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Simple Linear Regression: Probabilistic Models & The Least Squares Approach

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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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Presentation Transcript

  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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