04-11-2011 Linear Regression

04-11-2011 Linear Regression - 4/13/2011 Lecture 21: April...

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•4/13/2011 •1 Lecture 21: April 11, 2011 • Chapter 12: – Last class I covered through p. 498 – Today: pages 499-502 – Next Class: pages 502-middle page 505 • Homework No. 12: – Regression basics, through page 501 – Opens Wednesday. – Closes Friday at 9:00 a.m. One Thing We Did Last Class : We predicted Y for each value of X. • X Y Predicted Y = 6 + 0.20*X • 30 10 • 45 16 • 50 17 • 55 20 • 70 17 Important Notation As We Move Forward : predicted value of Y i •e i : a new item! – amount by which Y i and differ – The “e” stands for “error.” – It is prediction or forecast error associated with the model that we have constructed. • So, mathematically: –e i = Y i - i ˆ Y i ˆ Y i ˆ Y
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•4/13/2011 •2 Put X i , Y i , , and e i all in one table •X i Y i __ e i • 30 10 12 • 45 16 15 • 50 17 16 • 55 20 17 • 70 17 20 • Notice: each e i is a bad thing. • I’d love to eliminate them. i ˆ Y i ˆ Y I will now get a picture of e i for each observation. • To do this • I’ll return to my two dimensional graph. • I will put Y i and on the vertical axis. – Note: is a new addition to my picture. • I will put X i on the horizontal axis. – Attention! Each X i is now also paired with a . • I will connect the dots for all of the s. – This gives me my fitted regression line. • I will draw the vertical distance between each Y i and its corresponding . i ˆ Y i ˆ Y i ˆ Y i ˆ Y i ˆ Y Start: Put X i , Y i , , and e i all in one graph X i Y i __ e i 30 10 12 -2 45 16 15 1 50 17 16 1 55 20 17 3 70 17 20 -3 i ˆ Y i ˆ Y
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•4/13/2011 •3 Introduce a Line That Fits Through The Points X i Y i __ e i 30 10 12 -2 45 16 15 1 50 17 16 1 55 20 17 3 70 17 20 -3 i ˆ Y Measure the vertical distances. Each is an e i . X i Y i __ e i 30 10 12 -2 45 16 15 1 50 17 16 1 55 20 17 3 70 17 20 -3 i ˆ Y Numbers reported as vertical distances match the last column of numbers X i Y i __ e i 30 10 12 -2 45 16 15 1 50 17 16 1 55 20 17 3 70 17 20 -3 i ˆ Y
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•4/13/2011 •4 Summary Up To Now • Each value of X – call it X i – can be inserted into the regression equation to yield: = 6 + .2(X i ) • Interpretation of slope coefficient: b 1 = .2 – For each additional dollar of income, a household’s food expenditures go up by 20 cents. • Two important visual items: – The plot of (on the vertical axis) against X i (on the horizontal axis) yields a regression line .
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04-11-2011 Linear Regression - 4/13/2011 Lecture 21: April...

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