Transcription of 1 Simple Linear Regression I – Least Squares Estimation
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1 Simple Linear Regression I – Least Squares Estimation
1 Simple Linear Regression I Least Squares EstimationTextbook , we have worked with a random variablexthat comes from a population that isnormally distributed with mean and variance 2. We have seen that we can writexin termsof and a random error component , that is,x= + . For the time being, we are going tochange our notation for our random variable fromxtoy. So, we now writey= + . We will nowfind it useful to call the random variableyadependentorresponse variable. Many times, theresponse variable of interest may be related to the value(s) of one or more known or controllableindependentorpredictor variables. Consider the following situations:LR1A college recruiter would like to be able topredicta potential incoming student s first yearGPA (y) based on known information concerning high school GPA (x1) and college entranceexamination score (x2).
1.2 A Linear Probabilistic Model The adjustment people make is to write the mean response as a linear function of the predictor variable. This way, we allow for variation in individual responses (y), while associating the mean linearly with the predictor x. The model we fit is as follows: E(y|x)=β0 +β1x, and we write the individual responses as
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