Question

The following table represents the growing profits of a company that opened up in 2005.

# of years after opening in 2005	Profit
1	39,200
4	55,073
5	61,682
7	77,374

a. Find the best regression equation for this data. State how you know this is the best equation for this data.

b. How much does the company make in profit 3 years after they open?

c. What is the difference in profit from year 2007 to 2008?

d. In what year will the company's profit reach 100,000? You must show all work.

*This is for exam prep and i'm a bit lost, so any help with work shown would be a huge help, thank you.

Answer 1

$\\ Let\ x\ is\ year\ after\ 2005\\ and\ y\ represent\ profit \\ \\ Step\ 1: \ Find \ X\codt Y \ and \ X^ 2 \ as \ it \ was \ done \ in \ the \ table \ below.$

X	Y	X⋅Y	X⋅X
1	39200	39200	1
4	55073	220292	16
5	61682	308410	25
7	77374	541618	49

$\\ Step \ 2: \ Find \ the \ sum \ of \ every \ column:\\ \sum{X} = 17 ~,~ \sum{Y} = 233329 ~,~ \sum{X \cdot Y} = 1109520 ~,~ \sum{X^2} = 91\\\\ \ Step \ 3: \ Use \ the following \ equations \ to \ find \ a \ and \ b: \\ \begin{aligned} a &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 233329 \cdot 91 - 17 \cdot 1109520}{ 4 \cdot 91 - 17^2} \approx 31614.653 \\ \\b &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 4 \cdot 1109520 - 17 \cdot 233329 }{ 4 \cdot 91 - \left( 17 \right)^2} \approx 6286.493\end{aligned}\\\\ Step\ 4: \ Substitute \ a \ and\ b \ in \ regression \ equation \ formula\ \\ \begin{aligned} y~&=~a ~+~ b \cdot x \\y~&=~31614.653 ~+~ 6286.493 \cdot x\end{aligned}\\ \\ a)\ best \ regression \ equation \ for\ the\ data\ is\ \\ y=31614.653+6286.493x\\ \\$

$\\ b)\ company\ \ profit\ after\ 3\ year\ after\ they \ open\ is\ \ y(3)\\ y(3)=31614.653+6286.493(3)=31614.653+18859.479=50474.132\\ \\ so,\ profit = 50474\\ \\ \ c)\ for\ 2007\ \ x=2007-2005=2\\ {so,\ profit\ =y(2)=31614.653+6286.493(2)=31614.653+12572.986=441876.639}\\ \\ for\ 2008\ \ x=2008-2005=3\\ so,\ profit\ = y(3)=50474.132\\ \\ \therefore \ difference \ in\ profit =50474.132-441876.639=6286.493\approx 6286\\ \\ \\ d)\ let\ after\ x\ year\ from\ 2005\ profit\ reach\ 100,000\\ so,\ 100000=31614.653+6286.653x\\ \Rightarrow 6286.653x=100000-31614.653\\ \Rightarrow 6286.653x=68385.347\\ \Rightarrow x=10.8779\\ \\ so,\ year = 2005+10.8779=2015.8779\approx \boldsymbol{2016}\\$