The annual profits for a company are given in the following table, where $x$ represents the number of years since 2013 , and y represents the profit in thousands of dollars. Write the linear regression equation that represents this set of data, rounding all coefficients to the nearest hundredth. Using this equation, estimate the calendar year in which the profits would reach $28_{3}$ thousand dollars. \begin{tabular}{|c|c|} \hline Years since 2013 (x) & \begin{tabular}{c} Profits (y) \\ (in thousands of dollars) \end{tabular} \\ \hline 0 & 162 \\ \hline 1 & 170 \\ \hline 2 & 162 \\ \hline 3 & 208 \\ \hline 4 & 214 \\ \hline 5 & 226 \\ \hline \end{tabular}

Step 1 ：Calculate the slope (m) using the formula: \(m = \frac{N\Sigma(xy) - \Sigma x \Sigma y}{N\Sigma(x^2) - (\Sigma x)^2}\). Substitute the given values: \(m = \frac{6*2890 - 15*1142}{6*55 - 15^2} = \frac{17340 - 17130}{330 - 225} = \frac{210}{105} = 2\)

Step 2 ：Calculate the y-intercept (b) using the formula: \(b = \frac{\Sigma y - m\Sigma x}{N}\). Substitute the given values: \(b = \frac{1142 - 2*15}{6} = 170\)

Step 3 ：The linear regression equation is \(y = 2x + 170\)

Step 4 ：Estimate the calendar year when the profits would reach $283,000 by solving the equation \(2x + 170 = 283\) for x: \(2x = 283 - 170 = 113\), \(x = \frac{113}{2} = 56.5\)

Step 5 ：Since x represents the number of years since 2013, add 56.5 to 2013 to get the year: \(2013 + 56.5 = 2069.5\). Since we can't have a half year, round up to the nearest year, which is 2070

Step 6 ：\(\boxed{2070}\)