Least Squares Approx. for Life Expectancy

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SUMMARY

The discussion focuses on deriving a least squares approximation for life expectancy based on per capita income and average years of education using the equation life expectancy = x1 + x2 * income + x3 * education. The correct formulation of the normal equations for least squares analysis is ATAx = ATb, where A is the matrix of partial derivatives and b is the vector of life expectancy values. Participants emphasize the need to construct matrix A with 20 rows corresponding to the 20 countries in the dataset.

PREREQUISITES
  • Understanding of least squares approximation
  • Familiarity with matrix algebra
  • Knowledge of linear regression concepts
  • Basic statistics related to life expectancy and socioeconomic factors
NEXT STEPS
  • Study the construction of the matrix A for least squares problems
  • Learn about the normal equations in linear regression
  • Explore the application of least squares in real-world datasets
  • Investigate statistical software tools for performing regression analysis, such as R or Python's NumPy
USEFUL FOR

Mathematicians, data analysts, and researchers interested in statistical modeling and regression analysis, particularly those focusing on socioeconomic factors affecting life expectancy.

JoeSabs
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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
 
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JoeSabs said:
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:

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?