Question

Sir Francis Galton, in the late 1800s, was the first to introduce the statistical concepts of regression and correlation. He studied the relationships between pairs of variables such as the size of parents and the size of their offspring. Data similar to that which he studied are given below, with the variable x denoting the height (in centimeters) of a human father and the variable y denoting the height at maturity (in centimeters) of the father's oldest (adult) son. The data are given in tabular form and also displayed in the Figure 1 scatter plot, which gives the least-squares regression line as well. The equation for this line is y = 91.41+0.50x of father, Height Height of son, y in centimeters) Un centimeters) 191.3 181.7 162.1 182.2 185.4 194.3 189.9 167.0 176.5 186.3 181.5 210 170.9 161.6 191.6 158.6 201.9 74.0 190.6 174.2 90.1 178.9 88.6 172.3 87.4 173.3 176.7 202.0 192.9 175.5 188.6 Figure 1

Answer 1

1) Since there is a positive correlation between x and y

So heights of sons that are less than mean heights of sons are paired with heights of fathers that are less than mean height of fathers

2) Since there is a positive correlation between x and y

So as x increases y also increases.

So put Increase

3) Put $x=186.6$ in $hat{y}=91.41+0.50x$ we get

So height of son is 184.71 cm

4) Put

$x=185.4 \Rightarrow hat{y}=91.41+0.50(185.4)=184.11$

Height of son is 184.11cm