Save \& Exit Certify Lesson: 12.2 Linear Regression JEFFERY RIPKA Question 5 of 7, Step 6 of 6 $13 / 20$ 2 The table below gives the age and bone density for five randomly selected women. Using this data, consider the equation of the regression line, $\hat{y}=b_{0}+b_{1} x$, for predicting a woman's bone density based on her age. Keep in mind, the correlation coefficient may or may not be statistically significant for the data given. Remember, in practice, it would not be appropriate to use the regression line to make a prediction if the correlation coefficient is not statistically significant. \begin{tabular}{|c|c|c|c|c|c|} \hline Age & 33 & 36 & 42 & 43 & 53 \\ \hline Bone Density & 351 & 334 & 323 & 321 & 316 \\ \hline \end{tabular} Table Copy Data Step 6 of 6 : Find the value of the coefficient of determination. Round your answer to three decimal places. Tables Keypad Answer How to enter your answer (opens in new window) Keyboard Shortcuts Previous step answers

Step 1 ：Given the data of age and bone density, we are asked to find the coefficient of determination. The coefficient of determination, often denoted as r^2, is a statistical measure that shows the proportion of the variance for a dependent variable that's explained by an independent variable or variables in a regression model.

Step 2 ：To find the coefficient of determination, we first need to find the correlation coefficient (r), square it, and that will give us the coefficient of determination (r^2). The correlation coefficient can be calculated using the formula: \(r = \frac{\Sigma[(x_i - \bar{x})*(y_i - \bar{y})]}{\sqrt{\Sigma(x_i - \bar{x})^2 * \Sigma(y_i - \bar{y})^2}}\) where \(x_i\) and \(y_i\) are the individual sample points indexed with i, and \(\bar{x}\) and \(\bar{y}\) are the means of the x and y variables respectively.

Step 3 ：Given the age data as [33, 36, 42, 43, 53] and the bone density data as [351, 334, 323, 321, 316], we calculate the means of the x and y variables as \(\bar{x} = 41.4\) and \(\bar{y} = 329.0\) respectively.

Step 4 ：Substituting these values into the formula for the correlation coefficient, we get the numerator as -379.0 and the denominator as 429.5830536694854.

Step 5 ：Dividing the numerator by the denominator, we get the correlation coefficient as \(r = -0.8822508168387777\).

Step 6 ：Squaring the correlation coefficient, we get the coefficient of determination as \(r^2 = 0.7783665038126905\).

Step 7 ：Final Answer: The value of the coefficient of determination is \(\boxed{0.778}\).