A Belly Button Model

(Adapted from Mooney and Swift, A Course In Mathematical Modeling.)

Background

The Fibonacci sequence is a very famous sequence of numbers, with a history going back to Ancient Greece. It is defined as follows: the first two numbers in the sequence are 0 and 1, and each number after that is the sum of the previous two numbers. So the third number is 0+1 = 1, the fourth is 1+1 = 2, the fifth is 1+2 = 3, and so forth. Here are the first 20 numbers in the sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, ...

We can also look at the ratios of successive numbers in the sequence (i.e. the result of dividing each number by the one after it):

0, 1, 0.5, 0.667, 0.6, 0.625, 0.6154, 0.619, 0.6176, 0.6182, 0.61798, 0.61806, ...

You can see that these ratios are becoming increasingly uniform - in fact, as we move out in the sequence the ratios get closer and closer to a number called the golden ratio, about 0.618. This number has a long history, and appears in a wide range of natural phenomena (for reasons we don't really understand). The Greeks believed that a rectangle whose sides were in this ratio was the most esthetically pleasing - and if you measure some grocery items, you'll discover that the marketers who decide how to package products agree!

One claim is that the location of one's belly button is determined by the golden ratio. In particular, the claim is that the ratio of the height of one's navel (distance from the navel to the floor) to one's height is 0.618, subject to usual statistical variance. So the model is:

B = 0.618 ´ H

where H is height and B is distance between navel and floor. This project tests the validity of this model.

Project

1. Collect data (i.e. actually get out and measure some heights and belly button heights). If possible, collect at least 30 pieces of data.

2. Plot your data and the line y = 0.618x on the same graph. Does the proposed model seem reasonable?

3. Use linear regression to find the best least-squares line. How good is the fit of this line (discuss r and r²). Compare this model to the proposed one. How different are they?

4. Plot the residuals for the model you computed in part 3. What does this plot tell you about the model? Are there any outliers?

5. If you found any obvious outliers in part 4 (particularly if they had a strong influence on the regression line), remove them and repeat parts 3 and 4.

6. What are your conclusions about the relation between a person's height and the height of their belly button?

You may do this project as a group of up to 4 people, and turn in one report for the entire group. Each member of the group will receive the same grade.

Graded out of 35 points