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Instructions: Create a residual plot of the following data using the regression line [tex]\( y = 2.9 + 0.34x \)[/tex].

Given data:
[tex]\[ \{(5, 5), (7, 3), (9, 9), (11, 7), (13, 5), (15, 9)\} \][/tex]

Step One: Find [tex]\( \hat{y} \)[/tex] for the [tex]\( x \)[/tex]-values in the data set. (Round to the nearest tenth.)

[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $\hat{y}$ \\
\hline
5 & 4.6 \\
\hline
7 & 5.3 \\
\hline
9 & 6.4 \\
\hline
11 & 7.7 \\
\hline
13 & 9.1 \\
\hline
15 & 10.0 \\
\hline
\end{tabular}
\][/tex]

Calculate the residuals [tex]\( y - \hat{y} \)[/tex] for each data point and plot them to create the residual plot.

Sagot :

GinnyAnsweridn
Let's proceed step-by-step to complete the residual plot.

Step One: Find [tex]\(\hat{y}\)[/tex] for the given [tex]\(x\)[/tex]-values in the data set using the regression line equation [tex]\(y = 2.9 + 0.34x\)[/tex].

Given [tex]\(x\)[/tex]-values:
- [tex]\(x = 5\)[/tex]
- [tex]\(x = 7\)[/tex]
- [tex]\(x = 9\)[/tex]
- [tex]\(x = 11\)[/tex]
- [tex]\(x = 13\)[/tex]
- [tex]\(x = 15\)[/tex]

Let's compute [tex]\(\hat{y}\)[/tex] for each [tex]\(x\)[/tex]-value using the regression line equation [tex]\(y = 2.9 + 0.34x\)[/tex]. Additionally, we will round each [tex]\(\hat{y}\)[/tex] to the nearest tenth.

1. For [tex]\(x = 5\)[/tex]:
[tex]\[\hat{y} = 2.9 + 0.34 \times 5 = 2.9 + 1.7 = 4.6\][/tex]

2. For [tex]\(x = 7\)[/tex]:
[tex]\[\hat{y} = 2.9 + 0.34 \times 7 = 2.9 + 2.38 \approx 5.3\][/tex]

3. For [tex]\(x = 9\)[/tex]:
[tex]\[\hat{y} = 2.9 + 0.34 \times 9 = 2.9 + 3.06 \approx 6.0\][/tex]

4. For [tex]\(x = 11\)[/tex]:
[tex]\[\hat{y} = 2.9 + 0.34 \times 11 = 2.9 + 3.74 \approx 6.6\][/tex]

5. For [tex]\(x = 13\)[/tex]:
[tex]\[\hat{y} = 2.9 + 0.34 \times 13 = 2.9 + 4.42 \approx 7.3\][/tex]

6. For [tex]\(x = 15\)[/tex]:
[tex]\[\hat{y} = 2.9 + 0.34 \times 15 = 2.9 + 5.1 = 8.0\][/tex]

Now, let’s fill in the table with these calculated [tex]\(\hat{y}\)[/tex] values:

[tex]\[ \begin{tabular}{|c|c|c|} \hline $x$ & \multicolumn{2}{|c|}{$\hat{y}$} \\ \hline 5 & 4.6 & $\checkmark$ \\ \hline 7 & 5.3 & $\checkmark$ \\ \hline 9 & 6.0 & $\checkmark$ \\ \hline 11 & 6.6 & $\checkmark$ \\ \hline 13 & 7.3 & $\checkmark$ \\ \hline 15 & 8.0 & $\checkmark$ \\ \hline \end{tabular} \][/tex]

These are the [tex]\(\hat{y}\)[/tex] values for the given [tex]\(x\)[/tex]-values, as calculated:

- For [tex]\(x = 5\)[/tex], [tex]\(\hat{y} = 4.6\)[/tex]
- For [tex]\(x = 7\)[/tex], [tex]\(\hat{y} = 5.3\)[/tex]
- For [tex]\(x = 9\)[/tex], [tex]\(\hat{y} = 6.0\)[/tex]
- For [tex]\(x = 11\)[/tex], [tex]\(\hat{y} = 6.6\)[/tex]
- For [tex]\(x = 13\)[/tex], [tex]\(\hat{y} = 7.3\)[/tex]
- For [tex]\(x = 15\)[/tex], [tex]\(\hat{y} = 8.0\)[/tex]

Next step will be to plot the residuals, but this table completes Step One.
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