2. Two variables have a positive linear correlation. Is the slope of the regression line for the variables positive or negative?

Verified step by step guidance

1. Understand the concept of linear correlation: A positive linear correlation means that as one variable increases, the other variable also tends to increase.
2. Recall the relationship between correlation and the slope of the regression line: The slope of the regression line indicates the rate of change of the dependent variable with respect to the independent variable.
3. Recognize that a positive linear correlation implies a positive slope: Since the variables increase together, the regression line will have an upward trend.
4. Express the slope mathematically: The slope of the regression line is calculated as \( m = \frac{\text{Cov}(X, Y)}{\text{Var}(X)} \), where \( \text{Cov}(X, Y) \) is the covariance between the variables and \( \text{Var}(X) \) is the variance of the independent variable. A positive covariance leads to a positive slope.
5. Conclude that the slope of the regression line is positive: Based on the positive correlation and the mathematical relationship, the regression line will have a positive slope.

Key Concepts

Positive Linear Correlation

Positive linear correlation indicates that as one variable increases, the other variable also tends to increase. This relationship is quantified by the correlation coefficient, which ranges from 0 to 1 for positive correlations. A strong positive correlation suggests that the variables move together in the same direction.

Regression Line

A regression line is a straight line that best fits the data points in a scatter plot, representing the relationship between two variables. The equation of the line is typically expressed as y = mx + b, where m is the slope and b is the y-intercept. The slope indicates the direction and strength of the relationship between the independent and dependent variables.

Slope of the Regression Line

The slope of the regression line reflects the rate of change in the dependent variable for each unit change in the independent variable. In the case of a positive linear correlation, the slope will be positive, indicating that increases in the independent variable lead to increases in the dependent variable. This positive slope is a key indicator of the strength and direction of the relationship.