Question

5 from unit 3, lesson
at a restaurant, the total bill and the percentage of the bill left as a tip are represented by the scatter plot.
the best - fit line is represented by the equation ( y=-0.632x + 27.1 ), where ( x ) represents the total bill in dollars, and ( y ) represents the percentage of the bill left as a tip.
a. what does the best - fit line estimate for the percentage of the bill left as a tip when the bill is $15?
is this reasonable?
b. what does the best - fit line predict for the percentage of the bill left as a tip when the bill is $50?
is this reasonable?

Explanation:

Step1: Substitute \(x = 15\) into the equation

Given \(y=-0.632x + 27.1\), when \(x = 15\), we have \(y=-0.632\times15+27.1\).
First, calculate \(-0.632\times15=-9.48\). Then \(y=-9.48 + 27.1\).

Step2: Calculate the value of \(y\)

\(y=-9.48+27.1 = 17.62\). This is reasonable as \(17.62\) is within the range of the \(y -\) values (percentage of tip) shown in the scatter - plot.

Step3: Substitute \(x = 50\) into the equation

When \(x = 50\), \(y=-0.632\times50+27.1\). Calculate \(-0.632\times50=-31.6\). Then \(y=-31.6+27.1\).

Step4: Calculate the value of \(y\)

\(y=-31.6 + 27.1=-4.5\). This is not reasonable because the percentage of a tip ( \(y\) - value, which represents a proportion of the bill) cannot be negative.

Answer:

a. The best - fit line estimates the percentage of the bill left as a tip is \(17.62\%\) when the bill is \(\$15\), and this is reasonable.
b. The best - fit line predicts the percentage of the bill left as a tip is \(-4.5\%\) when the bill is \(\$50\), and this is not reasonable.