Question Show Examples The annual profits for a company are given in the following table, where $\mathrm{x}$ represents the number of years since 2004, 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, find the projected profit (in thousands of dollars) for 2016, rounded to the nearest thousand dollars. \begin{tabular}{|c|c|} \hline Years since 2004 (x) & \begin{tabular}{c} Profits (y) \\ (in thousands of dollars) \end{tabular} \\ \hline 0 & 144 \\ \hline 1 & 16 \\ \hline 2 & 163 \\ \hline 3 & 168 \\ \hline \end{tabular} Log Out Comy Values for Calculator
Solution
Step 1 ：Given the data, we can calculate the linear regression equation, which is of the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
Step 2 ：The slope \(m\) can be calculated using the formula \(m = \frac{n\Sigma xy - \Sigma x\Sigma y}{n\Sigma x^2 - (\Sigma x)^2}\), and the y-intercept \(b\) can be calculated using the formula \(b = \frac{\Sigma y - m\Sigma x}{n}\), where \(n\) is the number of data points, \(\Sigma xy\) is the sum of the product of each x and y, \(\Sigma x\) is the sum of all x, \(\Sigma y\) is the sum of all y, and \(\Sigma x^2\) is the sum of the square of each x.
Step 3 ：Given the data, we have \(x = [0, 1, 2, 3]\), \(y = [144, 16, 163, 168]\), \(n = 4\), \(\Sigma x = 6\), \(\Sigma y = 491\), \(\Sigma xy = 846\), and \(\Sigma x^2 = 14\).
Step 4 ：Substituting these values into the formulas, we get \(m = 21.9\) and \(b = 89.9\).
Step 5 ：We can use this linear regression equation to predict the profit for 2016. Since x represents the number of years since 2004, \(x = 2016 - 2004 = 12\) for the year 2016.
Step 6 ：Substituting \(x = 12\) into the equation, we get \(y = 352.7\).
Step 7 ：Rounding to the nearest thousand dollars, the projected profit for 2016 is \(\boxed{353}\) thousand dollars.
