Final answer:
Removing the point (50, 25) would possibly increase the correlation coefficient and the coefficient of determination, indicating a stronger linear relationship between the variables. The effects on the slope and y-intercept of the regression line would need recalculating to determine the specific changes.
Step-by-step explanation:
When considering the likelihood and impact of removing the point (50, 25) on the calculation of a new least-squares regression line, one must understand several concepts of regression analysis. These include the correlation coefficient (r), the coefficient of determination (r²), and the regression line equation. The correlation coefficient, r, measures the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where 1 indicates a perfect positive linear relationship, -1 indicates a perfect negative linear relationship, and values closer to 0 indicate a weaker linear relationship.
The coefficient of determination, r², represents the proportion of the variance in the dependent variable that is predictable from the independent variable and is the square of the correlation coefficient. Removing an outlier can increase r and r² if the outlier was causing a decrease in the strength of the linear relationship. Outliers can significantly affect the slope and the y-intercept of the regression line, as well as the strength of correlation between variables. In the given examples, after removing an outlier, the new regression equations and correlation values indicate that the outliers had a considerable influence, thereby making the new regression line a better fit for predicting outcomes.
Thus, removing an outlier such as the point (50, 25) in the context of the given examples would likely cause the correlation coefficient to increase, suggesting a stronger linear relationship. Depending on the position of the outlier in relation to the other data points, the y-intercept could either increase or decrease. The effects on the regression line are quantified after recalculating it without the outlier.