To determine the missing values [tex]\(a\)[/tex] and [tex]\(b\)[/tex] in the table, let's analyze the line of best fit [tex]\(y = 1.82x - 4.3\)[/tex].
1. Predicted Value for [tex]\(x = 3\)[/tex]:
Using the line of best fit formula:
[tex]\[
y_{predicted}(3) = 1.82 \times 3 - 4.3
\][/tex]
[tex]\[
y_{predicted}(3) = 5.46 - 4.3
\][/tex]
[tex]\[
y_{predicted}(3) = 1.16
\][/tex]
So, [tex]\(a\)[/tex] is the predicted value for [tex]\(x=3\)[/tex], and we found:
[tex]\[
a = 1.16
\][/tex]
2. Residual for [tex]\(x = 3\)[/tex]:
Given that the residual for [tex]\(x=3\)[/tex] is [tex]\(-0.06\)[/tex]. We can verify this:
[tex]\[
\text{Residual} = \text{Given} - \text{Predicted}
\][/tex]
[tex]\[
-0.06 = 1.1 - 1.16
\][/tex]
This confirms the calculated predicted value [tex]\(a = 1.16\)[/tex].
3. Predicted Value for [tex]\(x = 4\)[/tex]:
Again, using the line of best fit formula:
[tex]\[
y_{predicted}(4) = 1.82 \times 4 - 4.3
\][/tex]
[tex]\[
y_{predicted}(4) = 7.28 - 4.3
\][/tex]
[tex]\[
y_{predicted}(4) = 2.98
\][/tex]
The predicted value for [tex]\(x=4\)[/tex] is [tex]\(\approx 2.98\)[/tex].
4. Residual for [tex]\(x = 4\)[/tex]:
The given value for [tex]\(x=4\)[/tex] is [tex]\(3.1\)[/tex]. The residual is calculated as:
[tex]\[
\text{Residual} = \text{Given} - \text{Predicted}
\][/tex]
[tex]\[
b = 3.1 - 2.98
\][/tex]
[tex]\[
b \approx 0.12
\][/tex]
So, the missing values [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
[tex]\[
a = 1.16 \quad \text{and} \quad b = 0.12
\][/tex]
Thus, the correct choice is:
- [tex]\(a = 1.16\)[/tex] and [tex]\(b = 0.12\)[/tex]
