Question

The following table of values gives a company's annual profits in millions of dollars. Rescale the data so that the year 2005 corresponds to  x=1 .

Year	2005	2006	2007	2008	2009	2010
Profits (in millions of dollars)	51.8	63.4	66.3	65.8	62.1	63.8

Use the power regression model to predict the company's profits in 2012.

a) $70.2 million

b) $68.6 million

c) $71.5 million

d) $67.7 million

e) $66.6 million

f) None of the above

Answer 1

we have

power regression ----------

Y =a(x)^b

ln(y) = ln(a) + b*ln(x)

now, this is equivalent to linear equation Y= a + bx

x	y	ln(x)	ln(y)
1	51.8	0	3.94739
2	63.4	0.693147	4.149464
3	66.3	1.098612	4.19419
4	65.8	1.386294	4.18662
5	62.1	1.609438	4.128746
6	63.8	1.791759	4.155753

(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
1.20	0.03	0.20
0.16	0.00	-0.01
0.00	0.00	0.00
0.08	0.00	0.02
0.26	0.00	0.00
0.48	0.00	0.02

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	6.579251212	24.76216293	2.195482001	0.0	0.23
mean	1.10	4.13	SSxx	SSyy	SSxy

sample size ,   n =   6          
here, x̅ = Σx / n=   1.10   ,     ȳ = Σy/n =   4.13  
                  
SSxx =    Σ(x-x̅)² =    2.1955          
SSxy=   Σ(x-x̅)(y-ȳ) =   0.2          
                  
estimated slope , ß1 = SSxy/SSxx =   0.2   /   2.195   =   0.1030
                  
intercept,   ß0 = y̅-ß1* x̄ =   4.0141          
                  
so, regression line is   Ŷ =   4.0141   +   0.1030   *x

...............

for year 2012 x= 8

ln(8) = 2.07944

Predicted Y at X=   2.079441542   is                  
Ŷ =   4.01406   +   0.103024   *   2.079441542   =   4.228

now,

company's profit = antilog(4.228) = 68.6 ( excel formula =EXP(4.228) )

so answer is option (b)    $68.6 million