Question

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

x	given	predicted	residual
2	-2.9	2
3	-1.1	5.1
4	2.2	8.2
5	3.4	11.3

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

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

Explanation:

Step1: Calculate Residual for x=1

Residual = Given - Predicted = $-3.5 - (-1.1) = -3.5 + 1.1 = -2.4$

Step2: Calculate Residual for x=2

Residual = Given - Predicted = $-2.9 - 2 = -4.9$

Step3: Calculate Residual for x=3

Residual = Given - Predicted = $-1.1 - 5.1 = -6.2$

Step4: Calculate Residual for x=4

Residual = Given - Predicted = $2.2 - 8.2 = -6$

Step5: Calculate Residual for x=5

Residual = Given - Predicted = $3.4 - 11.3 = -7.9$

Step6: 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 options:
- Option 1: "Yes, the points have no pattern." – If residuals are random (no pattern), line of best fit is appropriate.
- Option 2: "No, the points are evenly distributed about the x - axis." – Even distribution is good, so this is wrong.
- Option 3: "No, the points are in a linear pattern." – If linear pattern, line of best fit is not appropriate, but our residuals (from calculation) don't show linear pattern.
- Option 4: "Yes, the points are in a curved pattern." – Curved pattern means line of best fit is not appropriate, so this is wrong.

First, let's list the residuals: for x=1: -2.4, x=2: -4.9, x=3: -6.2, x=4: -6, x=5: -7.9. Wait, maybe I made a mistake in residual calculation. Wait, residual is Given - Predicted or Predicted - Given? Wait, residual is observed (given) minus predicted. Wait, let's recalculate:

For x=1: Given = -3.5, Predicted = -1.1. Residual = -3.5 - (-1.1) = -3.5 + 1.1 = -2.4 (correct)
x=2: Given = -2.9, Predicted = 2. Residual = -2.9 - 2 = -4.9 (correct)
x=3: Given = -1.1, Predicted = 5.1. Residual = -1.1 - 5.1 = -6.2 (correct)
x=4: Given = 2.2, Predicted = 8.2. Residual = 2.2 - 8.2 = -6 (correct)
x=5: Given = 3.4, Predicted = 11.3. Residual = 3.4 - 11.3 = -7.9 (correct)

Now, let's think about residual plot pattern. A good residual plot (for linear model) has no pattern. Let's check the options again:

Option 1: "Yes, the points have no pattern." – If residuals are random (no pattern), then line of best fit is appropriate.

Option 2: "No, the points are evenly distributed about the x - axis." – Even distribution is a sign of good fit, so this is wrong.

Option 3: "No, the points are in a linear pattern." – Our residuals: -2.4, -4.9, -6.2, -6, -7.9. Let's see the trend: as x increases from 1 to 5, residuals are -2.4, -4.9 (more negative), -6.2 (more negative), -6 (less negative), -7.9 (more negative). Wait, maybe I miscalculated. Wait, maybe residual is Predicted - Given? Let's check:

If residual = Predicted - Given:

x=1: -1.1 - (-3.5) = 2.4

x=2: 2 - (-2.9) = 4.9

x=3: 5.1 - (-1.1) = 6.2

x=4: 8.2 - 2.2 = 6

x=5: 11.3 - 3.4 = 7.9

Ah! I think I had the residual formula wrong. Residual is usually Predicted - Observed (Given) or Observed - Predicted? Wait, the formula for residual is $e_i = y_i - \hat{y}_i$, where $y_i$ is observed (given) and $\hat{y}_i$ is predicted. So $e_i =$ Given - Predicted. Wait, but when we plot residuals, we want to see if they are random. Wait, maybe the options are about the pattern. Let's re - evaluate the options:

The correct rule: A residual plot indicates that a linear model is appropriate if the residuals are randomly scattered (no pattern). If there is a pattern (linear, curved), then the linear model is not appropriate.

Now, let's check the options:

- Option 1: "Yes, the points have no pattern." – If residuals have no pattern, linear model is appropriate.

- Option 2: "No, the points are evenly distributed about the x - axis." – Even distribution about x - axis is a good thing for residuals, so this is wrong.

- Option 3: "No, the points are in a linear pattern." – If residuals are in linear pattern, linear model is not appropriate.

- Option 4: "Yes, the points are in a curved pattern." – Curved pattern means linear model is not appropriate, so this is wrong.

Now, let's calculate the residuals correctly (Given - Predicted):

x=1: -3.5 - (-1.1) = -2.4

x=2: -2.9 - 2 = -4.9

x=3: -1.1 - 5.1 = -6.2

x=4: 2.2 - 8.2 = -6

x=5: 3.4 - 11.3 = -7.9

Now, let's see the trend of these residuals: as x increases from 1 to 5, the residuals are -2.4, -4.9 (more negative), -6.2 (more negative), -6 (less negative), -7.9 (more negative). Wait, maybe there's a linear trend here? Wait, no, maybe I made a mistake in the residual formula. Let's use the other definition: residual = Predicted - Given.

x=1: -1.1 - (-3.5) = 2.4

x=2: 2 - (-2.9) = 4.9

x=3: 5.1 - (-1.1) = 6.2

x=4: 8.2 - 2.2 = 6

x=5: 11.3 - 3.4 = 7.9

Now, let's list these residuals: 2.4, 4.9, 6.2, 6, 7.9. Now, let's see the pattern: from x=1 to x=3, the residuals are increasing (2.4, 4.9, 6.2), then at x=4, it decreases to 6, then increases to 7.9 at x=5. Wait, maybe the intended answer is based on the fact that if the residuals have no pattern, the line of best fit is appropriate.

Wait, the first part is to find residual values. Let's first complete the residual column:

For x = 1:
Residual = Given - Predicted = $-3.5-(-1.1)=-3.5 + 1.1=-2.4$

For x = 2:
Residual = Given - Predicted = $-2.9 - 2=-4.9$

For x = 3:
Residual = Given - Predicted = $-1.1-5.1=-6.2$

For x = 4:
Residual = Given - Predicted = $2.2 - 8.2=-6$

For x = 5:
Residual = Given - Predicted = $3.4 - 11.3=-7.9$

Now, for the second part, about the residual plot:

The key concept is that a residual plot for a linear model should show residuals with no pattern (randomly scattered). If there is a pattern (linear, curved), the linear model is not appropriate.

- Option 1: "Yes, the points have no pattern." – If residuals are randomly scattered (no pattern), the linear model (line of best fit) is appropriate.

- Option 2: "No, the points are evenly distributed about the x - axis." – Even distribution about the x - axis is a desirable property of residuals, so this is wrong.

- Option 3: "No, the points are in a linear pattern." – We need to check if residuals have a linear pattern. Let's plot the residuals against x: x=1 ( - 2.4), x=2 ( - 4.9), x=3 ( - 6.2), x=4 ( - 6), x=5 ( - 7.9). Let's see the slope between x=1 and x=2: $\frac{-4.9-(-2.4)}{2 - 1}=\frac{-2.5}{1}=-2.5$; between x=2 and x=3: $\frac{-6.2-(-4.9)}{3 - 2}=\frac{-1.3}{1}=-1.3$; between x=3 and x=4: $\frac{-6-(-6.2)}{4 - 3}=\frac{0.2}{1}=0.2$; between x=4 and x=5: $\frac{-7.9-(-6)}{5 - 4}=\frac{-1.9}{1}=-1.9$. The slopes are not consistent, so there is no linear pattern. So this option is wrong.

- Option 4: "Yes, the points are in a curved pattern." – A curved pattern in residuals means the linear model is not appropriate, so this is wrong.

So the correct answer for the second part is "Yes, the points have no pattern." (Option 1)

For the residual values:

The completed table (residual column) is:

| x | Given | Predicted | Residual |
|---|---|---|---|
| 1 | - 3.5 | - 1.1 | - 2.4 |
| 2 | - 2.9 | 2 | - 4.9 |
| 3 | - 1.1 | 5.1 | - 6.2 |
| 4 | 2.2 | 8.2 | - 6 |
| 5 | 3.4 | 11.3 | - 7.9 |

Answer:

Residual values: -2.4, -4.9, -6.2, -6, -7.9.
For the residual plot appropriateness: Yes, the points have no pattern. (Option: Yes, the points have no pattern.)