2702 130 1562 13 4 44130 8 514 13 2 2 21 x xn y x xy n b Example

2702 130 1562 13 4 44130 8 514 13 2 2 21 x xn y x xy

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 2702 . 0 130 1562 13 4 . 44 130 8 . 514 13 2 2 2 1   x x n y x xy n b Example 713 . 0 10 2702 . 0 4154 . 3 1 0 x b y b i i x 2702 . 0 713 . 0 y ˆ 𝑛 = 13 σ ? = 130 σ ? = 44.4 σ ?? = 514.8 σ ? 2 = 1562 σ ? 2 = 171.3
X 2702 . 0 713 . 0 Y ˆ
Interpretation of the Intercept, b 0 b 0 is the estimated average value of Y when the value of X is zero (if x = 0 is in the range of observed x values) Here, no sparrow with age 0, so b 0 = 0.5816 has no logical interpretation, except that it is the part on the Y-axis where the regression line passes through X 2702 . 0 713 . 0 Y ˆ
Interpretation of the slope, b 1 b 1 measures the estimated change in the average value of Y as a result of a one-unit change in X Here, b 1 = 0.2791 tells us that, on the average, the wing length of sparrows increases by 0.2791 cm per day X 2702 . 0 713 . 0 Y ˆ
Least Squares Regression Properties The sum of the residuals from the least squares regression line is 0 ( ) The sum of the squared residuals is a minimum (minimized ) The simple regression line always passes through the mean of the y variable and the mean of the x variable The least squares coefficients are unbiased estimates of β 0 and β 1 0 ) ˆ ( y y 2 ) ˆ ( y y