Question

Age (yr)	18	28	38	48	58	68
Median Income ($)	17,480	30,650	37,440	41,230	37,570	21,390

Find an appropriate model for the data. Round values to the nearest hundredth.

Answer 1

A scatterplot of the given data is attached. The scatterplot resembles a downwards opening parabola. Let its equation be y = f(x) = ax²+bx+c, where x is the age in years, y is the median income in Dollars, and a,b,c are unknown real numbers.

On substituting x = 18 and y = 17480 in the equation, we get a(18)²+b(18)+c = 17480 or, 324a+18b +c = 17480…(1).

On substituting x = 48 and y = 41230 in the equation, we get a(48)²+b(48)+c = 41230 or, 2304a +48b +c = 41230…(2).

On substituting x = 68 and y = 21390 in the equation, we get a(68)²+b(68)+c = 21390 or, 4624a +68b +c = 21390…(3).

The augmented matrix of this linear system is A (say) =

324	18	1	17480
2304	48	1	41230
4624	68	1	21390

To determine a,b,c, we have to reduce A to its RREF which is

1	0	0	-5351/150
0	1	0	235958/75
0	0	1	-689794/25

Therefore, a = -5351/150= -35.67, b = 235958/75 = 3146.11 and c = -689794/25 = -27591.76 ( on rounding off to the nearest hundredth).

Hence the required model for the data is y = f(x) = -35.67x²+3146.11x-27591.76.