Fiona wrote the predicted and residual values for a data set using the line of best fit [tex]\( y=3.71x-8.85 \)[/tex].

[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
x & Given & Predicted & Residual \\
\hline
1 & -5.1 & -5.14 & 0.04 \\
\hline
2 & -1.3 & -1.43 & -0.13 \\
\hline
3 & 1.9 & 2.28 & -0.38 \\
\hline
4 & 6.2 & 5.99 & 0.21 \\
\hline
\end{tabular}
\][/tex]

Which statements are true about the table? Select three options.

A. The data point for [tex]\( x=1 \)[/tex] is above the line of best fit.

B. The residual value for [tex]\( x=3 \)[/tex] should be a positive number because the data point is above the line of best fit.

C. Fiona made a subtraction error when she computed the residual value for [tex]\( x=4 \)[/tex].

D. The residual value for [tex]\( x=2 \)[/tex] should be a positive number because the given point is above the line of best fit.

E. The residual value for [tex]\( x=3 \)[/tex] is negative because the given point is below the line of best fit.



Answer :

Let’s analyze each statement given the data and compute the necessary steps to confirm the validity of each statement.

1. The data point for [tex]\(x=1\)[/tex] is above the line of best fit.
- From the table: When [tex]\(x=1\)[/tex], the given value is [tex]\(-5.1\)[/tex] and the predicted value is [tex]\(-5.14\)[/tex].
- Since [tex]\(-5.1 > -5.14\)[/tex], the given value is indeed above the predicted value.
- Therefore, this statement is correct.

2. The residual value for [tex]\(x=3\)[/tex] should be a positive number because the data point is above the line of best fit.
- From the table: When [tex]\(x=3\)[/tex], the given value is [tex]\(1.9\)[/tex] and the predicted value is [tex]\(2.28\)[/tex].
- Since [tex]\(1.9 < 2.28\)[/tex], the given value is below the predicted value.
- Therefore, this statement is incorrect.

3. Fiona made a subtraction error when she computed the residual value for [tex]\(x=4\)[/tex].
- From the table: When [tex]\(x=4\)[/tex], the given value is [tex]\(6.2\)[/tex] and the predicted value is [tex]\(5.99\)[/tex]. The residual Fiona computed is [tex]\(0.21\)[/tex].
- Residual should be: [tex]\[ \text{Residual} = \text{Given} - \text{Predicted} = 6.2 - 5.99 = 0.21 \][/tex]
- Fiona's calculated residual is [tex]\(0.21\)[/tex], which matches the theoretically calculated one.
- Therefore, this statement is incorrect.

4. The residual value for [tex]\(x=2\)[/tex] should be a positive number because the given point is above the line of best fit.
- From the table: When [tex]\(x=2\)[/tex], the given value is [tex]\(-1.3\)[/tex] and the predicted value is [tex]\(-1.43\)[/tex].
- Since [tex]\(-1.3 > -1.43\)[/tex], the given value is above the predicted value.
- Therefore, the residual should indeed be positive.
- This means this statement is correct.

5. The residual value for [tex]\(x=3\)[/tex] is negative because the given point is below the line of best fit.
- From the table: When [tex]\(x=3\)[/tex], the given value is [tex]\(1.9\)[/tex] and the predicted value is [tex]\(2.28\)[/tex].
- Since [tex]\(1.9 < 2.28\)[/tex], the given value is below the predicted value.
- This indicates the residual is negative.
- Therefore, this statement is correct.

Based on this analysis, the three correct statements are:
- The data point for [tex]\(x=1\)[/tex] is above the line of best fit.
- The residual value for [tex]\(x=2\)[/tex] should be a positive number because the given point is above the line of best fit.
- The residual value for [tex]\(x=3\)[/tex] is negative because the given point is below the line of best fit.