Question

The and age of each child in a random sample of children was recorded The value of the correlation coefficient between height and the children in the sample was 0.8 . Based on the line created from the data to the height of a child based on age which of the following is a correct statement ?

Answer 1

Options :

A.) On average, the height of a child is 80% of the age of the child.

B.) The least-squares regression line of height versus age will have a slope of 0.8.

C.) The proportion of the variation in height that is explained by a regression on age is 0.64.

D.) The least-squares regression line will correctly predict height based on age 80% of the time.

E.) The least-squares regression line will correctly predict height based on age 64% of the time.

Answer:

C.) The proportion of the variation in height that is explained by a regression on age is 0.64.

Step-by-step explanation: from the scenario stated above, The correlation Coefficient R = 0.8, which shows the strength of relationship between age and height of children. From the value of correlation Coefficient Given, we can obtain the Coefficient of determination (R²) which gives the percentage of Variation which can be explained by the regression model.

Hence the Coefficient of determination (R²) = 0.8² = 0.64. Hence, the proportion of variation in height of children which can he due to age is 0.64 while 0.36 can be based on other factors.

Answer 2

Based on the line created from the data to the height of a child based on age the correct statement is given by: Option C.) The proportion of the variation in height that is explained by a regression on age is 0.64.

How can we get to know the variability of one variable to other from correlation coefficient?

For that, we can use the coefficient of determination.

The coefficient of determination is square of correlation coefficient and thus, ranges from 0 to 1.

The coefficient of determination tells for unit variation in one variable will lead to how much variation in other variable and vice versa.

Let we take:

X = age of children of the sample considered

Y = height of children of the sample considered

Then, as it is given that:

$r(X, Y) = 0.8$

Thus, the coefficient of determination is: $r^2 = 0.8^2 = 0.64$

Thus, we get that 0.64 of the variation in the height is explained by the age of the children.

Thus, the true statement is:

Option C.) The proportion of the variation in height that is explained by a regression on age is 0.64.

Learn more about coefficient of determination here:

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