Median Age of Women at First Marriage ; Linear Algebra application problem

Question

We are given an application problem:

The median age of women in the United States at first marriage is given below. Let t= 0 correspond to 1970 and also let tbe measured in decades. Let 0 ≤ t≤ 4.

Year| | | | | |Median age

1970| | | | | |20.8

1980| | | | | |22.0

1990| | | | | |23.9

2000| | | | | |25.1

2010| | | | | |26.1

part (a): Find the quartic polynomial of the form M(t) = a0 +a1t^1 + a2t^2 + a3t^3 + a4t^4 that fits the data precisely. Round the coefficients to four decimal places.

A0 has to be 20.8, but how would we go about finding the other a constants? I was thinking of setting up an augmented matrix, but don't we have too many unknowns and also the constant 20.8 to account for?

Answer 1

Guide: $$M(t) = \sum_{i=0}^4 a_i t^i$$

Let $t=0$, we obtain a linear equation. Similarly let $t=1,2,3,4$ and we can obtain $5$ equation in total.

Hence we will obtain $5$ linear equations in $5$ unknowns and we solve linear system to recover $a_i$.

Answer 2

Here is one way to start building up your function.

$f(T) = y(1970) + (y(1980)-y(1970))(\frac {T-1970}{10})+ (\frac {Y(1990)-2Y(1980)+Y(1970)}{10\cdot20})(T-1980)(T-1970)+\cdots$

Another would be as you have above

$f(t) = a_0 + a_1 t + \cdots + a_4 t^4\\ f(0) = a_0= 20.8\\ f(1) = a_0 + a_1 +a_2 +a_3 + a_4 = 22.0\\ f(2) = a_0 + 2a_1 +4a_2 +8a_3 + 16a_4 = 23.9\\ \cdots$

Using linear aglebra:

$\begin{bmatrix} 1&0&0&0&0\\1&1&1&1&1\\1&2&4&8&16\\1&3&9&27&81\\1&4&16&64&256\end{bmatrix}\mathbf a = \mathbf Y$