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Math IB SL BMI Portfolio

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Introduction

This portfolio is an investigation into how the median Body Mass Index of a girl will change as she ages. Body Mass Index (BMI) is a comparison between a person’s height (in meters) and weight (in kilograms) in order to determine whether one is overweight or underweight based on their height. The goal of this portfolio is to prove of disprove how BMI as a function of Age (years) for girls living in the USA in 2000 can be modeled using one or more mathematical equations. This data can be used for parents wanting to predict the change in BMI for their daughters or to compare their daughter’s BMI with the median BMI in the USA.

The equation used to measure BMI is:

image03.png

The chart below shows the median BMI for girls of different ages in the United States in the year 2000:

Age (years)

Median BMI

2

16.40

3

15.70

4

15.30

5

15.20

6

15.21

7

15.40

8

15.80

9

16.30

10

16.80

11

17.50

12

18.18

13

18.70

14

19.36

15

19.88

16

20.40

17

20.85

18

21.22

19

21.60

20

21.65

The values “height (m)” and “weight (kg)” can be discarded when trying to analyze this data because it is not consistent that a girl will have a fixed weight to height or vice versa when they are a certain age. For example, the median BMI for a 10 year old girl is 16.80. With only the data provided, one cannot isolate either the height variable or the weight variable using the formula for calculating BMI.

Using the computer program TI InterActive!™  to graph a scatterplot using the data, the resultant graph is as follows:

image04.pngimage00.png

...read more.

Middle

image28.png

b =image29.png

Using the approximated a, b, c and d values, the resultant equation is:

image30.png

Using TI InterActive!™  to graph that function, the resultant graph is as follows:

image05.pngimage00.png

Overlaid with the original data, and restricting the graph of the function to the same domain and range as the domain and range on the scatterplot, the graph of both the data and the function together is as follows:

image06.pngimage00.png

Comparing the two graphs, one can see that where BMI = 18.4 in the function is around 0.400 years off from the corresponding point on the scatterplot where BMI = 18.4. That can be fixed by adjusting the horizontal phase shift value of c to 12.4 to compensate.

There is also a noticeable discrepancy between the function graph and the scatter plot in the domain {image07.png} and {image08.png}. A slight stretch about the y axis would fit the curve closer to the data points in those areas, therefore the b value should be changed from π/15 to π/14. The number π/14 was chosen because reducing the denominator’s value by 1 will increase the b value by a slight amount, and an increased b value will shorten period and compress the graph about the y axis, resulting in the values within domain image09.png to be closer to that of the data points.

Making these changes, the revised formula for the sinusoidal function is as follows:

image10.png

Using TI InterActive!™  

...read more.

Conclusion

Appendix

"Body mass index." Wikipedia, the free encyclopedia. 16 Mar. 2009. 16 Mar. 2009 <http://en.wikipedia.org/wiki/Body_mass_index>.

"Logistic regression." Wikipedia, the free encyclopedia. 16 Mar. 2009. 16 Mar. 2009 <http://en.wikipedia.org/wiki/Logistic_regression>.

Mueller, William. "Logistic Functions." Exploring Precalculus. 16 Mar. 2009 <http://www.wmueller.com/precalculus/families/1_80.html>.

Shang, Lei, Yong-yong Xu, Xun Jiang, and Ru-lan Hou. "Body Mass Index Reference Curves for Children Aged 0-18  Years in Shaanxi, China." Department of Health Statistics 1.1 (2005). 12 Mar. 2009 <http://www.ijbs.org/User/ContentFullText.aspx?VolumeNO=1&StartPage=57&Type=pdf>.

"Sigmoid function." Wikipedia, the free encyclopedia. 16 Mar. 2009. 16 Mar. 2009 <http://en.wikipedia.org/wiki/Sigmoid_curve>.

Appleby, A., Letal, R., & Ranieri, G. (2007). Pure Math 30 Workbook. Calgary: Absolute Value Publications

Technology Used:

TI InterActive!™  Free Trial by Texas Instruments

TI 84-Plus Silver Edition Graphing Calculator

Microsoft Office Excel 2003

Microsoft Office Word 2003

*Extra:

Graph of Sinusoidal function in Blue followed by Logistic Regression function in Red, with the extraneous values removed.

image23.png

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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