Final answer:
To derive the coefficients for a linear regression model with two variables using Least Squares, we identify the dependent and independent variables, create a scatter plot, calculate the least-squares line, and assess the regression model's significance through the correlation coefficient (r).
Step-by-step explanation:
To derive the coefficients of a linear regression model with two independent variables using Least Squares (LS), we first identify the dependent (y) and independent variables (x1, x2). We then create a scatter plot to visualize the data and determine if a linear relationship exists. If there's an apparent linear relationship, we proceed to calculate the least-squares line using the formula ý = a + b1x1 + b2x2, where 'a' is the intercept and 'b1' and 'b2' are the slopes of the independent variables.
To ensure that the estimates are unbiased, we assume the data comes from a well-designed experiment or sample, and we use the standard deviation of the residuals to estimate the population standard deviation (s), which ensures our sample represents the population adequately. The least-squares criteria aim to minimize the sum of the squared errors (SSE), giving us the best-fit line for our data.
After calculating the least-squares line, we assess the significance of the model by computing the correlation coefficient (r). If the coefficient is significant, the model is useful for making predictions. To predict values for other data points, we apply the derived equation to the new x values.