Question 7 of 13, Step 3 of 3
Correct
A study of bone density on 5 random women at a hospital produced the following results.
\begin{tabular}{|c|c|c|c|c|c|}
\hline Age & 49 & 57 & 61 & 65 & 69 \\
\hline Bone Density & 355 & 345 & 340 & 325 & 320 \\
\hline
\end{tabular}

Step 3 of 3: Calculate the coefficient of determination, $r^{2}$. Round your answer to three decimal places.

Answer
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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 \(\boxed{0.950}\).