Question

from unit 3, lesson 6
a researcher creates a scatter plot that displays the relationship between the number of years in business, x, and the percentage of company business that is fair trade, y. the researcher creates a line of best fit, y = 0.091x + 0.060, and wants to find the residuals for the companies that have been in business for 3 years.
a. find the residuals for the two points representing companies that have been in business for 3 years, (3, 0.42) and (3, 0.3).
b. compare the residuals for the two companies that have been in business for 3 years. how are they different? how are they similar? what does the information about the residuals for the two companies tell you about how much of their business that is fair trade?

Explanation:

Step1: Calculate the predicted value

Substitute \(x = 3\) into the line - of - best - fit equation \(y=0.091x + 0.060\).

$$y=0.091\times3+0.060=0.273 + 0.060=0.333$$

Step2: Calculate the residual for the point \((3,0.42)\)

The formula for the residual is \(e=y_{actual}-y_{predicted}\).
For the point \((3,0.42)\), \(e_1 = 0.42-0.333 = 0.087\)

Step3: Calculate the residual for the point \((3,0.3)\)

For the point \((3,0.3)\), \(e_2=0.3 - 0.333=- 0.033\)

• Differences:
• The sign of the residuals is different. The residual for \((3,0.42)\) is positive (\(e_1 = 0.087\)), meaning the actual value \(y = 0.42\) is above the line of best fit. The residual for \((3,0.3)\) is negative (\(e_2=-0.033\)), meaning the actual value \(y = 0.3\) is below the line of best fit.
• The magnitudes are different (\(\vert0.087\vert>\vert - 0.033\vert\))
• Similarities:
• Both companies have been in business for the same number of years (\(x = 3\))
• What the residuals tell us:
• For the company with the point \((3,0.42)\), a positive residual indicates that a higher - than - predicted percentage of its business is fair trade. For the company with the point \((3,0.3)\), a negative residual indicates that a lower - than - predicted percentage of its business is fair trade.

Answer:

The residual for the point \((3,0.42)\) is \(0.087\) and the residual for the point \((3,0.3)\) is \(-0.033\)