Modeling and Decision-Making

Christine BelledinNC School of Science and MathematicsTeaching Contemporary Mathematics

January 2012

From the Common Core Standards

Modeling links classroom mathematics and statistics to everyday life, work, and decision-making. Modeling is the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions. Quantities and their relationships in physical, economic, public policy, social, and everyday situations can be modeled using mathematical and statistical methods. 

The Heart Catheter Problem

When a heart catheterization is done, a catheter is passed into the femoral artery in the leg and maneuvered into the heart. One problem facing the physician is determining the proper length of the catheter to be used in surgery. Physicians have used both the patient’s height and the patient’s weight to predict the appropriate size catheter. The data for ten young patients is given below. The patient’s height is measured in inches, their weight in pounds, and the length of the catheter needed in centimeters.

Which measurement (height or weight) is more useful in predicting the length of the catheter required?

Height 63.5 37.5 43 37 39.5 45.5 33 58 42.5 38.5Weight 93.5 35.5 38.5 33 30 52 21 79 40 17Catheter

50 34 37 34 36 43 38 47 37 28

Catheter Length vs. Height

x

y

Height (in)

Cath

eter

leng

th (c

m)

𝐶=0.5791h+13.0364

Residuals

From the Nayland School in New Zealand

Residuals for Catheter Length vs. Height

x

y

Height (in)

Resid

uals

(cm

)

Height Residual

% Error

63.5 0.19 0.437.5 -0.75 2.243 -0.93 2.537 -0.46 1.4

39.5 0.08 0.245.5 3.61 8.433 5.85 15.458 0.37 0.8

42.5 -0.64 1.838.5 -7.33 26.2

Catheter Length vs. Weight

x

y

Weight (lbs)

Cath

eter

leng

th (c

m)

𝐶=0.2457𝑤+27.6025

Residuals for Catheter Length vs. Weight

x

y

Weight (lbs)

Resid

uals

(cm

)

Weight Residual % Error

93.5 -0.57 1.235.5 -2.32 6.838.5 -0.06 0.233 -1.71 5.030 1.02 2.952 2.62 6.121 5.23 13.879 -0.01 0.040 -0.43 1.217 -3.77 13.5

Sample Student Response 1

Our initial hypothesis is that height would be the best predictor of catheter length. We assumed a taller person would have a longer femoral bone, which would then cause a need for a longer catheter to reach from the leg to the heart.

Sample Student Response 1

The maximum percent error for the weight vs. catheter length data set was 13.79%. The highest percent error for the height vs. catheter length data set was 26.18%.

Sample Student Response 1

The smallest percent error for the weight vs. catheter length data set was 0.023% while the smallest percent error for the height vs. catheter length was only 0.25%.

Sample Student Response 1

The average percent error for height vs. catheter length was 5.928% and the average percent error of the weight vs. catheter length was 5.05%.

Sample Student Response 1

We conclude that our hypothesis was not supported by our findings. Height did not provide the most accurate catheter length.

Sample Student Response 2

Given the following information and strategies my partner and I determined that height would be the most appropriate and efficient way to determine the length of the catheter.

Sample Student Response 2

From the graphs you see that the residuals for weight are close to zero, but more are farther away from zero.

Sample Student Response 2

Looking at the [residuals for the] height verses catheter length graph you can tell that there are some extreme outliers, but more points closer to zero.

Sample Student Response 2

Also, when examining the percent error from the residual plots, the points on the height plot were very small, almost all of them being under 10%.

Another argument…Height Residua

l %

Error63.5 0.19 0.437.5 -0.75 2.243 -0.93 2.537 -0.46 1.4

39.5 0.08 0.245.5 3.61 8.433 5.85 15.458 0.37 0.8

42.5 -0.64 1.838.5 -7.33 26.2

Weight Residual % Error

93.5 -0.57 1.235.5 -2.32 6.838.5 -0.06 0.233 -1.71 5.030 1.02 2.952 2.62 6.121 5.23 13.879 -0.01 0.040 -0.43 1.217 -3.77 13.5

Weight was a better predictor for more of the patients.

The Ticket ProblemThe senior class at Northview High School wants to raise money to support the athletic program by selling tickets that will allow a family to attend all athletic events at the school. The class officers are trying to decide the price for a single ticket. Some students argue for setting the price low, believing that a low price would bring a large response. Others want to set a higher price, so that even if not many tickets were sold, the school would still make money. The students decide to ask parents what they would be willing to pay for an all-sports ticket. A survey is sent to all 811 families with students in the school asking, “What is the most you would be willing to pay for an all-sports ticket good for this school year?” The results are given in the following table:

What price should the students set for each ticket in order to bring in the most money to the school?

Maximum Price $50 $75 $90 $95 $115 $135 $150 $175# Willing to Purchase

145 80 45 85 120 80 60 150

Is there a relationship between price and the number of families willing to purchase a ticket?

40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

160

Maximum Price (dollars)

Expe

cted

Tic

ket

Sale

s

A related dataset that may prove more helpful…

Maximum Price

Cumulative Sales

50 76575 62090 54095 495

115 410135 290150 210175 150

40 60 80 100 120 140 160 180 2000

100200300400500600700800900

Cumulative Ticket Sales

Ticket Price (dollars)

Cum

ulat

ive

Tick

et S

ales

𝑇𝑖𝑐𝑘𝑒𝑡𝑠=−5.079∙𝑃𝑟𝑖𝑐𝑒+996.853

Finding a Model for RevenueRevenue = Price per ticket · Number of tickets

purchased

p

R (98.13, 48912.97)

Price (dollars)

Re

venu

e (d

olla

rs)