 Estimator of 2 for a straight-line model

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**Assessing the Utility of the Model: Making Inferences about the Slope 1**

Sampling Distribution of

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**Assessing the Utility of the Model: Making Inferences about the Slope 1**

A Test of Model Utility: Simple Linear Regression One-Tailed Test Two-Tailed Test H0: β1=0 Ha: β1<0 (or Ha: β1>0) Ha: β1≠0 Rejection region: t< -tα (or t< -tα when Ha: β1>0) Rejection region: |t|> tα/2 Where tα and tα/2 are based on (n-2) degrees of freedom

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**Assessing the Utility of the Model: Making Inferences about the Slope 1**

A 100(1-α)% Confidence Interval for 1 where

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**The Coefficient of Correlation**

A measure of the strength of the linear relationship between two variables x and y

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**The Coefficient of Determination**

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**Using the Model for Estimation and Prediction**

Sampling errors and confidence intervals will be larger for Predictions than for Estimates Standard error of Standard error of the prediction

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**Using the Model for Estimation and Prediction**

100(1-α)% Confidence interval for Mean Value of y at x=xp 100(1-α)% Confidence interval for an Individual New Value of y at x=xp where tα/2 is based on (n-2) degrees of freedom

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