Question

How could you change this line to be a better line of best fit for the data plot?
• Make it a negative sloped line to better show the trend in the data.
• Keep it parallel and move it up through the center of the data.
• Keep it parallel and move it down so that the points are equal above and below the line.
• Make it steeper to hit the most points possible.
• Make it less steep to hit at least one of the data points.

Answer 1

When adjusting a line of best fit for a data plot, the overarching goal is to find the line that best represents the trend of the data, minimizing the distance of the data points from the line. A properly-fitted line will usually have approximately an equal number of points above and below it, as well as reflect the direction (positive or negative correlation) of the data. It isn't about 'hitting' the most points, but rather capturing the overall trend.

Here are the options you provided, with commentary on each:

1. **Make it a negative sloped line to better show the trend in the data.**
- If the data indicates a negative correlation (i.e., as one variable increases, the other decreases), then adjusting the slope to be negative would indeed provide a better representation.

2. **Keep it parallel and move it up through the center of the data.**
- If the current line is already parallel to the trend of the data but is offset, moving it up (or down) to be more centered with the data points can improve its fit.

3. **Keep it parallel and move it down so that the points are equal above and below the line.**
- Similar to the previous point, if the line does not cut through the center of the data distribution, moving it up or down (while keeping the slope the same) can make the line a better fit.

4. **Make it steeper to hit the most points possible.**
- Changing the steepness of the line changes its slope, which represents the rate of change between the variables. Adjusting the steepness is only appropriate if the current slope does not accurately represent the data's trend.

5. **Make it less steep to hit at least one of the data points.**
- Attempting to hit a specific data point is not usually the aim when determining a line of best fit, as this could lead to a misrepresentation of the overall trend of the data. The slope should be adjusted to reflect the trend of the data, not just to pass through particular points.

To determine the best approach, examine the scatter plot of the data:

- If the trend is negative, ensure the slope of the line is also negative.
- If there is a positive trend but the line has a negative slope or vice versa, the slope needs to be adjusted to reflect the actual trend.
- The line should be moved vertically to lie as close as possible to the center of the data, with roughly equal numbers of points above and below the line.
- Adjust the slope to ensure that it best represents the general trend without overly focusing on individual data points.

Analyzing the existing errors (the distances from the data points to the line) can also guide you on how to adjust the line for a better fit. In general, the approach that best captures the trend of the data and minimizes these errors is the optimal choice for the line of best fit.