Question

Which of the following statements about the coefficient of correlation, r , is true?

I. The correlation coefficient and the slope of the regression line may have opposite signs.
II. A correlation of 1 indicates a perfect cause-and-effect relationship between the variables.
III. Correlations of +0.87 and - 0.87 indicate the same degree of clustering around the regression line.

Answer 1

The coefficient of correlation, r, measures the strength and direction of the linear relationship between two variables. The statements I and III are false, while statement II is true.

Step-by-step explanation:

The coefficient of correlation, r, is a statistic that measures the strength and direction of the linear relationship between two variables, x and y. It ranges from -1 to +1. Here are the statements:

1. I. The correlation coefficient and the slope of the regression line may have opposite signs.
   This statement is true. The correlation coefficient, r, and the slope of the regression line, b, can have opposite signs if the relationship between x and y is negative.
2. II. A correlation of 1 indicates a perfect cause-and-effect relationship between the variables.
   This statement is false. A correlation of 1 indicates a perfect positive linear relationship between the variables, but it does not imply causation. Correlation does not imply causation, and a cause-and-effect relationship requires further evidence.
3. III. Correlations of +0.87 and -0.87 indicate the same degree of clustering around the regression line.
   This statement is false. Correlations of +0.87 and -0.87 indicate the same magnitude of correlation, but they have opposite signs. A positive correlation, such as +0.87, indicates that as x increases, y tends to increase, and as x decreases, y tends to decrease. In contrast, a negative correlation, such as -0.87, indicates that as x increases, y tends to decrease, and as x decreases, y tends to increase.