Final answer:
The R², or coefficient of determination, is the proportion of variance in the dependent variable (y) that can be predicted from the variance in the independent variable (x), reflected in the square of the correlation coefficient, r. For R² to be at least 0.50, the correlation coefficient must be at least ± 0.71.
Step-by-step explanation:
R² is the proportion of variance in the outcome variable that can be predicted from the independent variable's variance. In statistics, this is known as the coefficient of determination, which is the square of the correlation coefficient, r. To illustrate, if we have a correlation coefficient of 0.8, squaring this value yields an R² of 0.64. This means that 64 percent of the variation in the dependent variable (y) can be predicted by the independent variable (x) using the regression line.
For a coefficient of determination of at least 0.50, a correlation coefficient of at least ± 0.71 (√0.50 rounded to two decimal places) is necessary, indicating a moderately strong linear relationship. This implies that at least 50 percent of the variance in y can be explained by variation in x, signifying that the independent variable is a good predictor of the dependent variable.