To find the estimated slope [tex]\( b_1 \)[/tex] of the regression line, we first need to calculate the required intermediate values using the data provided. Here, we have the ages of five women and their corresponding bone densities.
Given data:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
\text{Age} & 34 & 45 & 48 & 60 & 65 \\
\hline
\text{Bone Density} & 357 & 341 & 331 & 329 & 325 \\
\hline
\end{array}
\][/tex]
Step-by-step solution to find the estimated slope [tex]\( b_1 \)[/tex]:
1. Calculate the mean of Age ([tex]\( \bar{x} \)[/tex]):
[tex]\[
\bar{x} = \frac{34 + 45 + 48 + 60 + 65}{5} = \frac{252}{5} = 50.4
\][/tex]
2. Calculate the mean of Bone Density ([tex]\( \bar{y} \)[/tex]):
[tex]\[
\bar{y} = \frac{357 + 341 + 331 + 329 + 325}{5} = \frac{1683}{5} = 336.6
\][/tex]
3. Compute the numerator of the slope (the sum of the products of the differences from the means):
[tex]\[
\sum (x_i - \bar{x})(y_i - \bar{y}) = (34 - 50.4)(357 - 336.6) + (45 - 50.4)(341 - 336.6) + (48 - 50.4)(331 - 336.6) + (60 - 50.4)(329 - 336.6) + (65 - 50.4)(325 - 336.6)
\][/tex]
From the calculations, we get:
[tex]\[
-587.2
\][/tex]
4. Compute the denominator of the slope (the sum of the squares of the differences from the mean of Age):
[tex]\[
\sum (x_i - \bar{x})^2 = (34 - 50.4)^2 + (45 - 50.4)^2 + (48 - 50.4)^2 + (60 - 50.4)^2 + (65 - 50.4)^2
\][/tex]
From the calculations, we get:
[tex]\[
609.2
\][/tex]
5. Calculate the estimated slope [tex]\( b_1 \)[/tex]:
[tex]\[
b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} = \frac{-587.2}{609.2} \approx -0.964
\][/tex]
Thus, the estimated slope [tex]\( b_1 \)[/tex] is [tex]\(\boxed{-0.964}\)[/tex] rounded to three decimal places.
