Question

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

Part 1: Calculating Residual Values

The formula for the residual is \( \text{Residual} = \text{Given} - \text{Predicted} \). We will calculate the residual for each \( x \)-value.

Step 1: Residual for \( x = 1 \)

Given value: \( -2.7 \), Predicted value: \( -2.84 \)
Residual \( = -2.7 - (-2.84) = -2.7 + 2.84 = 0.14 \)

Step 2: Residual for \( x = 2 \)

Given value: \( -0.9 \), Predicted value: \( -0.81 \)
Residual \( = -0.9 - (-0.81) = -0.9 + 0.81 = -0.09 \)

Step 3: Residual for \( x = 3 \)

Given value: \( 1.1 \), Predicted value: \( 1.22 \)
Residual \( = 1.1 - 1.22 = -0.12 \)

Step 4: Residual for \( x = 4 \)

Given value: \( 3.2 \), Predicted value: \( 3.25 \)
Residual \( = 3.2 - 3.25 = -0.05 \)

Step 5: Residual for \( x = 5 \)

Given value: \( 5.4 \), Predicted value: \( 5.28 \)
Residual \( = 5.4 - 5.28 = 0.12 \)

The completed table is:

\( x \)	Given	Predicted	Residual
2	-0.9	-0.81	-0.09
3	1.1	1.22	-0.12
4	3.2	3.25	-0.05
5	5.4	5.28	0.12

Part 2: Analyzing the Residual Plot

To determine if the line of best fit is appropriate, we analyze the residual plot. A good line of best fit has residuals with no clear pattern (randomly scattered around the \( x \)-axis).

• Option 1: "No, the points are in a curved pattern." – The residuals we calculated (0.14, -0.09, -0.12, -0.05, 0.12) do not form a curved pattern.
• Option 2: "No, the points are evenly distributed about the \( x \)-axis." – Even distribution about the \( x \)-axis is a sign of a good fit, so this reasoning is incorrect.
• Option 3: "Yes, the points are in a linear pattern." – A linear pattern in residuals is bad (it means the linear model is not appropriate), so this is incorrect.
• Option 4: "Yes, the points have no pattern." – The residuals are randomly scattered (0.14, -0.09, -0.12, -0.05, 0.12) with no clear pattern, so this is correct.

Answer:

s:

Part 1 (Residuals):

For \( x = 1 \): \( 0.14 \)
For \( x = 2 \): \( -0.09 \)
For \( x = 3 \): \( -0.12 \)
For \( x = 4 \): \( -0.05 \)
For \( x = 5 \): \( 0.12 \)

Part 2 (Residual Plot Appropriateness):

\(\boldsymbol{\text{Yes, the points have no pattern.}}\) (Corresponding option: "Yes, the points have no pattern.")