Park City Mathematics Institute Data Analysis, Statistics and Probability Project Abstracts Drafts of Project Files (password required)

Having a Boy
David Kapolka

I am working on writing a simulation to model the probability of a family having a specified number of children.
Problem: A husband and wife want to have a boy to pass on the family name. However, due to financial and medical concerns, they do not want to have more than 3 children. Using Fathom, write a simulation to model the probability that the family will have one boy. The solution will run several samples so as to keep track of the expected family size and the expected probability of having a boy.
There will be a worksheet and explanation to solve the problem with TI-83.

Do Women and Men Respond Differently to a Survey Question?
John Mahoney

This unit parallels a similar problem from the 2003 Advanced Placement Statistics exam. In the problem a group of 200 voters are asked how they feel about the question: "The mayor is doing an excellent job." The gender and response from each voter is recorded in a table. The question is whether the differences between the responses, based on gender, is statistically significant. Using the dynamic statistical software program, Fathom, this question is addressed by performing a Test for Independence of Categorical Attributes using a chi-square test. Fathom permits the user to randomize the sampling process to determine if the conclusions remain the same with different samples.

What is the relationship between the area of a rectangle and its perimeter?
John Mahoney

This activity is designed for teachers to use in algebra or geometry classes. The first six pages of the activity are fully approachable by students who have seen equations of parabolas, but less advanced students will still get a lot out of the activity. The last four pages, involving triangles, require a comfort with quadratic equations.

The activity relates to the relationship, initially, between the area and perimeters of rectangles. Using Fathom, the graph of the areas of 1000 random rectangles are plotted as a function of their perimeters. Students are led to discover the equations for the borders of this, somewhat surprising, graph. There are three extensions to this activity: squares, circles, and triangles. Students are also led to find the equations for the graphs for these figures.

Olympics
Marla Cortes
This lesson uses the least squares linear regression to explore the squares in least squares and minimize the areas of the squares built on residuals for a 100m freestyle Olympic comparison of Men's winning times and Women's winning times from 1948 to 1992. This lesson is intended for any student in Pre-AP Algebra I or an Algebra II class.
A Parabolic Path to a Best Best-Fit Line:
Finding the Least Squares Regression Line By Exploring the Relationship between Slope and Residuals

Jeff Reinhardt & Joe Simons

How does one determine a best-fit line for a set of data? One approach is to find the line with the smallest overall error in prediction, or more precisely, the line that minimizes the sum of the squared deviations from the observed data to that line. This activity uses dynamic statistical software (Fathom) to engage students in an exploration of this concept. Building on intuitive guess-and-check methods to estimate the least squares line, students experiment with the underlying quadratic relationship between slope and residuals to accomplish this goal more effectively. This optimization technique connects students' knowledge of quadratic functions with statistics, and in the process, deepens their understanding of the least squares linear regression line. The activity would be appropriate in a course introducing concepts in regression analysis or at a more advanced level (such as AP Statistics) to reinforce and extend these ideas.

Using Probability to Approximate Pi
David N. Hernandez
Mathematical content and domain: Algebra---probability
Short description: The objective of this project is to explore using probability to approximate the value of pi. To meet this objective, students will generate a random set of points lying in a quarter circle that is inscribed in a unit square. Students will initially use the TI-83 graphing calculator and then Fathom software to:
1. determine the geometric probability of a point lying in the quarter circle,
2. generate a random set of 20 points,
3. determine the experimental probability of a point lying in a quarter circle from the random set, and
4. estimate the value of pi from their experimental probabilities.
Reference: Algebra 2: Explorations and Applications, McDougal Littell, 1998, p. 616, # 15.
Translating Parabolas
David N. Hernandez
Mathematical content and domain: Algebra-quadratic functions
Short description: Objective of this project is to see how graphs of equations in the form y = a(x - h)2 + k change for different values of a, h, and k. Students will initially use the TI-83 graphing calculator to graph different equations and complete a worksheet on translating parabolas. Students will then use Fathom to graph the same equations and see the changes more dynamically using sliders.
Reference: Algebra 2: Explorations and Applications, McDougal Littell, 1998, p. 192, Exploration Activity.
Cereal Box
Beverly Farahani
Use this activity to help students look at simulations. A cereal company has decided to include a toy of a cartoon character. There are 6 different characters in all. Students will create simulations to illustrate collecting one of each of the toys by imitating the purchase of boxes of cereal. The choice of how the simulation is created is left up to the student. Various simulations are shown in the teacher notes.