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# Linear regression

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### Linear regression

1. 1. EXAMPLE • Example of simple linear regression which has one independent variable.
2. 2. Least Squares Estimation of b0, b1 • b0  Mean response when x=0 (y-intercept) • b1  Change in mean response when x increases by 1 unit (slope) • b0, b1 are unknown parameters (like m) • b0+b1x  Mean response when explanatory variable takes on the value x • Goal: Choose values (estimates) that minimize the sum of squared errors (SSE) of observed values to the straight-line: 2 n   2^ ^ ^  ^   ^ ^y  b 0 b1 x SSE  i 1  yi  y i   i 1  yi   b 0  b 1 xi   n     
3. 3. The least squares estimate of the slopecoefficient β1 of true regression line is β1= Σ(Xi-X’)(Yi-Y’) Σ (Xi-X’)2 The least squares estimate of the intercept β0 of true regression line is β0 = Y’ – β1x’
4. 4. X YTemperature Sales 63 1.52 70 1.68 73 1.8 75 2.05 80 2.36 82 2.25 85 2.68 88 2.9 90 3.14 91 3.06 92 3.24 75 1.92 98 3.4 100 3.28 92 3.17 87 2.83 84 2.58 88 2.86 80 2.26 82 2.14 76 1.98
5. 5. Ice Cream Sales 43.5 32.5 21.5 10.5 0 0 20 40 60 80 100 120
6. 6. THANK YOU