Final answer:
The value of R² always lies between 0 and 1, being the square of the correlation coefficient, r, which ranges from -1 to +1. The strength of a linear relationship is weakest when r is closest to 0 and strongest when r is near -1 or +1. To have a coefficient of determination of at least 0.50, the absolute value of r must be at least approximately 0.71.
Step-by-step explanation:
The value of R², known as the coefficient of determination, always lies between 0 and 1. It is the square of the correlation coefficient, r. The coefficient of determination, R², indicates the proportion of the variance in the dependent variable that is predictable from the independent variable(s).
The correlation coefficient, r, measures the strength of linear association between two variables and can range from -1 to +1. A correlation coefficient closest to 0 indicates the weakest relationship, while coefficients closer to -1 or +1 signal a stronger linear relationship.
If the goal is to have a coefficient of determination of at least 0.50, we need the absolute value of r to be at least sqrt(0.50), which is approximately 0.71. Therefore, a correlation between two variables must be either more than +0.71 or less than -0.71 to have an R² of at least 0.50.