Regression involves the study of equations. First we talk about some simple equations or linear models. The simplest mathematical model or equation is the equation of straight line.
Example: Suppose a shop keeper is selling pencils. He sells one pencil for 2 cents. Table as shown gives the number of pencils sold and the sale price of the pencils.
The information written above can be presented in some other forms as well. For example we can write an equation describing the above relation between and. It is very simple to write the equation. The algebraic equation connecting and is..
The main features of the graph in the figure are:
It is called the slope of the line and in general it is denoted by “”. The slope of the line is the same at all points on the line. The slope “” is equal to the change in for a unit change in. The relation is also called linear equation between and .
Example: Suppose a carpenter wants to make some wooden toys for the small children. He has purchased some wood and some other material for $. The cost of making each toy is $. Table gives the information about the number of toys made and cost of the toys.
Let denote the number of toys and denote the cost of the toys. What is the algebraic relation between and. When, . This is called fixed or starting cost and it may be denoted by “”. For each additional toy, the cost is $. Thus and are connected through the following equation:
Let us note some important features of the graph obtained in figure.
This ratio is denoted by “” in the equation of straight line. Thus the equation of straight line has the intercept and slope. In general, when the values of intercept and slope are not known, we write the equation of straight line as. It is also called linear equation betweenand, and the relation between andis called linear. The equation may also be called exact linear model between and or simply linear model between and. The value of can be determined completely when is given. The relation is therefore, called the deterministic linear model between and. In statistics, when we shall use the term “Linear Model”, we shall not mean a mathematical model as described above.