The aim of this question is to learn the **method of regression** in general and **linear regression in particular**.

**Regression** is defined as a procedure in **statistics** that tries to find the **mathematical relationship** between **two or more variables** through the use of **statistical data**. One of these variables is called the **dependent variable** **y** while others are called **independent variables** **xi**. In short, we are **trying to predict** the value of **y** based on certain given values of **xi**.

Regression has **wide applications in finance, data science,** and many other disciplines. There are **many types of regression** based on the type of **mathematical model (or equation )** used. The most common form of regression is linear regression.

In **linear regression**, we **try to fit a straight line** through the given data. Mathematically:

\[ \hat{ y } \ = \ a \ + \ b x_1 \ + \ c x_2 \ + \ … \ … \ … \ \]

where, $a, \ b, \ c, \ … \ $ are the **constants or weights**.

## Expert Answer

Given:

\[ a \ = \ -6 \]

And:

\[ b \ = \ 3 \]

We can **assume following linear regression model**:

\[ \hat{ y } \ = \ a \ + \ b x \]

Substituting values:

\[ \hat{ y } \ = \ -6 \ + \ 3 x \]

Since we need to predict $ y $ at:

\[ x \ = \ 4 \]

So the above model becomes:

\[ \hat{ y } \ = \ -6 \ + \ 3 ( 4 ) \]

\[ \Rightarrow \hat{ y } \ = \ -6 \ + \ 12 \]

\[ \Rightarrow \hat{ y } \ = \ 6 \]

## Numerical Result

\[ \hat{ y } |_{ x = 4 } \ = \ 6 \]

## Example

Using the **same model** given in the above question, **predict values at**:

\[ x \ = \ \{ \ 0, \ 1, \ 2, \ 3, \ 5, \ 6 \ \} \]

**Using the model:**

\[ \hat{ y } \ = \ -6 \ + \ 3 x \]

We have:

\[ \hat{ y } |_{ x = 0 } \ = \ -6 \ + \ 3 ( 0 ) \ = \ -6 \]

\[ \hat{ y } |_{ x = 1 } \ = \ -6 \ + \ 3 ( 1 ) \ = \ -3 \]

\[ \hat{ y } |_{ x = 2 } \ = \ -6 \ + \ 3 ( 2 ) \ = \ 0 \]

\[ \hat{ y } |_{ x = 3 } \ = \ -6 \ + \ 3 ( 3 ) \ = \ 3 \]

\[ \hat{ y } |_{ x = 5 } \ = \ -6 \ + \ 3 ( 5 ) \ = \ 9 \]

\[ \hat{ y } |_{ x = 6 } \ = \ -6 \ + \ 3 ( 6 ) \ = \ 12 \]