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Unformatted text preview: 1 MA 214: Applied Statistics Instructor : Remus Oşan Lecture 15 – July 22, 2010 Chapter 11: Simple linear regression 11.6 Using the Model for Estimation and Prediction 11.7 A Complete Example Review and Problems s 11.1 Probabilistic Models s 11.2 Fitting the Model: The Least Squares Approach s 11.3 Model Assumptions s 11.4 Assessing the Utility of the Model: Making Inferences About the Slope β 1 s 11.5 The Coefficients of Correlation and Determination s 11.6 Using the Model for Estimation and Prediction s 11.7 A Complete Example Chapter 11. Simple Linear Regression 2 11.6 Using the Model for Estimation and Prediction s Two uses: s For example, in the drug reaction problem, we can use our parameters to estimate the expectation value, namely reaction time, as a function of the amount of drug in the bloodstream (points from the data set) s We can also make predictions about y if a new amount of drugs in the bloodstream x would be induced. 3 11.6 Using the Model for Estimation and Prediction s A. We are attempting to estimate the mean value of y for a very large number of experiments at the given x value. s B: In the second case, we are trying to predict the outcome of a single experiment at the given x value. s Interesting question: which can be accomplished with the greater accuracy, finding y_bar or estimating y at a new value for x? 4 What does the model predict in relationship to the real data s Estimated mean reaction time for all people when x = 4 (the drug is 4% of the blood content) Both estimated mean and predicted values are 2.7 s x x y ˆ 7 . 1 . ˆ ˆ 1 +- = + = β β 7 . 2 4 7 . 1 . ˆ , 4 ˆ = ⋅ +- = = y x when 5 What does the model predict in relationship to the real data s As a matter of fact we CAN be more precise about predicting the mean expected response that about predicting the individual responses s Think about a population response: after we accumulate enough evidence we can compute the population mean fairly accurately s The variation of individual responses may be still quite high, therefore making it difficult to predict individual responses s The difference between these two uses of the model lies in the accuracies of the estimate and the prediction 6 2 What does the model predict in relationship to the real data s Sampling Errors for the Estimator of the Mean of y and the Predictor of an Individual New Value of y s 1. The standard deviation of the sampling distribution of the estimator y-hat of the mean value of y at a specific value of x, say x p is s where σ is the standard deviation of the random error ε . We refer to σ y-ha t as the standard error of y-hat. 7 xx p y SS x x n 2 ) ( 1- + = σ σ ) What does the model predict in relationship to the real data s The standard deviation of the prediction error for the predictor y-hat of an individual new y value at a specific value of x is s where σ is the standard deviation of the random error ε . We refer to σ y - y-hat as the standard error of prediction 8 xx...
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