Question

5 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: Find the predicted value

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

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Step2: Calculate the residuals

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

$$ e_1 = 0.42-0.333=0.087 $$

For the point \((3,0.3)\):

$$ e_2=0.3 - 0.333=- 0.033 $$

Answer:

a. The residuals are \(0.087\) and \(-0.033\)

b. Differences:

• The sign: One residual (\(0.087\)) is positive and the other (\(-0.033\)) is negative.
• The magnitude: \(|0.087|=0.087\) and \(|-0.033| = 0.033\), so \(0.087>0.033\)

Similarities:

• They are calculated using the same predicted value (\(y = 0.333\)) for \(x = 3\) from the line of best - fit equation \(y=0.091x+0.060\)

What the residuals tell about fair - trade business:

• For the company with the point \((3,0.42)\) and residual \(e = 0.087\), the actual percentage of fair - trade business (\(y = 0.42\)) is higher than what is predicted by the line of best - fit.
• For the company with the point \((3,0.3)\) and residual \(e=-0.033\), the actual percentage of fair - trade business (\(y = 0.3\)) is lower than what is predicted by the line of best - fit.