Discover a wealth of knowledge at Westonci.ca, where experts provide answers to your most pressing questions. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
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]
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]
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.