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 $\hat{y}=48.187 x+18,261.62$. Complete parts a and $b$. \begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|} \hline Issues, $\mathbf{x}$ & 403 & 454 & 700 & 480 & 479 & 399 & 54 & 60 & 196 & 152 \\ \hline \begin{tabular}{l} Proceeds, \\ $\mathbf{y}$ \end{tabular} & 17,691 & 28,467 & 43,141 & 31,918 & 66,858 & 66,432 & 21,677 & 10,669 & 30,509 & 27,983 \\ \hline \end{tabular} (a) Find the coefficient of determination and interpret the result. 0.289 (Round to three decimal places as needed.) How can the coefficient of determination be interpreted? The coefficient of determination is the fraction of the variation in proceeds that is unexplained and is due to other factors or sampling error. The remaining fraction of the variation is explained by the variation in issues. The coefficient of determination is the fraction of the variation in proceeds that can be explained by the variation in issues. The remaining fraction of the variation is unexplained and is due to other factors or to sampling error.

Step 1 ：First, we need to calculate the mean of x and y. The given values of x are [403, 454, 700, 480, 479, 399, 54, 60, 196, 152] and the given values of y are [17691, 28467, 43141, 31918, 66858, 66432, 21677, 10669, 30509, 27983]. The mean of x is 337.7 and the mean of y is 34534.5.

Step 2 ：Next, we calculate the correlation coefficient. The numerator is the sum of the product of (x - x_mean) and (y - y_mean), which is 19525052.5. The denominator is the square root of the product of the sum of (x - x_mean) squared and the sum of (y - y_mean) squared, which is 36331632.40818291. The correlation coefficient, r, is the numerator divided by the denominator, which is 0.5374119247007025.

Step 3 ：Finally, we calculate the coefficient of determination by squaring the correlation coefficient. The coefficient of determination, \( r^2 \), is 0.28881157681051356.

Step 4 ：The coefficient of determination can be interpreted as the fraction of the variation in the proceeds that can be explained by the variation in the number of issues. The remaining fraction of the variation is unexplained and is due to other factors or to sampling error.

Step 5 ：Final Answer: The coefficient of determination is approximately \(\boxed{0.289}\). This means that about 28.9% of the variation in the proceeds can be explained by the variation in the number of issues. The remaining 71.1% of the variation is unexplained and is due to other factors or to sampling error.