This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 yhat 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 yha t as the standard error of yhat. 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 yhat 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  yhat as the standard error of prediction 8 xx...
View
Full
Document
This note was uploaded on 08/09/2011 for the course STATISTICS  taught by Professor Montilla during the Spring '11 term at Universidad Iberoamericana.
 Spring '11
 Montilla
 Statistics, Least Squares, Linear Regression

Click to edit the document details