Question

find the residual values, and use the graphing calculator tool to make a residual plot.

x	given	predicted	residual
2	-0.9	-0.81
3	1.1	1.22
4	3.2	3.25
5	5.4	5.28

does the residual plot show that the line of best fit is appropriate for the data?

• no, the points are in a curved pattern.
• no, the points are evenly distributed about the x - axis.
• yes, the points are in a linear pattern.
• yes, the points have no pattern.

Explanation:

Step1: Recall Residual Formula

Residual = Given - Predicted, so \( \text{Residual} = y - \hat{y} \).

Step2: Calculate Residual for \( x=1 \)

Given \( y = -2.7 \), \( \hat{y} = -2.84 \).
Residual = \( -2.7 - (-2.84) = -2.7 + 2.84 = 0.14 \).

Step3: Calculate Residual for \( x=2 \)

Given \( y = -0.9 \), \( \hat{y} = -0.81 \).
Residual = \( -0.9 - (-0.81) = -0.9 + 0.81 = -0.09 \).

Step4: Calculate Residual for \( x=3 \)

Given \( y = 1.1 \), \( \hat{y} = 1.22 \).
Residual = \( 1.1 - 1.22 = -0.12 \).

Step5: Calculate Residual for \( x=4 \)

Given \( y = 3.2 \), \( \hat{y} = 3.25 \).
Residual = \( 3.2 - 3.25 = -0.05 \).

Step6: Calculate Residual for \( x=5 \)

Given \( y = 5.4 \), \( \hat{y} = 5.28 \).
Residual = \( 5.4 - 5.28 = 0.12 \).

Step7: Analyze Residual Plot Appropriateness

A residual plot shows a line of best fit is appropriate if residuals have no pattern (randomly scattered). Let's check the residuals: \( 0.14, -0.09, -0.12, -0.05, 0.12 \). These are randomly scattered (no curve, no linear trend). So the correct option is "Yes, the points have no pattern."

Answer:

(Residuals):
For \( x=1 \): \( 0.14 \); \( x=2 \): \( -0.09 \); \( x=3 \): \( -0.12 \); \( x=4 \): \( -0.05 \); \( x=5 \): \( 0.12 \).
For the residual plot question: Yes, the points have no pattern.