Final answer:
The correlation coefficient (r) measures the strength and direction of the linear relationship between two variables, and its square gives us the coefficient of determination (r²). To have an r² of at least 0.50, the correlation coefficient needs to be at least ±0.7071.
Step-by-step explanation:
The correlation coefficient, often denoted as r, quantifies the strength and direction of the linear relationship between two variables. The value of r ranges from -1 to +1, where a value close to +1 or -1 indicates a strong relationship, and a value near 0 indicates a weak relationship. The coefficient of determination, denoted as r², is the square of the correlation coefficient and indicates the proportion of the variance in the dependent variable that is predictable from the independent variable.
When asked about the correlation needed to have a r² of at least 0.50, we are essentially seeking an r whose square meets this threshold. Since r² = 0.50 corresponds to an r value where r = ±0.7071 (the square root of 0.50), a correlation coefficient of at least 0.7071 or -0.7071 (to three decimal places) is necessary to achieve a r² of at least 0.50.