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Shanti wrote the predicted values for a data set using the line of best fit [tex]y=2.55x-3.15[/tex]. She computed two of the residual values.

\begin{tabular}{|c|c|c|c|}
\hline
[tex]$x$[/tex] & Given & Predicted & Residual \\
\hline
1 & -0.7 & -0.6 & -0.1 \\
\hline
2 & 2.3 & 1.95 & 0.35 \\
\hline
3 & 4.1 & 4.5 & [tex]$a$[/tex] \\
\hline
4 & 7.2 & 7.05 & [tex]$b$[/tex] \\
\hline
\end{tabular}

What are the values of [tex]$a$[/tex] and [tex]$b$[/tex]?

A. [tex]$a=0.4$[/tex] and [tex]$b=-0.15$[/tex]
B. [tex]$a=-0.4$[/tex] and [tex]$b=0.15$[/tex]
C. [tex]$a=8.6$[/tex] and [tex]$b=14.25$[/tex]
D. [tex]$a=-8.6$[/tex] and [tex]$b=-14.25$[/tex]


Sagot :

To determine the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] based on the above table detailing the given values, predicted values, and residuals, let’s follow these steps systematically.

First, let's understand how residuals are calculated. A residual is the difference between the given (observed) values and the predicted values from the line of best fit.

The residual can be computed using the formula:
[tex]\[ \text{Residual} = \text{Given value} - \text{Predicted value} \][/tex]

Given the line of best fit:
[tex]\[ y = 2.55x - 3.15 \][/tex]

We already have:
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ \text{Given value} = -0.7, \quad \text{Predicted value} = -0.6, \quad \text{Residual} = -0.7 + 0.6 = -0.1 \][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ \text{Given value} = 2.3, \quad \text{Predicted value} = 1.95, \quad \text{Residual} = 2.3 - 1.95 = 0.35 \][/tex]

We need to compute the residuals for:
- [tex]\( x = 3 \)[/tex]
- [tex]\( x = 4 \)[/tex]

### Step-by-Step Calculation:

1. For [tex]\( x = 3 \)[/tex]:
- Substituting [tex]\( x = 3 \)[/tex] into the line of best fit equation:
[tex]\[ y = 2.55(3) - 3.15 = 7.65 - 3.15 = 4.5 \][/tex]
- The given value for [tex]\( x = 3 \)[/tex] is [tex]\( 4.1 \)[/tex].
- Thus, the residual [tex]\( a \)[/tex] is:
[tex]\[ a = 4.1 - 4.5 = -0.4 \][/tex]

2. For [tex]\( x = 4 \)[/tex]:
- Substituting [tex]\( x = 4 \)[/tex] into the line of best fit equation:
[tex]\[ y = 2.55(4) - 3.15 = 10.2 - 3.15 = 7.05 \][/tex]
- The given value for [tex]\( x = 4 \)[/tex] is [tex]\( 7.2 \)[/tex].
- Thus, the residual [tex]\( b \)[/tex] is:
[tex]\[ b = 7.2 - 7.05 = 0.15 \][/tex]

Hence, the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are:
[tex]\[ a = -0.4 \][/tex]
[tex]\[ b = 0.15 \][/tex]

The correct answer is:
[tex]\[ \boxed{a = -0.4 \text{ and } b = 0.15} \][/tex]