11 Chapter Linear regression Example Old Faithful Geyser in Yellowstone National Park, Wyoming, derives its name and its considerable fame from regularity (and beauty) of its eruptions. As they do with most geysers in the park, rangers post the predicted times of eruptions on signs nearby, and people gather beforehand to witness the show. R.A. Hutchinson, a park geologist, collected measurements of the eruption durations (X, in minutes) and the subsequent intervals before next eruption (Y , in minutes) over an 8 day period. This data is as follows 4 75 4 1 78 4.4 2 80 4.3 3 76 4.5 4 73 3.7 1 74 3.9 2 56 1.7 3 82 3.9 4 67 3.7 1 68 4 2 80 3.9 3 84 4.3 4 68 4.3 1 76 4 2 69 3.7 3 53 2.3 4 86 3.6 1 80 3.5 2 57 3.1 3 86 3.8 4 72 3.8 2 90 4 3 51 1.9 1 84 4.1 4 75 3.8 1 50 2.3 2 42 1.8 3 85 4.6 4 75 3.8 1 93 4.7 2 91 4.1 3 45 1.8 4 66 2.5 1 55 1.7 2 51 1.8 3 88 4.7 4 84 4.5 1 76 4.9 2 79 3.2 3 51 1.8 4 70 4.1 3 80 4.6 1 58 1.7 2 53 1.9 4 79 3.7 1 74 4.6 2 82 4.6 3 49 1.9 4 60 3.8 1 75 3.4 2 51 2 3 82 3.5 4 86 3.4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 71 67 81 76 83 76 55 73 56 83 57 71 72 77 4 2.3 4.4 4.1 4.3 3.3 2 4.3 2.9 4.6 1.9 3.6 3.7 3.7 6 6 6 6 6 6 6 6 6 6 6 6 6 6 55 75 73 70 83 50 95 51 82 54 83 51 80 78 1.8 4.6 3.5 4 3.7 1.7 4.6 1.7 4 1.8 4.4 1.9 4.6 2.9 7 7 7 7 7 7 7 7 7 7 7 7 7 81 53 89 44 78 61 73 75 73 76 55 86 48 3.5 2 4.3 1.8 4.1 1.8 4.7 4.2 3.9 4.3 1.8 4.5 2 8 8 8 8 8 8 8 8 8 8 8 8 8 77 73 70 88 75 83 61 78 61 81 51 80 79 4.2 4.4 4.1 4.1 4 4.1 2.7 4.6 1.9 4.5 2 4.8 4.1
1. Linear y equation: = ax + b . b y-intercept a slope. Example 1 y = 2 3x. a) Compute y for x = 1, 2. b) Are (1, 0) and (2, 4) on the line? c) If x is changed from 1 to 2, what will the change in y be? 2. Fit data to a line Example 2 (continue) Let x 1 2 3 4 y 1 1 2 3 Draw scatter diagram 3. Fit data into a line (least squares technique) Least squares technique The coecient n xi y i xi y i , b = y a . x a= n x2 ( xi )2 i a is called the sample regression coecient y = ax + b is called the line of regression of y on x. Example 2 (continue) Let x 1 2 3 4 y 1 1 2 3 a) Find a and b. Write down the regression line. b) What value would you predict if x = 6? SOLUTION a) a = 0.7 , b = 0 . y = 0.7x . b) 4.2. 4. Correlation coecient (Coecient of linear correlation): Motivation Formula (Pearson product-moment formula) (xi x)(yi y ) (xi x)2 (yi y )2 n n 1 r 1. x2 ( i x i yi xi yi
2 yi (
r...