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Data with two variables

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.