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 denoting the height at maturity (in centimeters) of the father's oldest son. The data are given in tabular form and also displayed in the Figure 1 scatter plot. Also given are the products of the heights of fathers and heights of sons for each of the fifteen pairs. (These products, written in the column labelled "xy," may aid in calculations.)

Height of father, x (in centimeters)	Height of son, y (in centimeters)		xy
182.2	178.9		32595.58
180.7	189.6		34260.72
190.0	194.8		37012
160.3	172.0		27571.6
170.6	181.7		30998.02
161.7	165.6		26777.52
190.0	189.2		35948
185.9	186.9		34744.71
171.8	171.0		29377.8
176.5	174.8		30852.2
192.5	189.0		36382.5
173.1	176.5		30552.15
201.3	191.2		38488.56
155.9	174.2		27157.78
188.3	176.1		33159.63

Send data to Excel

y

150

160

170

180

190

200

210

x

150

160

170

180

190

200

210

Figure 1

Answer the following. Carry your intermediate computations to at least four decimal places, and round your answer as specified below.

What is the value of the sample correlation coefficient for these data? Round your answer to at least three decimal places.

Answer 1

we have given the data set

Height of father, x	Height of son, y
(in centimeters)	(in centimeters)
182.2	178.9
180.7	189.6
190	194.8
160.3	172
170.6	181.7
161.7	165.6
190	189.2
185.9	186.9
171.8	171
176.5	174.8
192.5	189
173.1	176.5
201.3	191.2
155.9	174.2
188.3	176.1

using excel>data>data analysis>Correlation

we have

	X	Y
X	1
Y	0.790407	1

the value of the sample correlation coefficient for these data is 0.790