LAB 3: Proportionality/Geometric Similarity Modeling and Model Fitting
1. PURPOSE: To provide the student the opportunity to develop proportionality arguments and test them using technology.
2. OBJECTIVE: Determine if the hearts of mammals are geometrically similar using your knowledge of proportionality models and technology. Provide a summary of your analysis using the following data to support or refute your argument.
Heart WeightLength of cavity of
Animal (in grams)left ventricle (in)
Mouse0.13 0.55
Rat0.64 1.00
Rabbit5.80 2.20
Dog 102.00 4.00
Sheep210.006.50
Ox 2030.00 12.00
Horse3900.00 16.00
3. PROCEDURE:
a. Develop the proportionality model relating heart weight (HW) to the length (L) of the cavity of the left ventricle. You should get HW L3.
b. Test the proportionality model using the data provided. Testing a proportionality model often involves the following steps:
(1) Enter the raw observed data into your calculator and computer.
(2) Plot the raw data to check for smoothness and potential "outliers", and to give you a "feel" of the type of proportionality you might find.
(3) Make any necessary transformations to the data for the model.
(4) Plot the transformed data to test the proportionality (It must form a straight line through the origin).
(5) Estimate the constant of proportionality (eyeball a straight line through the origin and use slope =rise/run to find the constant).
(6) The use a least squares fit on the model and record the output of the parameter (the slope of the line through the origin) If you have covered Model Fitting.
(7) Rewrite the model in its original form, i.e., untransform the model.
(8) Create a column of "modeled y values" using your model.
(9) Calculate the error and relative error between the actual data and your proportionality model.
(10) Obtain an overlay of the actual vs. computed data to see if your model properly captures the trend of the data. Comment about the visual fit.
(11) Plot the residuals of the model versus the independent variable. Check for ‘model reasonableness’
.
The TI-83 Plus
Press STAT>EDIT
Enter the independent variable (L) in L1 and the dependent variable (HW) into L2.
2nd STAT PLOT
Turn on Plot1 as a scatter plot of L1 versus L2.
Set WINDOW [0,20,.5],[0,4000,25]
Press GRAPH
The plot is concave up and increasing.
We want to plot the proposed "proportionality" argument, HW L3.
Press STAT>EDIT
Place the cursor up into L3. Type (L1)^3. This will place the cube of each L1 value into L3.
We now want to obtain a plot of HW versus L3.
Press 2nd STAT PLOT
Turn off PLOT1
Turn ON Plot2 as L3 versus L2. We want to see if it looks like a "reasonable" straight line.
Change the window.
WINDOW [0,4100,1],[0,4000,1]
Press GRAPH
This plot looks like a line through (0,0).
Pick two points and find the slope. (0,0) and (4096,3900).
Slope = 3900/4096 = 0.952148
Press Y=
Enter the following function, 0.952148x^3.
Turn off PLOT2
Turn on PLOT1
Change WINDOW back [0,16,.5],[0,4000,25]
Press GRAPH. This provides an overlay of the function and the data.
Looks reasonable.
We can obtain the residuals, Ya-Yp.
Press STAT>EDIT
Go to L4. In L4, .952148*L1^3.
Go to L5. In L5, (L2-L4).
You can obtain a residual plot, by plotting (STATPLOT) L1 versus L5.
MODEL FITTING
Since the least squares models internal to the TI-graphing calculators all are complete equations, for example,
y= ax +b
y=ax2+bx+c
We need a way to compute the least squares slope for a power functions through the origin, like for y = k xn , where n is known.
We will take advantage of the power model result to solve for the unknown parameter k.
K=xny/ x2n
The following programs is provided for the instructor and students.
Program 1.
Disp "FINDS THE SLOPE FOR A POWER MODEL"
Disp "Y SHOULD BE IN L1"
Disp "X IN L2 AND X^POWER IN L3"
0I
Disp "ENTER N, THE NUMBER OF DATA PAIRS"
Prompt N
L1*L3L5
L3^2L6
1-Var Stats L5
xB
1-Var Stats L6
xc
(B/CK
Disp "SLOPE IS "
Disp K
Stop
Or Program 2
Disp "ENTER P"
Prompt P
Disp "X IN L1,Y IN L2"
L1^PL6
1-Var Stats L6
x2H
L6*L2L5
1-Var Stats L5
xF
F/HZ
Disp Z
Each program calculates the least squares slope.
You can then plot the line with the data and calculate the residuals.
LAB : Residual Analysis
- Definition: A residual (error, deviation) is defined to be the difference between the original value of the dependent variable and the value provided by the model,
Ya-Yp.
- PURPOSE: This lab will expose you a method to collect and analyze residuals in order to help you to determine the adequacy of your model.
- ADEQUACY: See the attached handout on the examination of residuals. The bottom line is that we want no pattern (totally random) in the residual plot (residuals versus fitted values) for a specific model. If there is no discernable pattern, then the model is deemed adequate. If a pattern (or trend) is visible then we will conclude that the model may not be adequate. Additionally, the residuals should be Normally distributed (a bell shaped curve). In a normalized residual plot, if the residuals are truly normally distributed then the plot will appear linear.
- LAB EXERCISE:
(a)Using the data for weight of an animal’s heart (W) and the length of the left ventricle (l), determine if the model W l3 is adequate.
Lab Assignment:
SCENARIO: A modeler is investigating the relationship between the terminal velocity of a round object dropped from a tall building and its mass. The modeler cannot decide upon the proper assumption concerning the relationship between exposed surface area, velocity, and drag force.
The two assumptions the modeler wants to examine are:
1) FD Sv
2) FD Sv2
The modeler has collected the following data:
Mass of Ball(kg)Terminal Velocity(m/s)
.5226
.8239
1.2260
1.7275
2.3 285
2.8300
3.4306
REQUIRED:
1. Develop the proportionality models between mass and terminal velocity for each of the two assumptions above. Use Newton's Law which states that the sum of the forces acting on a body equals mass times acceleration and the knowledge the acceleration = 0 at terminal velocity.
2. After you develop each model. Test each model and determine which assumptions appear the more accurate.