AMS 5 Lecture 10: Class 10 - Regression: Additional Comments

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Applied Math and Statistics
Yonatan Katznelson

AMS 5 Lecture 10 4/26/2017 (8:00-9:05) Regression: Additional Comments Interpreting the coefficients • Example: o x(j) = mathSAT score of jth student o y(j) = Freshman GPA _ o x = 550, SD(x) = 80 _ o y = 2.6, SD(y) = 0.6, r = 0.4 o Regression equation ▪ Slope coefficient: B1 = r*SD(y)/SD(x) = 0.003 _ _ ▪ Intercept coefficient: B0 = y – B1*x o Equation: ^y(j) = .95 + .003x(j)  predicted average statistics of all students who scored x(j) on math SAT. ▪ .95 and .003 = sample statistics o B1 = .003 the average increase in GPA for every 1pt percentage in math SAT score. ▪ Remember: Correlation does not mean causation o B(0) = .95  Avg. GPA of all students who scored 0 on math SAT.  that’s meaningless, since you can’t score 0 on the SAT; lowest score is 200. o Regression based on this window (ex: 200-800)…Doesn’t help with areas outside the window  aka, you can’t predict trends way outside your range with regression • Example: o In a hypothetical BP on Cig/Day regression ▪ ^y(j) = 119.48 + .56x(j) o y(j) = BP of jth man o x(j) = # cigs/day smoked by jth man o 0.56 = avg. increase in BP for each cig./day smoked o 119.48 = Avg BP of men who do not smoke • Given a set of paired data, you can compute the correlation coefficient o The correlation between x and y is the same as the correlation between y and x over r. ▪ There are two regression equations! • Example: o SAT/GPA example ▪ ^y(j) = .95 + 0.003x(j) ▪ ^x(j) = U(0) + U(1)y(j) ▪ U(1) = r*SD(x)/SD(y)
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