Following his mentor Francis Galton, the mathematician/statistician Karl Pearson conducted comprehensive studies comparing hereditary traits between members of the same family. Figure 5.10 depicts the measurements of the heights of 1078 fathers and their adult sons (one son per father). The data is available from the book's GitHub site as pearson.csv.

(a) Show that sons are on average 1 inch taller than the fathers.

(b) We could try to "explain" the height of the son by taking the height of his father and adding 1 inch. The prediction line \(y=x+1\) (red dashed) is given Figure 5.10. The black solid line is the fitted regression line. This line has a slope less than 1 , and demonstrates Galton's "regression" to the average. Find the intercept and slope of the fitted regression line.

Transcribed Image Text:

15 75 Height Son (in) 65 65 70 70 60 60 58 60 62 64 66 68 70 72 74 76 Height Father (in) Figure 5.10: A scatterplot of heights from Pearson's data.