Question

A linear correlation coefficient of 0.92 suggests a ________________ linear relationship than a linear correlation coefficient of -0.86.

The value of the ___________________ always lies between -1 and 1, inclusive.

If the linear correlation coefficient of the regression line is negative, then the ____________________ of the least squares (linear) regression line must be negative.

Give a detailed interpretation of the slope of a least squares (linear) regression line.

Give a detailed interpretation of the intercept of a least squares (linear) regression line.

Give a detailed interpretation of the coefficient of determination (R^2).

Give two different methods to determine if a linear regression line fits data best. Describe in detail within each method what trends or information would constitute a lack of fit for any linear regression model.

Answer 1

A linear correlation coefficient ranges from -1 to +1. A linear correlation coefficient value of -1 represents a perfect negative correlation between the two variables while a +1 correlation value represents a perfect positive correlation between the variables. Also a value close to 0 represents no linear correlation between the variables.

Therefore A linear correlation coefficient of 0.92 suggests a stronger linear relationship than a correlation coefficient of -0.86.

This is because the strength is depicted by the magnitude of the coefficient while the direction is represented by the sign of the coefficient.

For a negative correlation coefficient, the slope of the regression line is negative.

The interpretation of the slope of the least square regression line is that for a unit increase in the independent variable value, the expected increase in the dependent variable value is equal to the slope, while for a unit decrease in the independent variable value, the expected change is a decrease in dependent variable value equal to slope of the line.

The intercept of the regression line interpretation is that for a 0 value of independent variable value, the value of dependent variable is expected to be equal to the intercept.

The coefficient of determination of regression is defined as the proportion of total variation in the dependent variable that is explained by regression.

Two methods to check a linear fit of the regression line fit are given as:

• T test for the significance of individual independent variables: The line is not a good fit if the test is not significant.
• Test for correlation coefficient of the regression line. Again a non significant correlation leads to a poor fit of the line.