To determine the coefficient of determination, we first carry out a linear regression analysis on the given data sets for age and bone density. Here is the step-by-step process:
1. Given Data:
- Age: [tex]\( [34, 45, 48, 60, 65] \)[/tex]
- Bone Density: [tex]\( [357, 341, 331, 329, 325] \)[/tex]
2. Perform Linear Regression:
- To fit a linear regression model, we find the best-fitting line of the form [tex]\(\hat{y} = b_0 + b_1 x\)[/tex].
3. Calculate Regression Parameters:
- Slope ([tex]\(b_1\)[/tex]): -0.9638870650032828
- Intercept ([tex]\(b_0\)[/tex]): 385.1799080761655
4. Correlation Coefficient ([tex]\(r\)[/tex]): -0.9266111039699972
5. P-value: 0.023601515119593738 (which is used to determine the statistical significance of the correlation coefficient, though not directly required for finding [tex]\(R^2\)[/tex]).
6. Calculate Coefficient of Determination ([tex]\(R^2\)[/tex]):
- The coefficient of determination is the square of the correlation coefficient. [tex]\( R^2 = r^2 \)[/tex].
[tex]\[
R^2 = (-0.9266111039699972)^2
\][/tex]
[tex]\[
R^2 = 0.8586081380004971
\][/tex]
7. Round to Three Decimal Places:
- The rounded value of [tex]\( R^2 \)[/tex] to three decimal places is [tex]\( 0.859 \)[/tex].
Thus, the value of the coefficient of determination, rounded to three decimal places, is [tex]\( \boxed{0.859} \)[/tex].
