To calculate the coefficient of determination $R^2$, we first need to determine the linear relationship between the variables, using the following steps:

Step 1: Calculate the Mean of the Variables

The first step is to find the means of both the Age and Bone Density values.

Given:

• Age: $33, 37, 45, 49, 57$
• Bone Density: $345, 335, 330, 325, 310$

The formula for the mean is: $\text{Mean} = \frac{\sum X}{n}$ Where:

• $X$ is the variable
• $n$ is the number of data points.

Mean of Age:

$\text{Mean of Age} = \frac{33 + 37 + 45 + 49 + 57}{5} = \frac{221}{5} = 44.2$

Mean of Bone Density:

$\text{Mean of Bone Density} = \frac{345 + 335 + 330 + 325 + 310}{5} = \frac{1645}{5} = 329$

Step 2: Calculate the Slope $b$ and Intercept $a$ of the Regression Line

To find the slope $b$ and intercept $a$ of the best-fit line, we use the following formulas:

1. Slope $b$: $b = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2}$ Where:

• $X_i$ and $Y_i$ are the data points for Age and Bone Density, respectively.
• $\bar{X}$ and $\bar{Y}$ are the means of Age and Bone Density, respectively.

2. Intercept $a$: $a = \bar{Y} - b \cdot \bar{X}$

Step 3: Calculate the Total Sum of Squares (SST), Explained Sum of Squares (SSE), and Residual Sum of Squares (SSR)

Once we have the regression line, we can calculate $R^2$ using the formula: $R^2 = \frac{\text{SSE}}{\text{SST}}$ Where:

• $\text{SSE}$ is the Explained Sum of Squares
• $\text{SST}$ is the Total Sum of Squares

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