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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.