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Simple Linear Regression: Probabilistic Models & The Least Squares Approach

This chapter explores the general form of probabilistic models in simple linear regression, including the deterministic component and random error. It covers the five steps of fitting a model, the least squares approach, model assumptions, assessing utility, and using the model for estimation and prediction.

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Simple Linear Regression: Probabilistic Models & The Least Squares Approach

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  1. Chapter 11 Simple Linear Regression

  2. Probabilistic Models • General form of Probabilistic Models • Y = Deterministic Component + Random Error • where • E(y) = Deterministic Component

  3. Probabilistic Models • First Order (Straight-Line) Probabilistic Model

  4. 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

  5. Fitting the Model: The Least Squares Approach

  6. 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

  7. Fitting the Model: The Least Squares Approach

  8. 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

  9. An Estimator of 2 • Estimator of 2 for a straight-line model

  10. Assessing the Utility of the Model: Making Inferences about the Slope 1 • Sampling Distribution of

  11. Assessing the Utility of the Model: Making Inferences about the Slope 1

  12. Assessing the Utility of the Model: Making Inferences about the Slope 1 • A 100(1-α)% Confidence Interval for 1 • where

  13. The Coefficient of Correlation • A measure of the strength of the linear relationship between two variables x and y

  14. The Coefficient of Determination

  15. 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

  16. 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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