Can anyone help me to answer this? Thank you. Recall the father son height data

Question

Can anyone help me to answer this? Thank you.

Recall the father son height data (Pearson's data) that we saw in class during the regression review. The average height of the fathers in this dataset was 67.7 inches, and the SD of the fathers' height was 2.74 inches. The average height of the sons was 68.7 inches and the SD was 2.81 inches. The correlatiorn between the heights of the fathers and the sons was 0.501. As- sume that these data are bivariate normal. That is, if you fix one of the values, say father's height at 70 inches, then the dis- tribution of the sons' heights (conditioned on this height for the father) will be normally distributed. True or false: because the sons average an inch taller than the fathers, if the father is 72 inches tall, it's 50-50 whether the son is taller than 73 inches.

Explanation / Answer

Given

Father's height Mean=67.7 inch

Father's height SD=2.74 inches

Son's height Mean=68.7 inches

Son's height SD =2.81 inches

correlation =0.501

The data are bivariate normal

A bivariate normal distribution is the sum of 2 independent normal distribution which on addition gives a normal distribution

False

Bivariate normal distribution is the sum of 2 independent norml distributions. Hence the Father's distribution is independent of Son's distribution

Hence the distribution of height of father will have no effect on the distribution of Sons height even though the correlation coefficient is 0.501 ie the strength of relation of father's height and Son's height which is a weak relation

Hence if father's height is 72 inches we cannot tell anything about the distribution of height of Son

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