Final answer:
Bivariate regression analyses the relationship between two independent and dependent variables, while multiple regression involves several independent variables influencing a single dependent variable. Least-squares regression is used to fit the best line in bivariate regression, and the correlation coefficient 'r' measures the strength and direction of the relationship. The model's suitability depends on the relationship's nature and the significance of 'r'.
Step-by-step explanation:
Bivariate regression refers to regression analyses that involve two variables: one independent variable and one dependent variable. It aims to establish the relationship between these two by fitting a linear equation to observed data. The equation of a simple linear regression is in the form ý = a + bx, where 'a' represents the y-intercept and 'b' represents the slope of the line. The correlation coefficient, denoted as 'r', measures the strength and direction of the linear relationship between the variables.
Multiple regression, on the other hand, involves more than one independent variable. This type of regression analysis is used to understand the relationship between several independent variables and a single dependent variable. The equation of a multiple regression takes the form ý = a + b1x1 + b2x2 + … + bnxn, where each 'b' represents the slope coefficients for each independent variable.
When deciding which variable should be the independent variable and which the dependent, one typically considers causality or the direction of the influence. For example, in a study involving third exam scores and final exam scores, the third exam score may be considered the independent variable (x), and the final exam score the dependent variable (y).
Scatter plots are commonly used to visualize the relationship between two variables in bivariate regression. Least-squares regression is a method to calculate the line that minimizes the sum of the squares of the vertical distances of the points from the line (the residuals).
After fitting the regression line, the significance of the correlation coefficient can be tested. A significant 'r' indicates that there is a statistically significant linear relationship between the variables.
Whether a linear relationship is appropriate can often be determined by visual inspection of the scatter plot and by considering the value and significance of the correlation coefficient. If there is no clear linear relationship, or if the relationship is curvilinear, linear regression may not be suitable.