Question

the annual profits for a company are given in the following table, where x represents the number of years since 2006, 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 2014, rounded to the nearest thousand dollars. years since 2006 (x) profits (y) (in thousands of dollars) 0 96 1 88 2 113 3 155 4 183 5 217

Explanation:

Step1: Calculate the mean of \(x\) and \(y\)

\(\bar{x}=\frac{0 + 1+2+3+4+5}{6}=\frac{15}{6}=2.5\)
\(\bar{y}=\frac{96 + 88+113+155+183+217}{6}=\frac{852}{6}=142\)

Step2: Calculate the slope \(m\)

$$
LATEXBLOCK0
$$

Step3: Calculate the \(y -\)intercept \(b\)

\(b=\bar{y}-m\bar{x}=142-26.63\times2.5=142 - 66.58=75.42\)
The linear regression equation is \(y = 26.63x+75.42\)

Step4: Find \(x\) for 2014

Since \(x\) represents the number of years since 2006, for 2014, \(x=2014 - 2006=8\)

Step5: Calculate the projected profit

\(y=26.63\times8+75.42=213.04+75.42=288.46\approx288\) (in thousands of dollars)

Answer:

The linear regression equation is \(y = 26.63x + 75.42\) and the projected profit for 2014 is \(288\) (in thousands of dollars)