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To find the coefficient of determination (\( R^2 \)), we need to first determine the linear regression equation that best fits the data and then calculate \( R^2 \), which measures the proportion of the variance in the dependent variable \( y \) that can be explained by the independent variable \( x \).

The steps for solving this are as follows:

1. **Calculate the mean of \( x \) and \( y \)**:
   \[
   \bar{x} = \frac{1 + 2 + \ldots + 12}{12} = 6.5
   \]
   \[
   \bar{y} = \frac{58 + 62 + \ldots + 90}{12}
   \]

2. **Compute the slope \( m \) of the regression line**:
   \[
   m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}
   \]

3. **Find the intercept \( b \) using**:
   \[
   b = \bar{y} - m\bar{x}
   \]

4. **Determine the predicted \( y \)-values using the regression equation**:
   \[
   \hat{y} = mx + b
   \]

5. **Calculate \( R^2 \) using the formula**:
   \[
   R^2 = 1 - \frac{\sum{(y_i - \hat{y})^2}}{\sum{(y_i - \bar{y})^2}}
   \]

By performing these calculations, the best approximation for the coefficient of determination will be identified. I’ll go through these calculations step-by-step now.The calculated coefficient of determination (\( R^2 \)) is approximately 0.715. Since none of the provided answer choices exactly match this value, the closest approximation would be option **B. 0.534**.

Would you like more details on the calculations, or do you have any other questions?

Here are some related questions to expand your understanding:
1. What is the significance of the slope and intercept in a linear regression equation?
2. How does the coefficient of determination (\( R^2 \)) relate to the correlation coefficient?
3. What are some limitations of using linear regression for modeling relationships?
4. How would you interpret an \( R^2 \) value close to 1 or close to 0?
5. What is the impact of outliers on the \( R^2 \) value?

**Tip:** When interpreting \( R^2 \), remember that it represents the proportion of variance in the dependent variable that is predictable from the independent variable. The closer \( R^2 \) is to 1, the better the model explains the data.

Given the data shown below, which of the following is the best approximation of the coefficient of determination (R^2)? The data points are (1, 58), (2, 62), (3, 63), (4, 59), (5, 47), (6, 68), (7, 77), (8, 81), (9, 84), (10, 79), (11, 82), (12, 90).

Learn how to approximate the coefficient of determination (R^2) for a set of data points using linear regression. This problem involves calculating R^2 to measure the strength of the linear relationship between the variables. Suitable for students in Grades 10-12.

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