Final answer:
The assumptions underlying simple regression analysis include a linear relationship between variables, normally distributed y values around the regression line, the same variance of y values (homoscedasticity), and independent residuals. These assumptions are necessary for the validity of the regression model.
Step-by-step explanation:
Assumptions Underlying Simple Regression Analysis
The assumptions underlying simple regression analysis are critical for the validity of its results. These assumptions ensure that the statistical methods used to estimate the relationship between variables are appropriate for the data. Here are the key assumptions:
- There must be a linear relationship between the independent variable (x) and the dependent variable (y) in the population, with the sample regression line serving as the best estimate of this relationship.
- The y values for any given x should be normally distributed around the regression line, indicating a higher density of y values close to the line and fewer as distance from the line increases.
- All the normal distributions of y values should have the same shape and spread, with commonly called homoscedasticity, meaning that the variance around the regression line is the same for all values of x.
- The residual errors must be independent, showing no pattern when plotted against x or time, ensuring that the observations are not correlated with each other.
Moreover, the residuals must also meet other criteria such as being randomly distributed as per the design of the study, thus satisfying the random condition. It is also required that the standard deviation of the y values is the same for each x value, meeting the condition of equal variance.
The accuracy of the slope and y-intercept of the regression equation, the correlation coefficient (r), and the coefficient of determination (R²) are dependent on these assumptions being met.