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Modeling and Decision-Making Christine Belledin NC School of Science and Mathematics Teaching Contemporary Mathematics January 2012

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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.

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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 patients height and the patients weight to predict the appropriate size catheter. The data for ten young patients is given below. The patients 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? Height63.537.5433739.545.5335842.538.5 Weight93.535.538.533305221794017 Catheter50343734364338473728

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Catheter Length vs. Height Height (in) Catheter length (cm)

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Residuals From the Nayland School in New Zealand

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Residuals for Catheter Length vs. Height Height (in) Residuals (cm) HeightResidual % Error 63.5 0.190.4 37.5 -0.752.2 43 -0.932.5 37 -0.461.4 39.5 0.080.2 45.5 3.618.4 33 5.8515.4 58 0.370.8 42.5 -0.641.8 38.5 -7.3326.2

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Catheter Length vs. Weight Weight (lbs) Catheter length (cm)

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Residuals for Catheter Length vs. Weight Weight (lbs) Residuals (cm) WeightResidual % Error 93.5-0.571.2 35.5-2.326.8 38.5-0.060.2 33-1.715.0 301.022.9 522.626.1 215.2313.8 79-0.010.0 40-0.431.2 17-3.7713.5

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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.

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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%.

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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%.

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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%.

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Sample Student Response 1 We conclude that our hypothesis was not supported by our findings. Height did not provide the most accurate catheter length.

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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.

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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.

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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.

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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%.

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Another argument… HeightResidual % Error 63.5 0.190.4 37.5 -0.752.2 43 -0.932.5 37 -0.461.4 39.5 0.080.2 45.5 3.618.4 33 5.8515.4 58 0.370.8 42.5 -0.641.8 38.5 -7.3326.2 WeightResidual % Error 93.5-0.571.2 35.5-2.326.8 38.5-0.060.2 33-1.715.0 301.022.9 522.626.1 215.2313.8 79-0.010.0 40-0.431.2 17-3.7713.5 Weight was a better predictor for more of the patients.

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The Ticket Problem The 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 Purchase14580 45851208060150

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

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A related dataset that may prove more helpful… Maximum Price Cumulative Sales 50765 75620 90540 95495 115410 135290 150210 175150

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Finding a Model for Revenue (98.13, 48912.97) Price (dollars) Revenue (dollars)

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