slide5 - Regression The probability theorist wants to sell...

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1 Regression The probability theorist wants to sell his  house. So, he wants to estimate the  expected price. He found some historic  data about houses sold recently in his  area. The data are shown in the  following table.
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2 Regression house 1 2 3 4 5 6 Area, x  1000  sq.ft. 1.8 2.1 2.3 3.0 3.4 3.6 Price, x  1000$ 202 243 251 289 341 374
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3 Regression Assuming there is a linear relationship    between the house size and its price, find the  estimate of its parameters. The PT’s house is 2800 sq.ft. Find the  estimated expected price and 95% confidence  intervals on the expected price and the actual  sale price. A P + = 1 0 β β
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4 Regression The linear relation is usually not exact.   A realistic model is: Where  β 0  and  β 1  are  regression  coefficients. i i i x Y ε β β + + = 1 0
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5 Random Error Properties of distribution A mean of zero Symmetry around zero An assignment of greater probability to  small errors than to larger ones Errors are assumed to be: Independent Have same variance ( homoscedasticity ) i ε
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6 Method of Least Squares Consider the simple formula: Where the random errors are  independent samples from N(0,    ) How to find the estimators of  β 0  and  β 1 ε σ ε β β + + = x Y 1 0
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7 Least Squares Estimates  Need an index to measure discrepancy  between points and line Focus on vertical disparities between  points and line Sum of the square of the deviations: = = - - = = n 1 i n 1 i 2 i 1 0 i 2 i 1 0 ) x y ( ) , ( L β β ε β β
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