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: Find Residual Values

Residual is calculated as \( \text{Residual} = \text{Given} - \text{Predicted} \)

Step 1: For \( x = 1 \)

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

Step 2: For \( x = 2 \)

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

Step 3: For \( x = 3 \)

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

Step 4: For \( x = 4 \)

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

Step 5: For \( x = 5 \)

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

Part 2: Residual Plot Appropriateness

A residual plot shows if a linear model (line of best fit) is appropriate. If residuals have no pattern (randomly scattered around the x - axis), the linear model is appropriate. If there is a pattern (like curved or linear), the model is not appropriate.

• Option 1: "No, the points are in a curved pattern" → Residuals here don't show a curved pattern. Eliminate.
• Option 2: "No, the points are evenly distributed about the x - axis" → Even distribution (no pattern) means the model is appropriate, so this is wrong. Eliminate.
• Option 3: "Yes, the points are in a linear pattern" → A linear pattern in residuals means the linear model is not appropriate. Eliminate.
• Option 4: "Yes, the points have no pattern" → Since residuals (0.14, - 0.09, - 0.12, - 0.05, 0.12) are randomly scattered (no pattern), the line of best fit is appropriate.

Final Answers for Residuals:

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

Final Answer for Residual Plot Question:

D. Yes, the points have no pattern (assuming the options are labeled as A, B, C, D with D being "Yes, the points have no pattern")

Answer:

Part 1: Find Residual Values

Residual is calculated as \( \text{Residual} = \text{Given} - \text{Predicted} \)

Step 1: For \( x = 1 \)

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

Step 2: For \( x = 2 \)

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

Step 3: For \( x = 3 \)

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

Step 4: For \( x = 4 \)

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

Step 5: For \( x = 5 \)

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

Part 2: Residual Plot Appropriateness

A residual plot shows if a linear model (line of best fit) is appropriate. If residuals have no pattern (randomly scattered around the x - axis), the linear model is appropriate. If there is a pattern (like curved or linear), the model is not appropriate.

• Option 1: "No, the points are in a curved pattern" → Residuals here don't show a curved pattern. Eliminate.
• Option 2: "No, the points are evenly distributed about the x - axis" → Even distribution (no pattern) means the model is appropriate, so this is wrong. Eliminate.
• Option 3: "Yes, the points are in a linear pattern" → A linear pattern in residuals means the linear model is not appropriate. Eliminate.
• Option 4: "Yes, the points have no pattern" → Since residuals (0.14, - 0.09, - 0.12, - 0.05, 0.12) are randomly scattered (no pattern), the line of best fit is appropriate.

Final Answers for Residuals:

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

Final Answer for Residual Plot Question:

D. Yes, the points have no pattern (assuming the options are labeled as A, B, C, D with D being "Yes, the points have no pattern")