Question

the annual profits for a company are given in the following table, where x represents the number of years since 2014, 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, estimate the calendar year in which the profits would reach 144 thousand dollars.

years since 2014 (x) profits (y) (in thousands of dollars)
0 46
1 57
2 84
3 76

Explanation:

Step1: Input data into calculator

Use a graphing calculator or statistical software to input the \(x\) - values (\(0,1,2,3\)) and \(y\) - values (\(46,57,84,76\)).

Step2: Find linear regression equation

The linear regression formula is \(y = ax + b\). After calculation, \(a\approx16.7\) and \(b = 46.3\). So the equation is \(y=16.7x + 46.3\).

Step3: Solve for \(x\) when \(y = 144\)

Set \(144=16.7x + 46.3\).
Subtract \(46.3\) from both sides: \(144−46.3 = 16.7x\), so \(97.7=16.7x\).
Divide both sides by \(16.7\): \(x=\frac{97.7}{16.7}\approx5.85\).

Step4: Find the calendar year

Since \(x\) is the number of years since 2014, the year is \(2014 + 6=2020\) (rounding \(x = 5.85\) up to the next whole number as we are looking for when the profit reaches the value).

Answer:

The linear regression equation is \(y = 16.7x+46.3\). The profits would reach 144 thousand dollars in the year 2020.