Let's start by calculating the residuals for each data point. The residual for each data point is calculated as the difference between the given value and the predicted value.
The formula for the residual is:
[tex]\[ \text{Residual} = \text{Given} - \text{Predicted} \][/tex]
Let's apply this formula to find the residuals for each data point.
1. For [tex]\( x = 1 \)[/tex]:
[tex]\[ \text{Residual} = -2.7 - (-2.84) = -2.7 + 2.84 = 0.14 \][/tex]
2. For [tex]\( x = 2 \)[/tex]:
[tex]\[ \text{Residual} = -0.9 - (-0.81) = -0.9 + 0.81 = -0.09 \][/tex]
3. For [tex]\( x = 3 \)[/tex]:
[tex]\[ \text{Residual} = 1.1 - 1.22 = -0.12 \][/tex]
4. For [tex]\( x = 4 \)[/tex]:
[tex]\[ \text{Residual} = 3.2 - 3.25 = -0.05 \][/tex]
5. For [tex]\( x = 5 \)[/tex]:
[tex]\[ \text{Residual} = 5.4 - 5.28 = 0.12 \][/tex]
So the table with the residuals filled in looks like this:
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
$x$ & Given & Predicted & Residual \\
\hline
1 & -2.7 & -2.84 & 0.14 \\
\hline
2 & -0.9 & -0.81 & -0.09 \\
\hline
3 & 1.1 & 1.22 & -0.12 \\
\hline
4 & 3.2 & 3.25 & -0.05 \\
\hline
5 & 5.4 & 5.28 & 0.12 \\
\hline
\end{tabular}
\][/tex]
Next, we use these residuals to create a residual plot. The residual plot will have the [tex]\( x \)[/tex] values on the x-axis and the corresponding residuals on the y-axis. Plotting these points:
[tex]\[ (1, 0.14), (2, -0.09), (3, -0.12), (4, -0.05), (5, 0.12) \][/tex]
After plotting the points, we analyze the residual plot to determine if the line of best fit is appropriate. We look for patterns in the residual plot.
- If the points display no obvious pattern and are randomly scattered around the x-axis, the line of best fit is appropriate.
- If the points show a specific pattern (such as a curve), then the line of best fit is not appropriate.
In this case, since the residuals do not form a clear pattern (like a curve or linear trend) but are rather scattered around the x-axis without any systematic arrangement, it indicates that the line of best fit is appropriate for the data.
Therefore, the correct interpretation is:
Yes, the points have no pattern.
