The following data represent travel times in a downtown area of a certain city. The independent, or input, variable is the distance to be traveled.
| Distance(miles) |
5 |
1 |
1.5 |
2 |
3 |
4 |
5 |
6 |
8 |
10 |
| Travel Time(minutes) |
15 |
15.1 |
16.5 |
19.9 |
27.7 |
29.7 |
26.7 |
35.9 |
42 |
49.4 |
Assuming a linear relationship of the form
Y = α + βx + e
between Y , the travel time, and x, the distance, how should we estimate α and β? To utilize the weighted least squares approach we need to know, up to a multiplicative constant, the variance of Y as a function of x. We will now present an argument that Var(Y) should be proportional to x.