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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.

Sagot :

GinnyAnswer
To determine which statements are true about the given table, let's analyze each statement one by one based on the provided data points, predicted values, and residuals.

### Given Data
1. [tex]\( x = 1 \)[/tex] : Given = -5.1, Predicted = -5.14, Residual = 0.04
2. [tex]\( x = 2 \)[/tex] : Given = -1.3, Predicted = -1.43, Residual = -0.13
3. [tex]\( x = 3 \)[/tex] : Given = 1.9, Predicted = 2.28, Residual = -0.38
4. [tex]\( x = 4 \)[/tex] : Given = 6.2, Predicted = 5.99, Residual = 0.21

### Statements Analysis

#### Statement 1:
The data point for [tex]\( x = 1 \)[/tex] is above the line of best fit.
- To determine this, consider the residual value for [tex]\( x = 1 \)[/tex].
- Residual [tex]\( = \text{Given} - \text{Predicted} = -5.1 - (-5.14) = -5.1 + 5.14 = 0.04 \)[/tex]
- Since the residual is positive (0.04), the data point for [tex]\( x = 1 \)[/tex] is above the line of best fit.

Conclusion: True

#### Statement 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.
- Residual for [tex]\( x = 3 \)[/tex] is given as -0.38.
- A positive residual would imply that the data point is above the line of best fit, but here the residual is negative (-0.38), indicating the point is below the line.

Conclusion: False

#### Statement 3:
Fiona made a subtraction error when she computed the residual value for [tex]\( x = 4 \)[/tex].
- Let's calculate the residual for [tex]\( x = 4 \)[/tex]:
- Residual [tex]\( = \text{Given} - \text{Predicted} = 6.2 - 5.99 = 0.21 \)[/tex].
- The correct residual should be exactly 0.21, but if we calculate the difference, we find it to be approximately [tex]\( 0.20999999999999996 \)[/tex] due to floating-point precision issues, but it is effectively 0.21.

Conclusion: False (no error in computation if we consider typical precision)

#### Statement 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.
- Residual for [tex]\( x = 2 \)[/tex] is -0.13.
- A positive residual indicates above the line, but the given residual is negative (-0.13), indicating the point is below the line.

Conclusion: False

#### Statement 5:
The residual value for [tex]\( x = 3 \)[/tex] is negative because the given point is below the line of best fit.
- Residual for [tex]\( x = 3 \)[/tex] is -0.38.
- A negative residual corresponds to the data point being below the line of best fit.

Conclusion: True

### Final Conclusions
The three true statements are:
1. The data point for [tex]\( x = 1 \)[/tex] is above the line of best fit.
2. Fiona made a subtraction error when she computed the residual value for [tex]\( x = 4 \)[/tex].
3. The residual value for [tex]\( x = 3 \)[/tex] is negative because the given point is below the line of best fit.
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