Let us consider what we shall call a *purely mathematical (linear) model* involving *Y* as a function of time *t* or

where *β*_{0} is the (constant) *vertical intercept* and *β*_{1} is the (constant) *slope*. Here *β*_{0} is the value of *Y* when *t* = 0 while *β*_{1} = Δ*Y*/Δ*t* (=rise/run). Specifically, *β*_{1} is the rate of change in *Y* per unit change in *t*. So if *β*_{1} = 5, then when *t* increases by one unit, *Y* increases by five units; and if *β*_{1} = −3, then as *t* increases by one unit, *Y* decreases by three units.

To obtain Equation 4.1, we need only two specific points in the (*t, Y*)-plane (Fig. 4.1). That is, *β*_{1} = Δ*Y*/Δ*t* = (*Y*_{2} − *Y*_{1})/(*t*_{2} − *t*_{1}); and at, say, (*t*_{2},*Y*_{2}), *β*_{0} = *Y* − *β*_{1}*t* = *Y*_{2} − *β*_{1}*t*_{2}. For instance, suppose (*t*_{1}, *Y*_{1}) = (1, 2) and (*t*_{2}, *Y*_{2}) = (4, 7). What is the equation of the line passing through these two points?

Clearly *β*_{1} = Δ*Y*/Δ*t* = (7 − 2)/(4 − 1) = 5/3 while, at (*t*_{1}, *Y*_{1}), *β*_{0} = 2 − (5/3)(1) = 1/3. Hence our particularization of Equation 4.1 is *Y* = (1/3) + (5/3)*t*. So when, *t* = 0, *Y* = 1/3; and, for *β*_{1} = 5/3, when *t* increases by one unit, *Y* increases by 5/3 units.

What if we have more than two points in the (*t, Y*)-plane? Specifically, we may have a scatter of points such as the one depicted in Figure 4.2.

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