Let's find the residuals for each [tex]\( x \)[/tex] value by calculating the difference between the given value and the predicted value. A residual is given by the formula:
[tex]\[ \text{Residual} = \text{Given Value} - \text{Predicted Value} \][/tex]
We'll apply this formula to each pair of given and predicted values from the table:
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]\[ \text{Residual} = 1.1 - 1.22 = 1.1 - 1.22 = -0.12 \][/tex]
4. For [tex]\( x = 4 \)[/tex]:
[tex]\[ \text{Residual} = 3.2 - 3.25 = 3.2 - 3.25 = -0.05 \][/tex]
5. For [tex]\( x = 5 \)[/tex]:
[tex]\[ \text{Residual} = 5.4 - 5.28 = 5.4 - 5.28 = 0.12 \][/tex]
Summarizing these results in the table, we get:
[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, let’s use these residuals to analyze whether the line of best fit is appropriate. Graphically, a residual plot can help us visualize this. Ideally, the residuals should be evenly distributed around the horizontal axis (i.e., the residuals should randomly scatter around zero without forming any specific pattern) to suggest that the line of best fit is appropriate.
- The residuals are: [tex]\(0.14, -0.09, -0.12, -0.05, 0.12\)[/tex].
By examining these residuals, we can determine the nature of their distribution:
- They do not consistently increase or decrease.
- They are scattered relatively randomly around the horizontal axis.
Based on this analysis, the residual plot indicates that the line of best fit is appropriate for the data because the residuals do not form a specific pattern and are evenly distributed around the x-axis.
So, the correct conclusion is:
Yes, the points have no pattern.
