# Linear Model

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.

 Number of pencils sold $0$ $1$ $2$ $3$ $4$ $5$ Sales Prices (Cents) $0$ $2$ $4$ $6$ $8$ $10$

Let us examine the two variables given in table. For the sake of our convenience, we can give some names to the variables given in the table. Let $X$ denote the number of pencils sold and $S$ ($S$ for sale) denote the amount realized by selling $X$ pencils. Thus,

 $X$ $0$ $1$ $2$ $3$ $4$ $5$ $S$ $0$ $2$ $4$ $6$ $8$ $10$

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 $X$ and $S$. It is very simple to write the equation. The algebraic equation connecting $X$ and $S$ is $S = 2X$.

It is called mathematical equation or mathematical model in which $S$ depends upon $X$. Here $X$ is called independent variable and $S$ is called dependent variable. Cent $4$. Neither less than$4$ nor more than $4$. The above model is called deterministic mathematical model because we can determine the value of $S$ without any error by putting the value of $X$ in the equation. The sale $S$ is said to be function of $X$. This statement in symbolic form is written as: $S = f\left( X \right)$.

It is read as “$S$ is function of $X$”. It means that $S$ depends upon $X$ and only $X$ and no other element. The data in the table can be presented in the form of a graph as shown in the figure.

The main features of the graph in the figure are:

1. The graph lies in the first quadrant because all the values of $X$ and $S$ are positive.
2. It is an exact straight line. But all graphs are not in the form of a straight line. It could be some curve also.
3. All the points (pair of $X$ and $S$) lies on the straight line.
4. The line passes through the origin.
5. Take any point $P$ on the line and draw a perpendicular line $PQ$ which joins $P$ with the X-axis. Let us find the ratio $\frac{{PQ}}{{OQ}}$. Here $PQ = 6$ units and $OQ = 3$ units. Thus $\frac{{PQ}}{{OQ}} = \frac{6}{3} = 2$ units.

It is called the slope of the line and in general it is denoted by “$b$”. The slope of the line is the same at all points on the line. The slope “$b$” is equal to the change in $Y$ for a unit change in $X$. The relation $S = 2X$ is also called linear equation between $X$ and $S$.

Example: Suppose a carpenter wants to make some wooden toys for the small children. He has purchased some wood and some other material for 20$. The cost of making each toy is$$5$. Table gives the information about the number of toys made and cost of the toys.

 Number of Toys $0$ $1$ $2$ $3$ $4$ $5$ Cost of Toys $20$ $25$ $30$ $35$ $40$ $45$

Let $X$ denote the number of toys and $Y$ denote the cost of the toys. What is the algebraic relation between $X$ and $Y$. When $X = 0$, $Y = 20$. This is called fixed or starting cost and it may be denoted by “$a$”. For each additional toy, the cost is Dollar $5$. Thus $Y$ and $X$ are connected through the following equation: $Y = 20 + 5X$

It is called equation of straight line. It is also mathematical model of deterministic nature. Let us make the graph of the data in given table. Figure as shown is the graph of the data in table.

Let us note some important features of the graph obtained in figure.

1. The line $AB$ does not pass through the origin. It passes through the point $A$ on Y-axis. The distance between $A$ and the origin $0$is called the intercept and is usually denoted by “$a$”.
2. Take any point $P$ on the line and complete a triangle $PQA$ as shown in the figure. Let us find the ratio between the perpendicular $PQ$ and the base $AQ$ of this triangle. The ratio is, $\frac{{PQ}}{{AQ}} = \frac{{15}}{3} = 5$units.

This ratio is denoted by “$b$” in the equation of straight line. Thus the equation of straight line $Y = 20 + 5X$has the intercept $a = 20$and slope $b = 5$. In general, when the values of intercept and slope are not known, we write the equation of straight line as $Y = a + bX$. It is also called linear equation between $X$ and $Y$, and the relation between $X$ and $Y$is called linear. The equation $Y = a + bX$ may also be called exact linear model between $X$ and $Y$ or simply linear model between $X$ and $Y$. The value of $Y$ can be determined completely when $X$ is given. The relation $Y = a + bX$ is therefore, called the deterministic linear model between $X$ and $Y$. In statistics, when we shall use the term “Linear Model”, we shall not mean a mathematical model as described above.