Question

the actual 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 hundredth. using this equation, find the projected profit (in thousands of dollars) for 2024.
years since 2014 (x) | profits (y) (in thousands of dollars)
0 | 107
1 | 134
2 | 132
3 | 170
4 | 200

Explanation:

Step1: Calculate sums for regression formula

Let \(n = 5\). Calculate \(\sum x=0 + 1+2 + 3+4=10\), \(\sum y=107 + 134+132+170+200 = 743\), \(\sum xy=0\times107+1\times134 + 2\times132+3\times170+4\times200=0 + 134+264+510+800 = 1708\), \(\sum x^{2}=0^{2}+1^{2}+2^{2}+3^{2}+4^{2}=0 + 1+4+9+16 = 30\).

Step2: Calculate slope \(m\)

The formula for the slope \(m\) of the regression - line is \(m=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}}\). Substitute the values: \(m=\frac{5\times1708-10\times743}{5\times30 - 10^{2}}=\frac{8540 - 7430}{150 - 100}=\frac{1110}{50}=22.2\).

Step3: Calculate y - intercept \(b\)

The formula for the y - intercept \(b\) is \(b=\frac{\sum y-m\sum x}{n}\). Substitute \(m = 22.2\), \(\sum x = 10\), \(\sum y=743\) and \(n = 5\): \(b=\frac{743-22.2\times10}{5}=\frac{743 - 222}{5}=\frac{521}{5}=104.2\).

Step4: Write the regression equation

The regression equation is \(y = mx + b\), so \(y=22.2x + 104.2\).

Step5: Find the projected profit for \(x = 7\)

Substitute \(x = 7\) into the regression equation: \(y=22.2\times7+104.2=155.4+104.2 = 259.6\).

Answer:

259.6