Part 1: Curve fitting the AIDS data.

The table below gives the number of new AIDS cases reported each year in the United States for the years 1981-1992.

(1) Examine the data in the chart and write several interpretive statements about your observations.

(2) Enter the data in your graphing calculator (using 1, 2, ... for the years) and construct a scatterplot. Discuss how the scatterplot supports your interpretations of the data, and describe the growth of the number of new cases of AIDS in mathematical terms.

(3) Make two additional scatterplots of the data, by manipulating the view window of your calculator, so that your three scatterplots support three different interpretations of the data.

(4) Record the windows for your three scatterplots. Discuss how the concept of slope is visually and numerically connected to these windows and their graphs.

(5) What interpretations of the data are supported by each of these scatterplots? For what purposes would someone choose to display the data in each of these windows?

Part 2: Comparing New and Cumulative Data

(6) Look at the numerical data and the scatterplot for the new AIDS case data for the years 1981-1992, and roughly sketch by hand a scatterplot of the cumulative AIDS cases for these years. Explain how you made this sketch.

(7) Use your calculator to generate a list of cumulative AIDS cases for the years 1981-1992, and to draw a scatterplot of this cumulative data. Compare this scatterplot with your sketch. What relationships do you see between the new case scatterplot and the cumulative case scatterplot?

Part 3: Curve-fitting AIDS Data

(8) It has been said that the number of new AIDS cases in the US is growing at a steady rate, and it has often been reported to be growing at an exponential rate. What mathematical models are consistent with these growth rates? Fit the new AIDS data with a linear model, an exponential model, and two other models that seem viable. Explain your choice of models, and record the equations of all of these models.

(9) Discuss the appropriateness of the assumptions that AIDS is growing at a steady rate and at an exponential rate appropriate? Do other models better fit the data? Which model(s) makes most sense? Discuss your answers to these questions.

(10) Fit several viable regression models to the cumulative AIDS data. Discuss your choice of models and choose the ones that most sensibly fit the data.

(11) Given that the cumulative data is determined by accumulating the new data, discuss possible mathematical relationships that should exist between models for the new data and models for the cumulative data. Do your "best fit" models for each have any such relationship?

Part 4: Predicting and Assessing Predictions

(12) Examine the 1981-92 numerical data and the scatterplots, and then predict the number of new AIDS cases that were reported in the years 1993-1996. Record your predictions.

(13) From two of the mathematical models determined in Task (8) predict the number of new AIDS cases for the years 1993 to 1996. Record these predictions and reconcile them with those made in Task(12). Consider your numerical, graphical, and algebraic predictions, and then make best guess predictions for these years. Record these predictions. Discuss how you made these predictions, and comment on the reasonableness of these predictions and on your confidence in them.

(14) Observe the data for the years 1993-1996.