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). She feels that the student s first year GPA will be related to thevalues of these two known marketer is interested in theeffectof changing shelf height (x1) and shelf width (x2)onthe weekly sales (y) of her brand of laundry detergent in a grocery psychologist is interested in testing whether the amount of time to become proficient in aforeign language (y) is related to the child s age (x).
1.3 Least Squares Estimation of β0 and β1 We now have the problem of using sample data to compute estimates of the parameters β0 and β1. First, we take a sample of n subjects, observing values y of the response variable and x of the predictor variable. We would like to choose as estimates for β0 and β1, the values b0 and b1 that
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