To find the estimated [tex]\( y \)[/tex]-intercept ([tex]\( b_0 \)[/tex]) of the regression line, we need to follow these steps:
1. Calculate the means of [tex]\( X \)[/tex] and [tex]\( Y \)[/tex]:
[tex]\[
\bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i
\][/tex]
[tex]\[
\bar{Y} = \frac{1}{n} \sum_{i=1}^{n} Y_i
\][/tex]
Here, [tex]\( n \)[/tex] is the number of data points, [tex]\( X_i \)[/tex] represents the age values, and [tex]\( Y_i \)[/tex] represents the bone density values.
2. Calculate the slope ([tex]\( b_1 \)[/tex]):
[tex]\[
b_1 = \frac{\sum_{i=1}^{n} (X_i - \bar{X})(Y_i - \bar{Y})}{\sum_{i=1}^{n} (X_i - \bar{X})^2}
\][/tex]
3. Calculate the [tex]\( y \)[/tex]-intercept ([tex]\( b_0 \)[/tex]):
[tex]\[
b_0 = \bar{Y} - b_1 \bar{X}
\][/tex]
Given the numerical solution:
- The [tex]\( y \)[/tex]-intercept ([tex]\( b_0 \)[/tex]) is determined to be 385.18 when rounded to three decimal places.
- The slope ([tex]\( b_1 \)[/tex]) is approximately -0.9638871 (though not directly needed for determining [tex]\( b_0 \)[/tex], it's useful for understanding the whole equation).
Hence, after performing the calculations and following the steps to find the estimated [tex]\( y \)[/tex]-intercept for the regression line, we obtain:
[tex]\[
b_0 = 385.180
\][/tex]
Therefore, the estimated [tex]\( y \)[/tex]-intercept of the regression line, rounded to three decimal places, is 385.180.
