Question

A 1954 study of children in California found that heights of 9-year-old girls have a mean of 135 centimeters and a standard deviation of 5.6 centimeters, while weights of 9-year-old girls have a mean of 31.6 kilograms and a standard deviation of 5.8 kilograms. The correlation between heights and weights is 0.73. Assume that heights and weights follow a bivariate normal distribution.

(a) What is the expected weight of a 9-year-old girl whose height is 125 centimeters?

(b) If a 9-year-old girl weighs 30 kilograms, what is the probability that her height is between 130 and 140 centimeters?

(c) What would be the correlation between heights and weights if heights were measured in inches (one inch is 2.54 centimeters)?

Answer 1

Let X is a random variable shows the height and Y is a random variable shows the weight. Here we have,

(a)

The expected weight of a 9-year-old girl whose height is x = 125 centimeters is

Answer: 24.04 Kilogram

(b)

Condiitonal distribution X|Y=30 will be normal distribution with mean

and standard deviation

z-score for X=130 in this case is

z-score for X=140 in this case is

So required probability is

(c)

The correlation coefficient remain unaffected by change of unit of measurement of variables. So the correlation between heights and weights if heights were measured in inches will be 0.73.