# Least Squares Approx. for Life Expectancy

• JoeSabs
In summary, the question is asking for 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. To find the values of x1, x2, and x3 that give the least squares approximation of the data, the normal equations for least squares analysis is used (ATAx = ATb). The matrix A contains the fitting function partial derivatives and the vector b contains the life expectancy values. The first row of A contains the values 1, i1, and e1, corresponding to the first country, and there will be 20 rows in total since
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

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?

## 1. What is the least squares approximation method for life expectancy?

The least squares approximation method for life expectancy is a statistical technique used to estimate the relationship between life expectancy and other variables. It finds the line of best fit that minimizes the sum of the squared differences between the observed and predicted values.

## 2. How is the least squares approximation method used in predicting life expectancy?

The least squares approximation method is used to predict life expectancy by fitting a linear regression model to historical data on life expectancy and other relevant predictors such as economic, social, and environmental factors. The resulting model can then be used to make predictions about future life expectancy based on changes in these predictors.

## 3. What are the assumptions of the least squares approximation method for life expectancy?

The assumptions of the least squares approximation method for life expectancy include linearity, normality, independence, and homoscedasticity. This means that the relationship between life expectancy and the predictors is linear, the errors are normally distributed, the observations are independent of each other, and the errors have a constant variance.

## 4. What are the limitations of the least squares approximation method for life expectancy?

The limitations of the least squares approximation method for life expectancy include the assumption of linearity, which may not hold in all cases, and the reliance on historical data, which may not accurately reflect future trends. Additionally, the method may not capture the full complexity of factors that influence life expectancy, such as cultural and genetic factors.

## 5. How can the accuracy of the least squares approximation method for life expectancy be improved?

The accuracy of the least squares approximation method for life expectancy can be improved by including more relevant predictors in the model, such as healthcare access and quality, lifestyle factors, and environmental factors. Additionally, using more recent data and regularly updating the model can help improve its accuracy. It is also important to carefully assess the assumptions of the method and address any violations that may affect the accuracy of the results.