Let's go through the problem step-by-step:
### Step 1: Calculate the Residuals
The residuals are calculated by subtracting the predicted values from the given values. Here are the values provided in the table:
- For [tex]\( x = 1 \)[/tex]:
- Given: -2.7
- Predicted: -2.84
- Residual: [tex]\( -2.7 - (-2.84) = -2.7 + 2.84 = 0.14 \)[/tex] (approximately 0.13999999999999968)
- For [tex]\( x = 2 \)[/tex]:
- Given: -0.9
- Predicted: -0.81
- Residual: [tex]\( -0.9 - (-0.81) = -0.9 + 0.81 = -0.09 \)[/tex] (approximately -0.08999999999999997)
- For [tex]\( x = 3 \)[/tex]:
- Given: 1.1
- Predicted: 1.22
- Residual: [tex]\( 1.1 - 1.22 = -0.12 \)[/tex] (approximately -0.11999999999999988)
- For [tex]\( x = 4 \)[/tex]:
- Given: 3.2
- Predicted: 3.25
- Residual: [tex]\( 3.2 - 3.25 = -0.05 \)[/tex] (approximately -0.04999999999999982)
- For [tex]\( x = 5 \)[/tex]:
- Given: 5.4
- Predicted: 5.28
- Residual: [tex]\( 5.4 - 5.28 = 0.12 \)[/tex] (approximately 0.1200000000000001)
Updating the table with these residuals:
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
$x$ & Given & Predicted & Residual \\
\hline
1 & -2.7 & -2.84 & 0.13999999999999968 \\
\hline
2 & -0.9 & -0.81 & -0.08999999999999997 \\
\hline
3 & 1.1 & 1.22 & -0.11999999999999988 \\
\hline
4 & 3.2 & 3.25 & -0.04999999999999982 \\
\hline
5 & 5.4 & 5.28 & 0.1200000000000001 \\
\hline
\end{tabular}
\][/tex]
### Step 2: Plot the Residuals
Next, we plot these residuals on a graph with [tex]\( x \)[/tex] on the x-axis and the residuals on the y-axis. The points to plot are:
- (1, 0.14)
- (2, -0.09)
- (3, -0.12)
- (4, -0.05)
- (5, 0.12)
### Step 3: Analyze the Residual Plot
The important aspect of analyzing a residual plot is to determine if there is any discernible pattern:
- If the points display no obvious pattern and are scattered randomly around the x-axis, it implies that the line of best fit is appropriate.
- If there is a clear pattern, such as a curve or a systematic arrangement, it suggests that the line of best fit may not be the best model for the data.
Given our residuals:
[tex]\[
[0.13999999999999968, -0.08999999999999997, -0.11999999999999988, -0.04999999999999982, 0.1200000000000001]
\][/tex]
### Conclusion
On evaluating these points, they are scattered randomly around the x-axis without showing a specific pattern.
Thus, for the given problem, the correct choice is:
Yes, the points have no pattern.
This indicates that the line of best fit is appropriate for the data provided.
