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2010-07-20-Lecture-15-Ch-11-6-Review-problems-handouts

2010-07-20-Lecture-15-Ch-11-6-Review-problems-handouts -...

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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 square6 11.1 Probabilistic Models square6 11.2 Fitting the Model: The Least Squares Approach square6 11.3 Model Assumptions square6 11.4 Assessing the Utility of the Model: Making Inferences About the Slope β 1 square6 11.5 The Coefficients of Correlation and Determination square6 11.6 Using the Model for Estimation and Prediction square6 11.7 A Complete Example Chapter 11. Simple Linear Regression 2 11.6 Using the Model for Estimation and Prediction square6 Two uses: square6 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) square6 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 square6 A. We are attempting to estimate the mean value of y for a very large number of experiments at the given x value. square6 B: In the second case, we are trying to predict the outcome of a single experiment at the given x value. square6 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 square6 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 . 0 1 . 0 ˆ ˆ 1 0 + - = + = β β 7 . 2 4 7 . 0 1 . 0 ˆ , 4 ˆ = + - = = y x when 5 What does the model predict in relationship to the real data square6 As a matter of fact we CAN be more precise about predicting the mean expected response that about predicting the individual responses square6 Think about a population response: after we accumulate enough evidence we can compute the population mean fairly accurately square6 The variation of individual responses may be still quite high, therefore making it difficult to predict individual responses square6 The difference between these two uses of the model lies in the accuracies of the estimate and the prediction 6
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2 What does the model predict in relationship to the real data square6 Sampling Errors for the Estimator of the Mean of y and the Predictor of an Individual New Value of y square6 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 square6 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 square6 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 square6 where σ is the standard deviation of the random error ε . We refer to σ y - y-hat as the standard error of
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