I think the curve over the given range of data can be modeled with a cubic curve. I looked at the fourth degree curve and found that the coefficient of the x^4 term was so small, it would not have a significant impact on the fit of the curve. My problem with using the cubic curve to model the data is that it cannot be used to predict BMIs after the age of 20 since the curve drops off sharply after the age of 20 (and becomes zero and negative). A similar problem would be found with a fourth degree except that it rises too sharply after 20 and would also not be a good predictor. A sine curve would suggest an oscillating BMI which would also not make sense. The other problem I have encountered is that my students have not been able to find data for BMIs for ages past 20 to help them find a model function that works past 20. I have suggested to students that they try to suggest what may happen to the BMI after age 20 and then find a function that would model this. I have also suggested that they could use one function to model the situation over the given range of data and a second function to model what the predicted BMI might be.
. Simple analytical methods do not produce sufficient consistency for any obvious curve (the initial numerical/algebraic analysis that students are supposed to do prior to using higher order use of technology). It seems to me that the task intends the students to actually make an "educated" guess on the curve, then try it out and refine it - as opposed to having some fairly consistent analysis that regression tools confirm. Furthermore, none of the curves actually work beyond around age 32 (I even had a go at a sine curve - on the basis that if the period was large enough it could be justified!!) As a task that "ticks" all the boxes for the criteria it is good but as a task that is supposed to give at least some consistency in the initial analysis it has the potential to be frustrating for students unless they are warned that what they have to show is a process toward a model but immediate "solutions" in the first stage are not available. I would recommend that students first do some research into BMI for this task - it is vital to understand the constraints bearing in mind the BMI data is median data and should therefore not be more than around 30 (the cutoff for overweight) The Wikipedia article is good as an introduction to BMI and the following website (which I only found last night) actually shows the various curves: http://www.scielo.br/scielo.php?pid=S0021-75572006000500007&script=sci_arttext&tlng=en If you would like a copy of what I have done (although, to my mind, it needs modification) let me know and I will upload it to the resources
go to google and enter : bmi females data