Question

the annual profits for a company are given in the following table, where x represents the number of years since 2008, 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 tenth. using this equation, find the projected profit (in thousands of dollars) for 2016, rounded to the nearest thousand dollars.

years since 2008 (x)	profits (y) (in thousands of dollars)
1	156
2	151
3	154

Explanation:

Step1: Calculate sums

Let \(n = 4\) (number of data - points).
\(\sum_{i = 1}^{n}x_i=0 + 1+2 + 3=6\), \(\sum_{i = 1}^{n}y_i=135 + 156+151+154 = 606\), \(\sum_{i = 1}^{n}x_i^2=0^2 + 1^2+2^2 + 3^2=0 + 1+4 + 9 = 14\), \(\sum_{i = 1}^{n}x_iy_i=0\times135+1\times156 + 2\times151+3\times154=0 + 156+302+462 = 920\).

Step2: Calculate slope \(m\)

The formula for the slope \(m\) of the linear - regression line is \(m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^2-(\sum_{i = 1}^{n}x_i)^2}\).
Substitute the values: \(m=\frac{4\times920-6\times606}{4\times14 - 6^2}=\frac{3680 - 3636}{56 - 36}=\frac{44}{20}=2.2\).

Step3: Calculate y - intercept \(b\)

The formula for the y - intercept \(b\) is \(b=\frac{\sum_{i = 1}^{n}y_i-m\sum_{i = 1}^{n}x_i}{n}\).
\(b=\frac{606-2.2\times6}{4}=\frac{606 - 13.2}{4}=\frac{592.8}{4}=148.2\).
The linear - regression equation is \(y = 2.2x+148.2\).

Step4: Find \(x\) for 2016

Since \(x\) represents the number of years since 2008, for 2016, \(x = 2016 - 2008=8\).

Step5: Predict profit

Substitute \(x = 8\) into the equation \(y = 2.2x+148.2\).
\(y=2.2\times8+148.2=17.6+148.2 = 165.8\approx166\).

Answer:

The linear regression equation is \(y = 2.2x+148.2\). The projected profit for 2016 is 166 (in thousands of dollars).