Final answer:
The student's queries focus on regression analysis and correlation in a statistical context, with an emphasis on interpreting the meaning of coefficients, residuals, and the fit of the regression model to the data provided.
Step-by-step explanation:
The questions provided all relate to creating and interpreting linear regression models and correlation coefficients. A multiple regression model, as described in the initial query, is used to predict paint sales based on promotional expenditures and selling price. To determine the relationship between independent and dependent variables, one would draw a scatter plot. The least-squares line, often represented as ŷ = a + bx, indicates the best fit for the data in a regression analysis. The correlation coefficient (r) measures the strength and direction of the linear relationship between two variables, and the coefficient of determination (r^2) indicates the proportion of the variance for the dependent variable that's explained by the independent variable(s) in the model.
For example, the regression equation ŷ = − 0.3031x + 31.93 for sparrow hawks might not be meaningful at the y-intercept since there cannot be 32 percent new sparrow hawks when there are no returning birds. The correlation coefficient r = -.7584 suggests a somewhat strong negative correlation between the variables. In prediction, residual analysis, such as finding that the data point (66, 6) produces a significant residual, is important. After removing certain data points, the slope and intercept may adjust, but the significance of the correlation and the fit of the model should always be reassessed.