To solve this problem, follow these steps:
### Step 1: Determine the Predicted Value of [tex]\( y \)[/tex]
Given the line of best fit [tex]\(\hat{y} = 8 + 2.2x\)[/tex], we need to find the predicted value of [tex]\( y \)[/tex] when [tex]\( x = 11 \)[/tex].
[tex]\[
\hat{y} = 8 + 2.2 \times 11
\][/tex]
Calculate:
[tex]\[
\hat{y} = 8 + 24.2 = 32.2
\][/tex]
So, the predicted value of [tex]\( y \)[/tex] for [tex]\( x = 11 \)[/tex] is 32.2.
### Step 2: Calculate the Residual
The residual is the difference between the observed value and the predicted value. The observed value of [tex]\( y \)[/tex] when [tex]\( x = 11 \)[/tex] is 33.7.
[tex]\[
\text{Residual} = \text{Observed } y - \text{Predicted } y = 33.7 - 32.2
\][/tex]
Calculate:
[tex]\[
\text{Residual} = 1.5
\][/tex]
So, the residual for the value [tex]\( x = 11 \)[/tex] is 1.5.
### Step 3: Interpret the Residual
To interpret the residual, we need to compare the observed value with the predicted value:
- If the observed value is greater than the predicted value, it indicates that the observed value is above the average value predicted by the model for that particular [tex]\( x \)[/tex] value.
- If the observed value is less than the predicted value, it indicates that the observed value is below the average value predicted by the model for that particular [tex]\( x \)[/tex] value.
In this case, the observed value of [tex]\( y = 33.7 \)[/tex] is greater than the predicted value of [tex]\( y = 32.2 \)[/tex].
Hence:
- The value 33.7 is above the average value for [tex]\( y \)[/tex] when [tex]\( x = 11 \)[/tex].
### Summary of Answers:
- Predicted value of [tex]\( y \)[/tex] for [tex]\( x = 11 \)[/tex]: 32.2
- Residual for the value [tex]\( x = 11 \)[/tex]: 1.5
- Best interpretation for the residual: The value 33.7 is above the average value for [tex]\( y \)[/tex] when [tex]\( x = 11 \)[/tex]
