Some time ago, I participated in a running contest. I wasn’t the best, but I was happy to see that my finishing time, of 51 minutes, was at least below the group average:
However, some friends pointed out that I had an advantage, since, compared to the other in the group, my body length is relatively high and my height is relatively low.
How can we know if my friends are right?
It is hard to find a group of people with exactly the same length and body masses as me to compared to.
Here’s the idea: instead of looking at the average finished time of the whole group, let’s try to find out what the average result would be if body length and mass were took into account.
That is, we assume the average of the finished time is a function of length and weight: $T_{\text{average}}=f(L,m)$
This graph is just an idea of how it would like: we don’t know its precise formula.
Since the variance of masses and length in the group is not that large, it is reasonable that the function can be approximated by a linear function. So our, model will be:
We assume that the function $T_{\text{average}}$ depends linearly on the mass and length variables. But, what are the coefficients?
We can use the data from the contest:
From this data we get a bunch of data points (the blue area). From here we try to get the coefficients $\beta_0,\beta_1,\beta_2$ such that the graph is as close as possible to these data points.
This is exactly what we looked before: the difference is that now we have more than one independent variable in the model. However, the exact same idea is applied here.
First, we plug in the measurement data for each data in the model. This will give us a series of linear equations:
