Final answer:
The correlation coefficient r is -0.50, signifying a moderate negative linear relationship between the variables. The coefficient of determination r² reflects the proportion of variance in one variable that can be explained by the other. To have an r² of at least 0.50, the absolute value of r must be at least ±0.7071.
Step-by-step explanation:
If the covariance between x and y is -0.20, with a standard deviation for x of 0.50 and a standard deviation for y of 0.80, the correlation coefficient r can be found using the formula:
r = covariance(x, y) / (std_dev(x) * std_dev(y))
Plugging in the values given:
r = -0.20 / (0.50 * 0.80)
r = -0.20 / 0.40
r = -0.50
The correlation coefficient r indicates the strength and direction of a linear relationship between two variables. In this case, r = -0.50 suggests a moderate negative linear relationship: as x increases, y tends to decrease and vice versa. To have a coefficient of determination of at least 0.50, the absolute value of r must be at least √0.50, which approximates to 0.7071. Therefore, r must be at least ±0.7071 to have an r² of at least 0.50.