The height and age of each child in a random sample of children was recorded. The value of the correlation coefficient between height and age for the children in the sample was . Based on the least-squares regression line created from the data to predict the height of a child based on age, which of the following is a correct statement?
The Correct Answer and Explanation is :
To provide a comprehensive explanation, I need the specific value of the correlation coefficient between height and age in your sample. The correlation coefficient, denoted as ( r ), quantifies the strength and direction of a linear relationship between two variables. Its value ranges from -1 to 1:
- ( r = 1 ): Perfect positive linear relationship. As one variable increases, the other increases proportionally.
- ( r = -1 ): Perfect negative linear relationship. As one variable increases, the other decreases proportionally.
- ( r = 0 ): No linear relationship between the variables.
In the context of predicting a child’s height based on age using the least-squares regression line, the correlation coefficient influences the slope of the regression line. The slope ( m ) of the least-squares regression line is calculated using the formula:
[ m = r \times \frac{s_y}{s_x} ]
where:
- ( s_y ) is the standard deviation of the dependent variable (height),
- ( s_x ) is the standard deviation of the independent variable (age),
- ( r ) is the correlation coefficient.
This formula indicates that the slope depends on both the strength and direction of the linear relationship between age and height. A positive ( r ) results in a positive slope, meaning that as age increases, height is expected to increase. Conversely, a negative ( r ) leads to a negative slope, indicating that as age increases, height decreases.
The correlation coefficient also affects the coefficient of determination, ( r^2 ), which represents the proportion of the variance in the dependent variable that is predictable from the independent variable. A higher ( r^2 ) value indicates a better fit of the regression model to the data.
Without the specific value of ( r ), I cannot provide a precise statement regarding the relationship between height and age in your sample. If you can provide the value of the correlation coefficient, I can offer a more detailed interpretation.