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[Phuong Fulsom]

Lesson 2 Homework Practice: Lines of Best Fit

In this lesson, you will learn how to use a scatter plot to find a line of best fit that represents the relationship between two sets of data. A line of best fit is a straight line that comes closest to the points on a scatter plot. It can help you make predictions based on the data.

To find a line of best fit, you can use a method called linear regression, which is a way of finding the equation of a line that minimizes the sum of the squares of the distances from each point to the line. You can use a graphing calculator or an online tool to perform linear regression and get the equation of the line of best fit.

For example, suppose you have the following data about the number of hours students studied for a test and their test scores:

Hours

Score

1

72

2

80

3

84

4

88

5

92

You can make a scatter plot of this data and use linear regression to find the equation of the line of best fit. The equation is y = 4x + 68, where y is the score and x is the hours. You can draw this line on the scatter plot and see how well it fits the data.

You can use the equation of the line of best fit to make predictions based on the data. For example, if a student studies for 6 hours, you can plug in x = 6 into the equation and get y = 4(6) + 68 = 92. This means that the predicted score for a student who studies for 6 hours is 92.

However, you should be careful not to make predictions outside the range of the data, because the line of best fit may not be accurate for values that are too high or too low. For example, if a student studies for 10 hours, you can plug in x = 10 into the equation and get y = 4(10) + 68 = 108. This means that the predicted score for a student who studies for 10 hours is 108. But this prediction may not be realistic, because it is higher than the highest score in the data set. Therefore, you should only use the line of best fit to make predictions within the range of the data.

To practice finding lines of best fit, you can try some exercises at this link: www.khanacademy.org/math/statistics-probability/describing-relationships-quantitative-data/introduction-to-trend-lines/e/line-of-best-fit

Another way to find a line of best fit is to use a method called the median-median line, which is a way of finding the equation of a line that passes through the medians of three groups of points on a scatter plot. The median-median line is easier to find by hand than the linear regression line, but it may not be as accurate.

To find the median-median line, you need to follow these steps:

• Divide the data into three groups of equal size. If the number of points is not divisible by three, put the extra point(s) in the middle group.

• Find the median x-value and the median y-value for each group. These are the coordinates of the group medians.

• Plot the group medians on the scatter plot and connect them with two line segments. These are the median lines.

• Find the slope of each median line by using the formula m = (y₂ - y₁) / (x₂ - x₁), where (x₁, y₁) and (x₂, y₂) are the endpoints of the median line.

• Average the slopes of the two median lines to get the slope of the median-median line.

• Find the y-intercept of the median-median line by using the formula b = y - mx, where (x, y) is any point on the median-median line and m is its slope.

• Write the equation of the median-median line in slope-intercept form: y = mx + b.

For example, suppose you have the same data as before about the number of hours students studied for a test and their test scores:

Hours

Score

1

72

2

80

3

84

4

88

5

92

You can make a scatter plot of this data and use the median-median line method to find the equation of the line of best fit. Here are the steps:

• The data has five points, so you can divide it into three groups: Group 1: (1, 72), Group 2: (2, 80), (3, 84), Group 3: (4, 88), (5, 92).

• The group medians are: Group 1: (1, 72), Group 2: (2.5, 82), Group 3: (4.5, 90).

• You can plot these points on the scatter plot and connect them with two line segments:

• The slope of the left median line is m_L = (82 - 72) / (2.5 - 1) = 4. The slope of the right median line is m_R = (90 - 82) / (4.5 - 2.5) = 3.2.

• The slope of the median-median line is m = (m_L + m_R) / 2 = (4 + 3.2) / 2 = 3.6.

• The y-intercept of the median-median line is b = y - mx = 82 - 3.6(2.5) = 73.6.

• The equation of the median-median 51082c0ec5