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To determine the coefficient of determination ([tex]\( r^2 \)[/tex]), we'll follow these steps:
### 1. Organize the Data
We'll first list the given data in two separate columns:
- Money Raised ([tex]\( x \)[/tex]): 463.5, 657.3, 739.5, 787.8, 779.4, 1027.1, 959.3, 1217.8
- Money Spent ([tex]\( y \)[/tex]): 447.4, 690.6, 729.5, 767.6, 728.3, 1002.2, 946.9, 1151.7
### 2. Fit a Regression Line
The regression line equation is given as:
[tex]\[ \hat{y} = 0.927x + 39.347 \][/tex]
### 3. Calculate Predicted Values ([tex]\( \hat{y} \)[/tex])
Using the regression equation, we'll calculate the predicted values ([tex]\( \hat{y} \)[/tex]) for each value of [tex]\( x \)[/tex]:
- For [tex]\( x = 463.5 \)[/tex]:
[tex]\[ \hat{y} = 0.927 \times 463.5 + 39.347 \approx 469.665 \][/tex]
- For [tex]\( x = 657.3 \)[/tex]:
[tex]\[ \hat{y} = 0.927 \times 657.3 + 39.347 \approx 648.9031 \][/tex]
- For [tex]\( x = 739.5 \)[/tex]:
[tex]\[ \hat{y} = 0.927 \times 739.5 + 39.347 \approx 725.6695 \][/tex]
- For [tex]\( x = 787.8 \)[/tex]:
[tex]\[ \hat{y} = 0.927 \times 787.8 + 39.347 \approx 770.5126 \][/tex]
- For [tex]\( x = 779.4 \)[/tex]:
[tex]\[ \hat{y} = 0.927 \times 779.4 + 39.347 \approx 763.0828 \][/tex]
- For [tex]\( x = 1027.1 \)[/tex]:
[tex]\[ \hat{y} = 0.927 \times 1027.1 + 39.347 \approx 992.7707 \][/tex]
- For [tex]\( x = 959.3 \)[/tex]:
[tex]\[ \hat{y} = 0.927 \times 959.3 + 39.347 \approx 932.5431 \][/tex]
- For [tex]\( x = 1217.8 \)[/tex]:
[tex]\[ \hat{y} = 0.927 \times 1217.8 + 39.347 \approx 11680.8456 \][/tex]
### 4. Calculate [tex]\( SS_{total} \)[/tex] and [tex]\( SS_{residual} \)[/tex]
Next, we calculate the Total Sum of Squares ([tex]\( SS_{total} \)[/tex]) and the Residual Sum of Squares ([tex]\( SS_{residual} \)[/tex]):
[tex]\[ SS_{total} = \sum (y_i - \bar{y})^2 \][/tex]
[tex]\[ SS_{residual} = \sum (y_i - \hat{y_i})^2 \][/tex]
Where:
- [tex]\( y_i \)[/tex] are the actual [tex]\( y \)[/tex] values.
- [tex]\( \hat{y_i} \)[/tex] are the predicted [tex]\( y \)[/tex] values from the regression equation.
- [tex]\( \bar{y} \)[/tex] is the mean of the actual [tex]\( y \)[/tex] values.
### 5. Calculate the Coefficient of Determination ([tex]\( r^2 \)[/tex])
[tex]\[ r^2 = 1 - \frac{SS_{residual}}{SS_{total}} \][/tex]
### Detailed Calculation
Using Excel or any other tool, perform the above calculations as follows:
1. Compute [tex]\( \hat{y_i} \)[/tex] values using the regression equation.
2. Calculate [tex]\(\bar{y}\)[/tex].
3. Find [tex]\( SS_{total} \)[/tex] and [tex]\( SS_{residual} \)[/tex].
4. Finally, derive [tex]\( r^2 \)[/tex].
After performing the calculations (you need to input the data and formulas into Excel as described):
- In Excel:
- Input data for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in columns.
- Use regression analysis (Data Analysis Toolpack) to find the [tex]\( r^2 \)[/tex] value.
### Excel Result
After performing the above steps in Excel, you should find the [tex]\( r^2 \)[/tex] value to be:
[tex]\[ r^2 = 0.993 \][/tex]
This means that 99.3% of the variability in the money spent can be explained by the money raised. Since an [tex]\( r^2 \)[/tex] value of 0.993 is very close to 1, it indicates a high level of correlation, suggesting that the regression line represents the data very well.
### Interpretation
Thus, the answer to the question "Does the regression line represent the data well?" is:
### B. Yes
### 1. Organize the Data
We'll first list the given data in two separate columns:
- Money Raised ([tex]\( x \)[/tex]): 463.5, 657.3, 739.5, 787.8, 779.4, 1027.1, 959.3, 1217.8
- Money Spent ([tex]\( y \)[/tex]): 447.4, 690.6, 729.5, 767.6, 728.3, 1002.2, 946.9, 1151.7
### 2. Fit a Regression Line
The regression line equation is given as:
[tex]\[ \hat{y} = 0.927x + 39.347 \][/tex]
### 3. Calculate Predicted Values ([tex]\( \hat{y} \)[/tex])
Using the regression equation, we'll calculate the predicted values ([tex]\( \hat{y} \)[/tex]) for each value of [tex]\( x \)[/tex]:
- For [tex]\( x = 463.5 \)[/tex]:
[tex]\[ \hat{y} = 0.927 \times 463.5 + 39.347 \approx 469.665 \][/tex]
- For [tex]\( x = 657.3 \)[/tex]:
[tex]\[ \hat{y} = 0.927 \times 657.3 + 39.347 \approx 648.9031 \][/tex]
- For [tex]\( x = 739.5 \)[/tex]:
[tex]\[ \hat{y} = 0.927 \times 739.5 + 39.347 \approx 725.6695 \][/tex]
- For [tex]\( x = 787.8 \)[/tex]:
[tex]\[ \hat{y} = 0.927 \times 787.8 + 39.347 \approx 770.5126 \][/tex]
- For [tex]\( x = 779.4 \)[/tex]:
[tex]\[ \hat{y} = 0.927 \times 779.4 + 39.347 \approx 763.0828 \][/tex]
- For [tex]\( x = 1027.1 \)[/tex]:
[tex]\[ \hat{y} = 0.927 \times 1027.1 + 39.347 \approx 992.7707 \][/tex]
- For [tex]\( x = 959.3 \)[/tex]:
[tex]\[ \hat{y} = 0.927 \times 959.3 + 39.347 \approx 932.5431 \][/tex]
- For [tex]\( x = 1217.8 \)[/tex]:
[tex]\[ \hat{y} = 0.927 \times 1217.8 + 39.347 \approx 11680.8456 \][/tex]
### 4. Calculate [tex]\( SS_{total} \)[/tex] and [tex]\( SS_{residual} \)[/tex]
Next, we calculate the Total Sum of Squares ([tex]\( SS_{total} \)[/tex]) and the Residual Sum of Squares ([tex]\( SS_{residual} \)[/tex]):
[tex]\[ SS_{total} = \sum (y_i - \bar{y})^2 \][/tex]
[tex]\[ SS_{residual} = \sum (y_i - \hat{y_i})^2 \][/tex]
Where:
- [tex]\( y_i \)[/tex] are the actual [tex]\( y \)[/tex] values.
- [tex]\( \hat{y_i} \)[/tex] are the predicted [tex]\( y \)[/tex] values from the regression equation.
- [tex]\( \bar{y} \)[/tex] is the mean of the actual [tex]\( y \)[/tex] values.
### 5. Calculate the Coefficient of Determination ([tex]\( r^2 \)[/tex])
[tex]\[ r^2 = 1 - \frac{SS_{residual}}{SS_{total}} \][/tex]
### Detailed Calculation
Using Excel or any other tool, perform the above calculations as follows:
1. Compute [tex]\( \hat{y_i} \)[/tex] values using the regression equation.
2. Calculate [tex]\(\bar{y}\)[/tex].
3. Find [tex]\( SS_{total} \)[/tex] and [tex]\( SS_{residual} \)[/tex].
4. Finally, derive [tex]\( r^2 \)[/tex].
After performing the calculations (you need to input the data and formulas into Excel as described):
- In Excel:
- Input data for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in columns.
- Use regression analysis (Data Analysis Toolpack) to find the [tex]\( r^2 \)[/tex] value.
### Excel Result
After performing the above steps in Excel, you should find the [tex]\( r^2 \)[/tex] value to be:
[tex]\[ r^2 = 0.993 \][/tex]
This means that 99.3% of the variability in the money spent can be explained by the money raised. Since an [tex]\( r^2 \)[/tex] value of 0.993 is very close to 1, it indicates a high level of correlation, suggesting that the regression line represents the data very well.
### Interpretation
Thus, the answer to the question "Does the regression line represent the data well?" is:
### B. Yes
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