Question

If the correlation coethcient is 0 , the two regression lines are (o) parallel (b) perpendicular (c) coincident. (d) inclined at \(45^{\circ}\) to each other.

Short answer

The two regression lines are perpendicular.

Step-by-step solution

Understand the correlation coefficient

The correlation coefficient, often denoted by \( r \), is a measure of the linear relationship between two variables. If \( r = 0 \), it indicates that there is no linear correlation between the variables.

Recall the properties of regression lines

Regression lines are derived from minimizing the sum of squared deviations between actual and predicted values for each variable. If \( r = 0 \), it means the slopes of the regression lines are in opposite directions such that their product equals -1.

Analyze the relationship between the slopes

For two lines with slopes \( m_1 \) and \( m_2 \), if their product equals -1 \((m_1 \, m_2 = -1)\), the lines are perpendicular to each other. This is consistent with no correlation \((r = 0)\), because their directions are completely independent of each other.

Key concepts

Regression Lines

Regression lines are essential tools in statistics to determine the relationship between variables. They allow us to make predictions based on data. Each regression line represents the best possible fit for the data points on a graph, minimizing the distance between the data points and the line itself.

• This is done by reducing the sum of the squares of these vertical distances, which is known as the method of least squares.
• Two regression lines usually exist: one for predicting the dependent variable from the independent variable, and the other for the reverse prediction.

In certain cases, when the correlation coefficient is zero, the regression lines can be perpendicular because they describe completely unrelated directions of change. Understanding these lines provides insights into data patterns and helps in making predictions.

Perpendicular Lines

Perpendicular lines are an interesting concept in both geometry and statistics. In the context of regression and statistics, when we say two regression lines are perpendicular, we mean they meet at a right angle (90 degrees). This relationship is particularly important when analyzing datasets with no linear correlation.

• Mathematically, if two lines are perpendicular, the product of their slopes will equal -1.
• This perpendicularity implies total independence between the directions they represent.

Thus, in a scatter plot where the correlation coefficient is zero, the regression estimates can plot perpendicular lines, showing no definitive trend between the variables.

Linear Relationship

A linear relationship is fundamental in understanding how two variables interact. It indicates that the change in one variable is proportional to the change in another variable. This relationship can be visualized through straight lines on a graph, known as linear regression lines.

• If a linear relationship exists, a correlation coefficient comes into play, showing the strength and direction.
• A correlation coefficient of zero indicates no linear relationship, which also results in unrelated regression lines.

Grasping the concept of linear relationships helps us in predicting outcomes and understanding how changes in one entity might affect another.

Slope of a Line

The slope of a line is a core concept, both geometrically and statistically. In regression analysis, the slope determines how steep the regression line is and visually shows the rate of change between the dependent variable and the independent variable.

• In mathematical terms, the slope is defined as the change in the y-coordinate divided by the change in the x-coordinate ( \[ m = \frac{\Delta y}{\Delta x} \] ).
• A positive slope indicates that as one variable increases, the other also increases. A negative slope suggests the opposite.
• If there is no relationship between the variables, such as when the correlation coefficient is zero, the regression lines can have a slope that makes them perpendicular.

Understanding the slope can help determine the type of linear relationship that exists between the studied variables.