Final answer:
The simple linear regression results show a significant relationship between third exam scores and final exam scores, with the slope indicating an average increase of 4.83 points in the final score for every additional point in the third exam. The fit of the regression line is partially explained, with residuals indicating potential outliers or influential points.
Step-by-step explanation:
Interpreting Simple Linear Regression Results
Based on the provided results, we can interpret that there is a strong linear relationship between third exam scores (x, independent variable) and the final exam scores (y, dependent variable). This is evident from the significance of the correlation coefficient which suggests that as the score on the third exam changes, the final exam score also changes in a predictable way.
Understanding the Regression Equation
The least-squares regression line has been calculated as ŷ = 173.51 + 4.83x. The slope of this line, which is 4.83, indicates that for each additional point earned on the third exam, the final exam score is expected to increase by 4.83 points. The y-intercept, 173.51, represents the predicted final exam score when the third exam score is zero.
Fit of the Regression Line
The fit of the regression line can be evaluated by looking at how close the data points are to the line; this is often quantified by R² (Coefficient of determination). While the exact R² value isn't provided here, the principle remains that the closer R² is to 1, the better the line fits the data. However, part of the variation in final exam scores remains unexplained by third exam scores alone.
Residuals and Predictions
A residual is the difference between an observed value and the value predicted by the regression line. The point with the largest residual would be furthest from the predicted line which may indicate an outlier or an influential point if it has a substantial effect on the regression equation.
In the case of predicting sparrow hawk colony sizes, while not directly related to the student's school performance data, the regression approach would remain similar, involving the relationship between two variables, to make a prediction based on existing data.