Body Mass Index (BMI) IB Mathematics, Standard Level Internal Assessment: Type II Portfolio

Megan Elise Griffith Candidate Number: 00007 – 013

May 2007 Examinations IB Mathematics, Standard Level Frankfurt International School Teacher: Mr. John Poole

Griffith 2

Table of Contents COVER PAGE ............................................................................................................................................................................ 1 TABLE OF CONTENTS ........................................................................................................................................................... 2 ABSTRACT ............................................................................................................................................................................... 3 INTRODUCTION ..................................................................................................................................................................... 5 PLOTTING AND MODELING THE DATA .......................................................................................................................... 7 GRAPHING THE MODEL FUNCTION AND ORIGINAL GRAPH ................................................................................... 9 FINDING OTHER FUNCTIONS .......................................................................................................................................... 10 ESTIMATING (EXTRAPOLATING) FOR OLDER WOMEN ......................................................................................... 12 USING THE MODEL IN OTHER CONTEXTS .................................................................................................................. 13

Griffith 3

Abstract This study investigated whether gender influences the number of hours IDEA College Preparatory Donna seniors in the Class of 2013 spend on Facebook each week. First, the study mentions relevant literature regarding how gender affects communication and past studies showing that overall, women use Facebook more than men. Although the study had been done on a macro level (with all Facebook users), the study investigated the same question to see whether the different demographics of IDEA Donna seniors changed this scenario. Second, the study surveyed 75% of IDEA Donna seniors (74 out of 98 individuals); the survey itself and the raw data collected were included in the appendices. Finally, a chisquared test (using Yates’ Continuity Correction since there was only 1 degree of freedom) was implemented. To check for one other possible factor (availability of a computer with an Internet connection), another chi-squared test was implemented. The chi-squared test yielded the conclusion that among IDEA Donna 2013 seniors: • Gender and Facebook Usage are independent • Internet at Home and Facebook Usage are independent Sorry this is the abstract from studies… but you still need an abstract.

Griffith 4

Griffith 5

Introduction The table below gives the median BMI for females of different ages in the US in the year 2000. Age (years) (kg)2 BMI ( ) m 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 This information comes from the American Centers for Disease Control and Prevention (CDC), whose mission is to “create the expertise, information, and tools that people and communities need to protect their health – through health promotion, prevention of disease, injury and disability, and preparedness for new health threats” (cdc.gov). As a government source in a more economically developed country (MEDC), it should be reliable. The x-variable is age in years and ranges from 2 to 20, covering females from toddlers to the end of adolescence. In the middle of this period, we should expect changes and girls go through puberty throughout their early teenage years. The y-variable is body mass index (BMI) (kg)2 (kg)2 (kg)2 in square kilograms per meter ( ) and ranges from 15.20 to 21.65 . In adults, m m m there are standard BMI guidelines across all age groups, while for children and teens, doctors refer to percentiles in a chart from the Centers for Disease Control and Prevention, as seen on the next page:

Griffith 6

Griffith 7 As we can see, the ideal BMI varies in children and teens based on age; this is likely based on growth spurts and other such changes occurring over these ages.

Plotting and Modeling the Data

BMI (kg2/m)

Below is a graph plotting the data points from the table on page 4, developed with Mac Grapher:

Age in Years This data could be modeled using a sinusoidal curve since it decreases, then increases, and then seems to be leveling off in preparation to decrease again in a smooth curve. To model using a cosine curve, we need to find the amplitude, vertical shift, phase shift and (kg)2 (kg)2 period. For this, we need to know the maximum BMI (21.65 ), minimum BMI (15.20 m m ), period (2 x (20-5) = 30) and maximum occurs at x = 20: 21.65 !15.20 Amplitude = half the distance from min to max = = 3.225 2 Period = twice the time to go from min to max = 2 ! (20 " 5) = 30 15.2 + 21.65 Vertical Shift = average y-value = = 18.425 2

Griffith 8 Phase Shift = on cosine, when the max occurs = 20 Using this information, the model for y = a cos(bx ! c) + d would be y = !3.225cos(12x ! 20) +18.425 .

Griffith 9

Graphing the Model Function and Original Graph

BMI (kg2/m)

Below is the graph of the original data with the model function:

Age in Years Although the period, amplitude and vertical shift seem to match, I have evidently made a mistake in calculating the phase shift: I should have accommodated the changed period. Instead, I need to show that the graph has shifted to the right even more by increasing the value of c in y = a cos(bx ! c) + d . In fact, since the graph is reflected over its horizontal axis, we need to look at the phase shift as moving to 5. Since the period is compressed by 12, we need to multiply 5 x 12 to obtain c = 60: y = !3.225cos(12x ! 60) +18.425 . The new graph is shown on the next page:

BMI (kg2/m)

Griffith 10

Age in Years As we can see, the refinement of this model is much more accurate than the original model function.

Finding Other Functions As commented before, the best model will be one that matches the decrease in BMI between the ages of 2 and 5 as well as the increase in BMI from age 5 to 20, especially considering how this data starts to level off. This could possibly include quadratics (since they can decrease, then increase) and cubic functions. Using the TI-84 Plus calculator, we can find that the possible quadratic model for the data is y = .01911x 2 !.01590x +15.26 with R2 = .9498 (a strong, quadratic correlation). Here is the graph of this new possible model with the first model and original data:

BMI (kg2/m)

Griffith 11

Age in Years Clearly, the quadratic model does not match the data as well as the cosine model as it does not share the same minimum, and continues to rise instead of leveling off around 20 years. Next, we will try the cubic model. Using the TI-84 Plus calculator again, we can obtain the cubic model, which is y = !.004075x 3 +.1536x 2 !1.276x +18.27 with R2 value of 0.9991, a very strong, cubic correlation. Here is the graph of the cubic model with the original data (see next page):

BMI (kg2/m)

Griffith 12

Age in Years This model matches the data extremely well – slightly better than the cosine curve did originally, and definitely better than the quadratic curve. Though its superiority over the quadratic model is almost too obvious, the improvements over the cosine curve are slight: Both seem to hug the middle data and level off near 20 years, but the cubic curve swings up on the left-hand side of the graph to include the data below 5 years, especially 2 years.

Estimating (Extrapolating) for Older Women Now, we will assess the usefulness of our model for females outside of the 2 – 20 year old range by estimating the BMI of a 30-year-old woman. In that case, we would use x = 30 in our cubic model, as it matched the data closest: kg 2 y = !.004075(30)3 +.1536(30)2 !1.276(30) +18.27 = 8.1966 m Considering this would be less than half of the BMI threshold for “underweight” women, this is not a reasonable estimate for the BMI of a 30-year-old woman. It seems that our model is not useful for females older than 20 years of age.

Griffith 13

Using the Model in Other Contexts The table below gives the median BMI for females of different ages in Canada in the year 2000. Age (years) (kg)2 BMI ( ) m 2 15.70 3 15.40 4 15.25 5 15.20 6 15.25 7 15.40 8 15.75 9 16.10 10 16.60 11 17.20 12 18.00 13 18.60 14 19.60 15 20.20 16 20.75 17 20.85 18 21.22 19 21.25 20 21.40 This information comes from the World Health Organization (WHO) and from Canada (who.org). As we can see in the graph below, the Canadian model still mostly fits the data without making any changes. We should still expect it to be limited to girls 2 to 20 years of age, though.