Let's analyze the problem provided in the image:

Problem 4:

For the scatter plot shown, Lincoln's line of fit is represented by the equation $y = 0.2x + 6$.

Part (a)

Explain why Lincoln's line is a good line of fit.

Answer: Lincoln's line of fit $y = 0.2x + 6$ appears to be a good line of fit because it has a positive slope, matching the general upward trend of the data points in the scatter plot. This means that as $x$ values increase, $y$ values also tend to increase, aligning with the scatter plot's trend. The line also passes through or near several data points, indicating it approximates the data's trend reasonably well.

Part (b)

Draw a line that would be a better line of fit for the data. Write an equation of that line.

Answer: To find a better line of fit, observe where most of the data points lie relative to Lincoln's line. A better line of fit might have a slightly steeper slope or a different y-intercept to better capture the central tendency of the data points.

For example, if a better line of fit seems to be closer to the data points with a slope slightly higher than 0.2, we might try a line like: $y = 0.25x + 5$ This line has a steeper slope (0.25) and a lower y-intercept (5), which might align more closely with the observed data distribution.

Would you like more details on the process of determining a better line of fit, or any help with graphing it?

Additional Questions to Explore

1. How can you calculate the slope of a line from a scatter plot with given data points?
2. What methods can be used to determine the "best fit" line statistically?
3. Why is the slope important in understanding the trend of a line of fit?
4. How does the y-intercept affect the position of a line on a graph?
5. What is the difference between a line of fit and a line of best fit?

Tip: To create a better line of fit, try to balance the line so it minimizes the distance between the line and all points, both above and below it.