Q. 14 - The Practice of Statistics for AP

Question

The correlation between the age and height of children under the age of $12$ is found to be$r=0.60$. Suppose we use the age$x$ of a child to predict the height y of the child. What can we conclude?
(a) The height is generally $60\%$ of a child’s weight.
(b) About $60\%$ of the time, age will accurately predict height.
(c) The fraction of the variation in heights explained by the least-squares regression line of $y$on $x$ is $0.36$.
(d) The least-squares regression line of $y$ on $x$ has a slope of $0.6$.
(e) Thirty-six percent of the time, the least-squares regression line accurately predicts height.

Step-by-Step Solution

Answer

The correct answer is option (c) The fraction of the variation in heights explained by the least-squares regression line of $y$on $x$is $0.36.$

Step 1: Given information

The correlation between the age and height of  children under the age of $12$ is  $r=0.60.$ Let, the age $x$ of a child to predict the height $y$of the child.

Step 2: Explanation

Utilizing the given value of the correlation, the coefficient of determination is estimated as:

$$\begin{gathered} \text{ Coefficient of determination }=(\text{ Correlation }{)}^2 \\ ={(0.60)}^2 \\ =0.36 \end{gathered}$$

The coefficient of determination indicates the variation is defined by the predictor variable on the response variable. The value of the coefficient of determination indicates the $36\%$ of the variation on height is described by the least square line.

Hence, option (c) is correct.