Least Squares Approx. for Life Expectancy

1. Dec 6, 2009

JoeSabs

This test question is really boggling me and my math group. Any help would be appreciated. We know that AT*x=AT*b is the setup, but we're not so sure how to approach the problem most effectively. Here's the question:

Use the data below to find an approximate formula for the life expectancy in a country as a function of per capita income and average number of years of education. The formula should look like this:

life expectancy= x1+x2*income+x3*education

No set of values for x1, x2, and x3 will hold exactly for all 20 countries; instead you should find x1, x2[/SU], and x3 which give you the least squares approximation of the data.

Country Life Expect. Income (\$1000s) Years of Education
Afghanistan 44.64 0.76 1.14
Brazil 71.99 10.47 4.56
Burma 63.39 1.16 2.44
Congo 54.15 0.33 3.18
Dominican Rep. 73.70 8.62 5.17
Ghana 59.85 1.52 4.01
Guatemala 70.29 4.91 3.12
Ireland 78.24 42.11 9.02
Japan 82.12 34.12 9.72
Kazakhstan 67.87 11.43 9.03
Netherlands 79.40 40.56 9.24
New Zealand 80.36 27.08 11.52
Panama 77.25 11.36 7.90
Paraguay 75.77 4.77 5.74
Russia 66.03 15.95 10.49
Sri Lanka 75.14 4.59 6.09
Sudan 51.42 2.31 1.91
Swaziland 31.88 5.75 5.73
Syria 71.19 4.76 5.74
Thailand 73.10 8.24 6.10

2. Dec 8, 2009

hotvette

Note quite correct. The normal equations for least squares analysis is ATAx=ATb. You forgot an 'A'.

Re how to proceed, b is the vector containing the life expectancy values. The matrix A contains the fitting function partial derivatives. If we call e the education and i the income, then the first row of A will be [ 1 i1 e1] where 1 corresponds to the first country. There will be 20 rows because there is data for 20 countries. Does this help?