Type 2 Tasks: Body Mass Index
This task is about the BMI of American women of different ages in 2000.
BMI is the relationship of height and weight of a specific person. It is used to find out a person's health. It is calculated by dividing the weight by the height squared.
1. Using technology, plot the data points on a graph. Define all variables used and state any parameters clearly.
Through observation I know that the dependent variable is the BMI which is dependent on the independent variable, Age. Moreover I figured if there are any parameters. I came to the conclusion that there is one, which is a constant in the curve's equation that changes to “give in” a group consisting of similar curves.
On the graph below, the red dots present the mean BMI in relation to the age.
2. What type of function models the behavior of the graph? Explain why you chose this function. Create an equation (a model) that fits the graph.
In my opinion the sine function would fit the curve. Already when I saw the graph first I immediately thought of the sine curve and then I sat the goal to change my graph in a few aspects (stretching, shifting etc…) until it fit the sine curve.
The function for the sine graph is stated below:
Now I have to find the missing values and I started with getting A. A is the amplitude and to find it I must calculate the distance from the maximum point to the middle point of the graph. From looking at the data table above I can see that the maximum value is 21.65 and the minimum value is 15.20.
To find out the middle point of the graph I used the following calculation:
(Maximum + Minimum) ÷ 2
Therefore I calculated this:
(21.65+15.20) ÷ 2 = 18.425
To finally get the Amplitude I used the following calculation:
Maximum point – Middle point
As a result I got: 21.65-18.425= 3.225
Now I have to calculate B, which is the stretch (period) of the graph (sin x = 2π). Nevertheless, because the period on the data above is not completed it becomes sin x = 2π÷b andin this we have to find b.
The graph is most likely to be half of the period, therefore I have to multiply it by 2.
The whole calculation is shown below:
20 (maximum) – 5 (minimum)= 15
15 x 2 = 30
Now I replace b in the equation with 30 and get:
2π÷ 30 = 0.2094
Now I have to find out C which is the function's shift (C÷ B). When the graph is being shifted positively it moves to the right, in contrary, a negative shift means it moves to the left.
The first step I did to find out C is that I figured out the difference between the Maximum, which is 20, and the minimum, 5. points. So it is 20-5=15. After this we have to halve the difference and add the result to the minimum. Therefore I got: 7.5+5=12.5 While in a finished sine curve the period (in a whole) is 30, I got 12.5 for the period end in the sine curve under a normalized condition. Then I did 30-12.5 to get the difference and the result was 17.5.
Lastly, to get the final result for C I had to multiply 17.5 with B and got the following result: 17.5 x 0.2094 = 3.664
C = 3.664
Then D had to be calculated which is used to indicate upward movement of the function graph. To do this I used the formula where I calculated the midpoint:
(Maximum + Minimum) ÷ 2
I therefore got: (21.65+15.20) ÷ 2= 18.425
Now I have to see whether my calculated values and the values from the data are similar enough or not. To do this I will draw a graph that shows both, my model and the model of the data given.
During my research I found out what RMSE means. It stands for “root means square error”. This is very useful when two sets values are compared (in this case my values and the original ones). It shows the differences of the two models.
Below I drew a graph that shows my values and the original data (white points).
The red line indicates my function; the function is: y=3.225*sin(0.2094x+3.664)+18.425
3. On a new set of axes, draw your model function and the original graph. Comment on any differences. Refine your model if necessary.
I have changed my values slightly to reduce the RMSE and through this make both curves more similar.
4. Use technology to find another function that models the data. On a new set of axes, draw your model functions and the function you found using technology. Comment on any differences.
After research I chose the cubic function which is: y= ax3+bx2+cx+d
Its graph is shown below (blue); my model is red.
There is an enormous difference. The cubic function is much steeper.
5. Use your model to estimate the BMI of a 30-year-old woman in the US. Discuss the reasonableness of your answer.
6. Use the Internet to find BMI data for females from another country. Does your model also fit this data? If not, what changes would you need to make? Discuss any limitations to your model.
On the left side is a table that I found in the internet which shows the BMI values for young women in the UK.
In the following steps I am going to form a model (same steps then in question 2) from these values and compare it to the original data.
The function is again: Y=A*Sin(Bx+c)+d
A: (Maximum + Minimum) ÷ 2
25 + 17.3 = 42.3
42.3 ÷ 2 = 21.15
(Maximum – Middle Point)
25 – 21.15 = 3.85
A = 3.85
B: (Maximum - Minimum) * 2
25 – 17.3 = 7.7
7.7 * 2 = 15.4
2π ÷ 15.4 = 0.4080
B = 0.4080
C: (Maximum - Minimum) ÷ 2
7.7 ÷ 2 = 3.85
3.85 + 17.3 = 21.15
30 – 21.15 = 8.85
8.85 * 0.4080 = -0.2568
C = 3.6108
D: (Maximum + Minimum) ÷ 2
25 + 17.3 = 42.3
42.3 ÷ 2 = 21.15
D = 21.15
The graph next page shows the new model (y=3.85*sin(0.4080x+3.6108)+21.15; white) and my model (y=3.225*sin(0.2094x+3.664)+18.425; red).C:\Users\lukasS\Desktop\graphs\last.JPG
There is a very big difference between my model and the new one (UK values). To make it more similar I would need to change the values of the new model. These changes are that the A, B and D values would need to be decreased because currently they are higher than the ones of my model. On the other side, C is currently smaller and therefore it would be necessary to make that value bigger.
Limitations are the slightly different age groups (2-20 years for my model; 2-19.5 for the new model).
Through this coursework I have found out that when comparing two different models it is useful to look at the RMSE and change values to make them more similar. I also discovered the difference between women from the US and women from the UK. There are quite big differences between these groups.