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FACULTY NOTES
The LTAs and Spinoffs are designed so that each professor can implement them in a way that is consistent with his/her teaching style and course objectives. This may range from using the materials as out-of-class projects with minimal in-class guidance to doing most of the work in class. The LTAs and Spinoffs are amenable to small group cooperative work and typically benefit from the use of some learning technology. Since the objective of the LTAs and Spinoffs is to support the specific academic goals you have set for your students, the Faculty Notes are not intended to be prescriptive. The purpose of the Faculty Notes is to provide information that assists you to take full advantage of the LTAs and Spinoffs. This includes suggestions for instruction as well as answers for the exercises.
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FACULTY NOTES
LTA 15
Population Size Matters: Formulating a Mathematical Model for Scrub-Jay Population at the Kennedy Space Center
Level of Presentation: College Algebra, Precalculus
Mathematical Prerequisites: Linear Equations and Graphing
Technology Requirement: Scientific calculator with two-variable statistics, or a graphing calculator such as TI-83™, or computer software capable of regression analysis, such as Excel™.
Time Required: 2 Class Hours
Recommended Working Format: Short lecture, small groups, and student use of graphing calculator or appropriate software
Comments: The instructor may need to elaborate on some concepts that might be unfamiliar to students, such as the concept of a regression line and the coefficient of determination.
Your Turn
- Sketch a scatterplot of the data from Table 1 using an x-y coordinate system with year the independent variable and population size the dependent variable. Does the pattern of the plotted points appear to fit any of above models?
Table 1: Scrub-Jay Population
If we plot the data in the above table, we obtain the scatterplot shown below.
Year (t) Population Size (N) 1980 3697 1985 2512 1989 2176 1991 2100 1992 1922 1993 1857 1994 1860 1995 1689 1996 1603 1999 1127
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Power regression equation: N = 4066.18t-0.
Your Turn
- Sketch graphs of the quadratic regression, exponential regression, and power regression equations superimposed on the scatterplot of the scrub jay data. Visually inspect the graphs and describe how well each regression equation fits the data.
To see how closely the different models fit the scatterplot, we use the TI-83™ calculator to display graphs of the different regression equations superimposed on the scatterplot. From the following graphs we see that all of the regression equations appear to fit the data reasonably well, but the power regression model appears to provide the weakest fit. This would suggest that we eliminate the power regression equation as a candidate for the best mathematical model.
Linear Regression Quadratic Regression
Exponential Regression Power Regression
Your Turn
- Using your calculator or computer software, retrieve the values of the coefficient of determination for each of the models considered in Step 2. Use the coefficients of determination to decide which model best fits the data.
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The R^2 values are shown on theTI-83™ screens below:
The regression equations and their corresponding coefficients of determination are listed below:
Linear: N = 3537.7 - 119.63t r^2 = 0.
Quadratic: N = 2.5125t^2 - 171.538t + 3725.1 R^2 = 0.
Exponential : N = 3833.64(0.94754t) r^2 = 0.
Power: N = 4066.18t-0.3148397^ r^2 = 0.
Based on a comparison of the coefficients of determination, we should eliminate the power model as a candidate for the best model since the other values are closer to 1. In comparing the three remaining models (linear, quadratic, exponential), it might be tempting to simply select one with the largest r^2 value, but the values of 0.9601 and 0.9457 are not that far apart, and we should not exaggerate the importance of their difference. The differences among the linear, quadratic, and exponential models are relatively small; we should not simply select the model with the largest value of r^2.
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The linear and quadratic models both have r^2 values close to 1, so both models are very good. When making a prediction for year 7, the linear model results in 214 and the quadratic model results in 212, so both estimates are close.
- a) y = -846.733 + 436.4x (with r^2 = 0.728) b) y = 169.429x^2 - 749.6x + 734.6 (with R^2 = 0.962) c) y = 9.98671(2.50224)x^ (with r^2 = 0.999) d) y = 15.7435(x2.51242) (with r^2 = 0.970)
The exponential model appears best because it has the highest r^2 and because of the same reasoning used in the Scrub-Jay example discussed in Step 2 of the LTA, The estimated value of y when x = 7 is 6134.
- In the following equations, x = 1 represents 1980. a) Use y = 933185 - 47031.5x (with r^2 = 0.621) to get an estimate of 39,587. b) Use y = -899.986x^2 - 29931.8x + 876186 (with R^2 = 0.626) to get -17,413. c) Use y = (1203014)(0.886814)x^ (with r^2 = 0.726) to get 122,769. d) Use y = (1,538,310)(x-0.685964) (with r^2 = 0.534) to get 204,110.
The exponential model appears best because it has the highest r^2 and because of the same reasoning used in the Scrub-Jay example discussed in Step 2 of the LTA.
- Answers will vary depending on the data obtained from the Internet.
