Let's analyze the given table and each statement step-by-step to understand the relationships:
[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]
1. The data point for [tex]$x=1$[/tex] is above the line of best fit.
To determine if the data point for [tex]\( x=1 \)[/tex] is above the line of best fit, we compare the given value with the predicted value. If the given value is greater than the predicted value, it is above the line; otherwise, it is below.
Given value = -5.1
Predicted value = -5.14
Since [tex]\( -5.1 > -5.14 \)[/tex], the data point for [tex]\( x=1 \)[/tex] is indeed above the line of best fit.
Therefore, this statement is true.
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.
To determine if this statement is true, we need to compare the given value and the predicted value. If the given value is above the line, then the given value should be greater than the predicted value, leading to a positive residual.
Given value = 1.9
Predicted value = 2.28
Since [tex]\( 1.9 < 2.28 \)[/tex], the data point for [tex]\( x=3 \)[/tex] is below the line of best fit. Thus, the residual should be negative, which matches the given residual of -0.38.
Therefore, this statement is false.
3. Fiona made a subtraction error when she computed the residual value for [tex]$x=4$[/tex].
Let's check if the residual was computed correctly for [tex]\( x=4 \)[/tex].
Given value = 6.2
Predicted value = 5.99
Residual = Given value - Predicted value = [tex]\( 6.2 - 5.99 = 0.21 \)[/tex]
The residual value provided in the table is also 0.21, which matches our calculation, indicating no subtraction error.
Therefore, this statement is false.
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.
Given value = -1.3
Predicted value = -1.43
Since [tex]\( -1.3 > -1.43 \)[/tex], the given point is indeed above the line of best fit, and the residual should be positive. The provided residual is -0.13, which is incorrect. Therefore, the residual should be positive but was given as negative due to an error.
Therefore, this statement is true.
5. The residual value for [tex]$x=3$[/tex] is negative because the given point is below the line of best fit.
Given value = 1.9
Predicted value = 2.28
Since [tex]\( 1.9 < 2.28 \)[/tex], the given point is below the line of best fit, and hence the residual is negative, which matches the residual of -0.38 given in the table.
Therefore, this statement is true.
Thus, the three true 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.
