The number of initial public offerings of stock issued in a 10 -year period and the total proceeds of these offerings (in millions) are shown in the table. The equation of the regression line is $\hat{y}=47.644 x+18,221.88$. Complete parts a and b.
\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|}
\hline Issues, $\mathbf{x}$ & 409 & 461 & 695 & 486 & 485 & 390 & 64 & 60 & 195 & 160 \\
\hline \begin{tabular}{l}
Proceeds, \\
$\mathbf{y}$
\end{tabular} & 17,785 & 28,570 & 42,636 & 30,973 & 66,827 & 67,227 & 20,271 & 11,188 & 31,325 & 27,646 \\
\hline
\end{tabular}
(a) Find the coefficient of determination and interpret the result.
(Round to three decimal places as needed)

Step 1 ：Given the data for the number of issues, \(x = [409, 461, 695, 486, 485, 390, 64, 60, 195, 160]\) and the proceeds, \(y = [17785, 28570, 42636, 30973, 66827, 67227, 20271, 11188, 31325, 27646]\)

Step 2 ：Calculate the mean of \(x\) and \(y\), which are \(x_{mean} = 340.5\) and \(y_{mean} = 34444.8\) respectively

Step 3 ：Calculate the numerator of the correlation coefficient, which is the sum of the product of \(x - x_{mean}\) and \(y - y_{mean}\). The result is \(numerator = 18975173.000000004\)

Step 4 ：Calculate the denominator of the correlation coefficient, which is the square root of the product of the sum of \((x - x_{mean})^2\) and the sum of \((y - y_{mean})^2\). The result is \(denominator = 36320070.02690454\)

Step 5 ：Calculate the correlation coefficient \(r\) by dividing the numerator by the denominator. The result is \(r = 0.5224431832302061\)

Step 6 ：Calculate the coefficient of determination \(r^2\) by squaring the correlation coefficient. The result is \(r_{squared} = 0.27294687970371073\)

Step 7 ：The coefficient of determination is \(\boxed{0.273}\). This means that approximately 27.3% of the variance in the proceeds can be explained by the number of issues.