What tells you that the relation is best fit by a linear model? Use principles about what kinds of functions have consistency in which types of rates of change?

Question

Suppose a set of data about the spread (in hundreds of acres) of a food-borne bacteria (after t weeks) in lettuce has the following percentage rates of change: 0.9%, 1.3%, 1.1%, 1.0%, 1.3%, 1.0%; and the following average rates of change: 2.9, 2.4, 3.7, 3.1, 3.6, 3.2. Based only on the information given, and without doing any calculations, what type of model would you use? Why? (NOTE THAT the information given is not the actual data set. I have calculated the two types of rate of change for you. You should NOT BE CALCULATING ANYTHING!) Comment:What tells you that the relation is best fit by a linear model? Use principles about what kinds of functions have consistency in which types of rates of change?

Answer 1

Rates of change are measures of gradient (slope) of the curve of the data, and in this case the range of the rates, whether by percentage or rate of change of spread (area), is small, because percentage ranges from 0.9% to 1.3% and spread rate from 2.4 to 3.7 (hundreds of acres per week). I guess that if we have two areas, A₁ and A₂, and two corresponding time periods, t₁ and t₂, the average rate of change is (A₂-A₁)/(t₂-t₁) hundreds of acres per week, and the percentage rate is 100(A₂-A₁)/(A₁(t₂-t₁)) or 100((A₂/A₁-1)/(t₂-t₁) percent. The dataset would contain seven values for area of spread with a corresponding timescale, in order for the rates to be calculable. It appears that the dataset undulates and may resemble a sine wave, so my guess is that the model is not linear but cyclic (sinusoidal).