Step 1 :Given the ages and bone densities of 5 random women at a hospital, we are asked to calculate the coefficient of determination, denoted as \(r^{2}\). This statistical measure shows the proportion of the variance for a dependent variable that's explained by an independent variable or variables in a regression model. In this case, we are trying to determine how much of the variance in bone density can be explained by age.
Step 2 :To calculate \(r^{2}\), we first need to calculate the correlation coefficient (\(r\)) between age and bone density. The formula for \(r\) is: \(r = \frac{\Sigma[(x_{i} - \bar{x})(y_{i} - \bar{y})]}{\sqrt{[\Sigma(x_{i} - \bar{x})^{2} * \Sigma(y_{i} - \bar{y})^{2}]}}\), where \(x_{i}\) and \(y_{i}\) are the individual sample points indexed with \(i\), \(\bar{x}\) is the mean of the \(x\) values, and \(\bar{y}\) is the mean of the \(y\) values.
Step 3 :Given the ages \([49, 57, 61, 65, 69]\) and bone densities \([355, 345, 340, 325, 320]\), we calculate the mean age as 60.2 and the mean bone density as 337.0.
Step 4 :Substituting these values into the formula for \(r\), we get a numerator of -432.0 and a denominator of 443.3328320799171, which gives us \(r = -0.9744371919698602\).
Step 5 :Finally, we calculate \(r^{2}\) by squaring \(r\), which gives us \(r^{2} = 0.9495278410941063\).
Step 6 :Rounding to three decimal places, the final answer is \(\boxed{0.950}\).