Least square procedure

The least square procedure obtains estimates of the linear equation coefficients, a and b, in the model

by minimizing the sum of the squared residuals

The coefficients a and b are chosen so that the quantity

is minimized. It can be shown that the resulting estimates are

and

where and are the respective sample means.

The line

is called the sample regression line or the least squares regression line of y on x.

Example:

Find the least squares regression line for the data on incomes (in hundreds of dollars) and food expenditures of seven households given in the table below.

Household
Income x
Food expenditure y

Use income as an independent variable and food expenditure as a dependent variable.

Solution:

We are to find the values of a and b for the regression model .

The following table shows the calculations required for the computations of a and b.

Using data from the table 3.3 we find

;

Table3.3

Household	Income	Food expenditure (
Sums

Thus, our estimated regression model is

This regression line is called the least squares regression line. It gives the regression of food expenditure on income.

Using this estimated model, we can find the predicted value of y for a specific value of x. For example, suppose that we randomly select a household whose monthly income is $3500 so that (x denotes income in hundred of dollar in our example). The predicted value of food expenditure for this household is

hundred

In other words, based on our regression line, we predict that a household with a monthly income of $3500 is expected to spend $1038.84 per month on food.

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