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]
