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problems 2.1, 2.2 and 2.3 pages 88-92 How Fitting! – The Least Squares Line and How Fitting! – The Least Squares Exponential Fitting Models to Data

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YearAverage IQ 1932100 1947106 1952108 1972112 1978113 1997118 t (years since 1932) Average IQ 0100 15106 30108 40112 46113 65118 Data Driven Models What function (linear or exponential) describes the pattern made by the following IQ data?

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tIQ 0100 15106 30108 40112 46113 65118 Data Driven Models Plot data points and take a close look. Linear or Exponential or Neither ?

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Data Driven Models What function (linear) best describes the pattern made by the following IQ data? tIQ 0100 15106 30108 40112 46113 65118 m = (106-100)/15 =.40 b = 100 y1(t) = 100 +.4*t To measure the goodness of this model, we use the sum of the square errors (as outlined on pages 88 & 89).

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T (years since 1932) y(t) (IQ) y1(t) 100+.4t Error (y-y1) Square error (y-y1) 2 0100 00 15106 00 30108112416 40112116416 46113118.45.429.16 65118126864 Sum of Square Errors (SSE) 0 + 0 + 16 + 16 + 29.16 + 64 = 125.16 Goodness of Fit y1(t) = 100 +.4*t We compute the SSE.

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Data Driven Models What function (linear) best describes the pattern made by the following IQ data? tIQ 0100 15106 30108 40112 46113 65118 m = (118-113)/19 =.2631578947 y2(t) = 118 +.2631578947*(x-65) To measure the goodness of this model, we use the sum of the square errors (as outlined on pages 88 & 89).

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T (years since 1932) y(t) (IQ) y2(t)Error (y-y2) Square error (y-y2) 2 0100100.895.895.801 15106104.842-1.1581.341 30108108.790.789.623 40112111.421-.579.335 46113 00 65118 00 Sum of Square Errors (SSE).801 + 1.341 +.623 +.335 + 0 + 0 = 3.10 Goodness of Fit y2(t) = 118 +.2631578947*(x-65) We compute the SSE.

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SSE = 3.10SSE = 125.16 Goodness of Fit Compare Linear Models

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Data Driven Models How do we find the best (smallest SSE) linear model? Use Maple’s LinearFit command

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T (years since 1932) y(t) (IQ) y10(t)Error (y-y10) Square error (y-y10) 2 0100100.718.718.516 15106104.751-1.2501.561 30108108.783.783.613 40112111.471-.529.279 46113113.084.084.007 65118118.192.192.037 Sum of Square Errors (SSE).516 + 1.561 +.613 +.279 +.007 +.037 = 3.013 Goodness of Fit y10(t) = 100.718+.269*t We compute the SSE.

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Data Driven Models What function (exponential) best describes the pattern made by the following IQ data? tIQ 0100 15106 30108 40112 46113 65118 c = 100 a = (113/100) ^ (1/46) = 1.002660438 y3(t) = 100 * (1.002660438)^t To measure the goodness of this model, we use the sum of the square errors (as outlined on pages 88 & 89).

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T (years since 1932) y(t) (IQ) y3(t)Error (y-y3) Square error (y-y3) 2 0100 00 15106104.066-1.9343.741 30108108.297.297.088 40112111.213-.787.620 46113 00 65118118.851.851.724 Sum of Square Errors (SSE) 0 + 3.741 +.088 +.620 + 0 +.724 = 5.173 Goodness of Fit y3(t) = 100 * (1.002660438)^t We compute the SSE.

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Data Driven Models How do we find the best (smallest SSE) exponential model? Use Maple’s LinearFit command

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T (years since 1932) y(t) (IQ) y20(t)Error (y-y20) Square error (y-y20) 2 0100100.874.874.764 15106104.681-1.3191.739 30108108.632.632.399 40112111.348-.652.425 46113113.011.011.000 65118118.440.440.194 Sum of Square Errors (SSE).764 + 1.739 +.399 +.425 +.000 +.194 = 3.521 Goodness of Fit y20(t) = 100.874 * (1.002472751)^t We compute the SSE.

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Goodness of Fit CONCLUSIONS y10(t) = 100.718+.269*t with SSE: 3.013 y20(t) = 100.874 * (1.002472751)^t with SSE: 3.521 Linear Model is best! To predict IQ in 2008: y10(76) = 100.718 +.269*(76) = 121.162 HOMEWORK: See Maple Handout

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