Question

identifying the correct residual plot
a scatterplot consists of (1, 4.0), (2, 3.3), (3, 3.8), (4, 2.6), and (5, 2.7). the line of best fit used to model the data is $y = - 0.33x + 4.27$. which residual plot is correct?

Explanation:

Step1: Calculate residuals for each \(x\) value

The formula for residual \(e=y - \hat{y}\), where \(y\) is the observed value and \(\hat{y}=-0.33x + 4.27\) is the predicted value.

• For \(x = 1\): \(\hat{y}=-0.33\times1+4.27 = 3.94\), \(e = 4.0-3.94=0.06\)
• For \(x = 2\): \(\hat{y}=-0.33\times2 + 4.27=3.61\), \(e=3.3 - 3.61=-0.31\)
• For \(x = 3\): \(\hat{y}=-0.33\times3+4.27 = 3.28\), \(e=3.8 - 3.28 = 0.52\)
• For \(x = 4\): \(\hat{y}=-0.33\times4+4.27=2.95\), \(e=2.6-2.95=-0.35\)
• For \(x = 5\): \(\hat{y}=-0.33\times5+4.27=2.62\), \(e=2.7 - 2.62=0.08\)

Step2: Analyze the \(x\) (input) and \(e\) (residual) pairs

The \(x\) values are \(1,2,3,4,5\) and the corresponding residuals (approximate) are \(0.06,-0.31,0.52,-0.35,0.08\)

Answer:

The second residual plot (where for \(x = 1\) residual is near \(0\), \(x=2\) residual is negative, \(x = 3\) residual is positive, \(x=4\) residual is negative and \(x = 5\) residual is near \(0\)) is correct.