To determine which equation best models the given set of data, we will perform the following steps:
1. Calculate the linear regression line:
First, we need to find the best fit line in the form \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept.
Given:
[tex]\[
x = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
\][/tex]
[tex]\[
y = [32, 67, 79, 91, 98, 106, 114, 120, 126, 132]
\][/tex]
The calculated slope \( m \) and y-intercept \( b \) are:
[tex]\[
m = 9.67 \quad \text{(approximately)} \quad \text{and} \quad b = 53.00 \quad \text{(approximately)}
\][/tex]
So, the best fit line is:
[tex]\[
y = 9.67x + 53.00
\][/tex]
2. Generate y-values based on the regression line:
Using the equation \( y = 9.67x + 53.00 \), the predicted \( y \)-values are:
[tex]\[
[53, 62.67, 72.33, 82, 91.67, 101.33, 111, 120.67, 130.33, 140]
\][/tex]
3. Calculate the error term for each given equation:
We will use the sum of the squared errors (SSE) to determine how well each given equation fits the data. The equation with the smallest SSE will be the best fit.
[tex]\[
\text{Error} = \sum_{i=1}^{n} (y_{\text{actual},i} - y_{\text{predicted},i})^2
\][/tex]
Given options are:
- Option A: \( y = 33x - 32.7 \)
- Option B: \( y = 33x + 32.7 \)
- Option C: \( y = 33 \sqrt{x - 32.7} \)
- Option D: \( y = 33 \sqrt{x} + 32.7 \)
Calculated errors for each of these options:
- Error A: 49380.90
- Error B: 117396.90
- Error C: \( \text{(undefined due to square root of a negative number)} \)
- Error D: 4.66
4. Identify the best fit option:
From the errors calculated, we find that the equation with the smallest error is option D. This indicates that option D fits the given data set best.
Therefore, the equation that best models the given set of data is:
[tex]\[
\boxed{y = 33 \sqrt{x} + 32.7}
\][/tex]
