Question

The following table of values gives a company's annual profits in millions of dollars. Rescale the data so that the year 2003 corresponds to x = 0.

Year	2003	2004	2005	2006	2007	2008
Profits (in millions of dollars)	51.7	53.2	51.9	52.2	56.1	58.3

Find the quadratic regression model for this data.

a) y=0.45x2−0.19x+51.98

b) y=0.4339x2−0.9696x+52.3464

c) y=0.4339x2−1.8375x+53.75

d) y=0.7571x2−1.5886x+52.9743

e) y=0.4571x2−1.0486x+52.3743

f) None of the above

Answer 1

from given data:

y	x	x^2
51.7	0	0
53.2	1	1
51.9	2	4
52.2	3	9
56.1	4	16
58.3	5	25

Applying multiple regression from data analysis tab of excel :

below is regression equation:

y^ =0.4339*X²-0.9696X+52.3464

option B is correct

Answer 2

Answer with Calculation:

1. Rescale Years to $x=0$ starting at 2003:
   $x$-values: $0,1,2,3,4,5$.

2. Test Key Points for Each Option:

• Option a:
  $x=0$: $y=51.98$ (vs. 51.7) → Error: +0.28
  $x=1$: $y=52.24$ (vs. 53.2) → Error: -0.96
  Poor fit.

• Option b:
  $x=0$: $y=52.3464$ (vs. 51.7) → Error: +0.65
  Immediate mismatch.

• Option c:
  $x=0$: $y=53.75$ (vs. 51.7) → Error: +2.05
  Invalid.

• Option d:
  $x=0$: $y=52.9743$ (vs. 51.7) → Error: +1.27
  Poor start.

• Option e:
  $x=0$: $y=52.3743$ (vs. 51.7) → Error: +0.67
  $x=1$: $y=51.78$ (vs. 53.2) → Error: -1.42
  Does not align.

3. Conclusion:
   All options show significant errors (>0.5) at critical points (e.g., $x=0$ and $x=1$).

Final Answer:
                               f). None of the above