The Linear model for statistics

• If the relationship is linear, then a line of best fit is drawn on the graph. This line of best fit is called the linear model.
• The linear model can be written as:
  

  \[ \hat{y} = b_{0} + b_{1}x \]

  where \(b_{0}\) is the \(y\)-intercept and \(b_{1}\) is the gradient (or slope).
  

• This model is used to find the predicted \(\left(\hat{y}\right)\) value for any given value of the independent variable \((x)\).
• For the previous example the equation to the linear model is
  

  \[ \hat{y} = -4.24 +1.04 x\]

  where \(y\) is the true speed of the vehicle and \(x\) is the speedometer reading of the vehicle.

• A predicted true speed can be calculated if the the speedometer reads \(90\) km/h.
  
  \begin{eqnarray*}
  \hat{y} &=& -4.24 +1.04 x \\
  &=& -4.24 +1.04 \times 90 \\
  &\approx & 93.6
  \end{eqnarray*} Therefore, the predicted true speed is approximately \(93.6\) km/h.