Given four points, (2,1), (3,2), (4,2), and (5,3), what can be determined about the correlation coefficient for the least squares line?

To determine the correlation coefficient for the least squares line fitted to the given points (2,1), (3,2), (4,2), and (5,3), one must analyze the distribution of the points in relation to a possible linear trend.

The points exhibit a general upward trend where as the x-values increase, the y-values also tend to increase. Specifically, from (2,1) to (3,2) there is an increase of 1 unit in both x and y. The transition from (3,2) to (4,2) shows a flat line, where y doesn't change despite an increase in x, indicating some variation in the trend but still suggesting a tendency to rise. Lastly, from (4,2) to (5,3), we observe another increase of 1 unit in both x and y.

The correlation coefficient quantifies the strength and direction of a linear relationship between the x and y values. Given that the majority of the points fit together well in a line that tends to rise, this suggests a high positive correlation. Even with the flat segment, the overall trend maintains a noticeable positive inclination across the x-range from the lowest to highest x-value. As such, we can determine