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Accelerated Math I
Unit 6: Data Analysis and Probability – Teachers Edition
Table of Contents
Introduction
Task 1:
Regression Learning Task …………………………………………..…….1
Written by George R. Shirley, Jr.
Airline Online Instructions written by Susan Hotle, Edited
by George R. Shirley, Jr.
Summative Evaluation for Regression based on Climate Change……….19
Written by George R. Shirley, Jr.
Task 2:
Probabilities Learning Task……………………………………………...25
Written by Brittany Luken, Edited by George R. Shirley, Jr.
Task 3:
Correlation and its Meaning……………………………………………...30
Written by Brittany Luken, Edited by George R. Shirley, Jr.
Data Analysis and Probability Using Applications from the Airline
Industry with Evaluation Exploring Climate Change
The purpose of this unit is to introduce Data Analysis and Probability using real
data gleaned from the airline industry and simulations involving the airline industry. Task
1 uses a program simulating building and maintaining an airline to generate real data that
is used to explore the topic of regression analysis. Task 2 takes place during the
maintaining part of task 1 and explores the topic of probability by evaluating delayed
departures in the aviation industry. Task 3 uses departure times to explore linear models.
All of this material was developed for Dr. Laurie Garrow in the School of Civil and
Environmental Engineering at Georgia Tech as a part of a statistics project by Brittany
Luken and a G.I.F.T. project by George R. Shirley, Jr., with guidance from Brittany
Luken and Susan Hotle.
Task 1 - Regression Learning Task:
Regression Analysis through Airline Simulation
Accelerated Math I
Performance Standards:
MA1D5. Students will determine an algebraic model to quantify the association between
two quantitative variables.
a. Gather and plot data that can be modeled with linear and quadratic functions.
b. Examine the issues of curve fitting by finding good linear fits to data using simple
methods such as the median-median line and “eyeballing.”
c. Understand and apply the processes of linear and quadratic regression for curve fitting
using appropriate technology.
Process Standards:
MA1P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d.. Monitor and reflect on the process of mathematical problem solving.
MA1P2. Students will reason and evaluate mathematical arguments.
a. Recognize reasoning and proof as fundamental aspects of mathematics.
b. Make and investigate mathematical conjectures.
c. Develop and evaluate mathematical arguments and proofs.
d. Select and use various types of reasoning and methods of proof.
MA1P3. Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers,
and others.
c. Analyze and evaluate the mathematical thinking and strategies of others.
d. Use the language of mathematics to express mathematical ideas
precisely.
MA1P4. Students will make connections among mathematical ideas and to other
disciplines.
a. Recognize and use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to produce
a coherent whole.
c. Recognize and apply mathematics in contexts outside of
mathematics.
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MA1P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical
ideas.
b. Select, apply, and translate among mathematical representations to solve problems.
c. Use representations to model and interpret physical, social, and
mathematical phenomena.
I - Engage Activity
Materials: Smartboard/Whiteboard/ActivSlate
Activity Type: Small group, Full class
Timetable: This task should take one class period (30 to 45 minutes).
Instructions:
1. Divide students into small groups of no more than three students. This may be adjusted
depending on the needs of the teacher.
2. Give groups 6 minutes to develop a list of variables that influence the success of an airline
through brainstorming.
3. Following the completion of the teaks the teacher needs to discuss the explanatory/response
relationship of some of the variables and be sure to introduce (time, profit/loss) where time is
the number of iterations of the simulation.
4. If possible, it would be helpful to have someone from the airline industry come in and share
ideas related to this discussion.
Task(s):
1. Students are to develop an all inclusive list of the variables they identified through the
brainstorming process.
a. Provide students 3 minutes within their small groups to identify variables that
influence airline success
b. Provide students 3 minutes to intermingle and share group findings.
c. The teacher will make an extensive list on the board of the variables found by all
groups.
Answers will vary but may include: percentage of on-time flights, percentage of
delayed flights, airline name recognition, cost of airline fuel, pilot salaries, aircraft
maintenance costs, cost of terminal rental, cost of terminal ownership, food costs,
navigator salaries, stewardess salaries, cargo costs, ticket agent costs, etc.
2. Within their small groups, students are to identify each of the variables as qualitative or
quantitative and explain why they made each choice.
Qualitative
Quantitative
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Airline name recognition
Percentage of on-time flights
Percentage of delayed flights
Cost of airline fuel
Pilot salaries
Etc (any variable where finding the average
would have meaning)
3. Using class discussion, the teacher will label each variable on the board as qualitative or
quantitative.
Answers will be based on the student responses.
4. For quantitative variables, students are to group them and identify explanatory and response
variables if possible.
Answers will vary.
5. The teacher will discuss the findings from #4 while writing them on the board.
While discussing the answers in #4 the teacher needs to explain explanatory and response
variables.
II – Explore Activity
Materials: Airline Online1, Instruction packets2 as needed, Chart for recording simulations results
for each iteration of the task, the chart will have to be modified to meet the instructor’s timetable.
Activity Type: Small group
Timetable: This is an ongoing task that will take all of the first two days and parts of five or more
days thereafter, depending on the desire of the instructor.
Instructions:
1. Discuss simulation and its importance as a mathematical tool for generating data.
2. Each small group is to create an airline for use in the simulation task.
3. After developing the airline, a simulation iteration will be run to determine the profit and/or
loss generated by each airline.
4. Students will then record the information from the simulation and modify their airline to
improve its profitability.
Task(s):
1. Using the instruction packet as a guide, each group will create an airline they believe will
generate the greatest profit. [ 2 days ]
2. Each group will record the specifics of their airline and answer questions found in the
instruction packet regarding how and why they made particular decisions while creating their
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airline.
3. Following each simulation iteration, each group will modify their airline to improve its
profitability.
4. Each group must maintain a record of all modifications made during the improvement
process.
Instruction Packet:
This instruction packet is a modification of the one developed by Susan Hotle at the Georgia
Institute of Technology for use in guiding students through the process of setting-up and
maintaining an airline using the program Airline Online found at www.airlinesimulation.com/.
Airline Simulation for High School
This is a group (2-3 people) activity that uses the simulation program Airline Online. The
program puts your group in charge of the financial and operational decisions for your own airline
company. Your group will be competing against other groups to get the highest company
liquidated value after the simulation has run 3.5 years. Each group starts with $1 billion.
This packet contains:
Your groups website, username, and password
Airport Simulation Instructions (to be turned in at the due date for the project)
Three-letter codes for particular airports
Airplane Types
(Flight Charts will be created by your teacher and presented at the time of the simulation)
Usernames and Passwords
www.airlineonline.com/
Username
Password
These will be filled in according to the
needs of the class
and the set-up the
teacher uses.
Airport Simulation Instructions
Name of Airline/Username: _________________________________________________
Group Members: _________________________________________________________
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NOTE BEFORE STARTING: Decisions made during this simulation cannot be undone. This
program does not give the teacher or you the ability to reset your account or undo any actions
(including buying airplanes, hiring staff, and purchasing maintenance bases). To be safe, make
sure you have the approval of your group members before taking action.
It is highly recommended that you follow this order when creating your airline.
Throughout this process, this task will require that you answer questions. Please be specific and
complete in answering these questions, as this will be used in determining your grade for this
portion of the assignment. Questions will be printed in italics and underlined for convenience.
Day 1
Building an Airline (Part 1)
Go to the website, sign in and press Start Simulation. Make sure pop-ups are allowed on your
computer. After you get to the home page, in order to see all of the tabs at any point in the
simulation press Main at the top of the screen.
Staffing Tab – Go to Managers. Hire the managers you want. The number employed will stay
zero until the simulation is run. The following is a list of what each staff member does:
Maintenance Manager – Scheduling: provides advice to an airline on setting up A and B
checks during scheduling. This manager will also provide information on the check
effectiveness.
Maintenance Manager – Staffing: increases the effectiveness of staff by 20%.
Maintenance Manager – Purchasing: reduces final maintenance costs by 10%.
Maintenance Manager – Training: increases effectiveness of staff by 20% (reduces A/B
check times by 20% and can be used in addition to staffing to get 40%). However, this also
increases the maintenance staff costs.
Maintenance Manager – Reporting: provides access to maintenance reports. These can
be viewed from the maintenance report button on the maintenance screen.
Operations Analyst: provides you access to all financial reporting via the Analyst section
of the simulation.
Advertising Manager: Assists you when setting up an advertising campaign by providing
suggestions on effective budgeting and allocation of funds.
Cargo Manager: provides you with additional information, via the Analyst section, on
your cargo operations.
Explain which managers you chose and why.
Answers will vary. The teacher needs to remind students that once you hire a manager it will
cost you double to let them go later on. Although students will hire as they see fit, certain
managers may be more important to the success of the airline than others.
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Aircraft Tab – Go to Buy Used Aircraft. Because it takes years to obtain a new airplane and this
simulation is only for 3.5 years, you should only buy used aircrafts. Before buying, look at the
attached Airplane Types sheet. Keep in mind, you will have to purchase a maintenance base for
each type of engine (General Electric or Rolls-Royce) and train staff for each type of airplane
(Boeing or Regional) you buy. To buy an airplane, click view next to the airplane and then
purchase it. It is best that you have at least $300 million left over after buying airplanes to cover
future costs.
Explain your purchasing decisions and why they were made.
Students should explain why they purchased particular aircraft types, particular engine types,
and the number of aircraft.
Airports Tab – At the bottom of the screen, type in the three-letter code of the airport in which
you want to purchase space. Purchase offices, cargo handling centers, terminals, and
maintenance bases. These are not needed at every airport. You should look at the attached flight
chart for your specific airline before purchasing. Each airplane will need to go to an engine
overhaul centre multiple times a week. An aircraft will be grounded if it has been 7500 hours
since the last C check or if it has flown 22,000 hours. D checks reset flown hours to zero. Buying
a terminal is only needed if you feel the airport will be so busy that your airplane might not have
a terminal to go to.
Explain what you did and why. Where did you put your maintenance bases? Why?
Students should explain why they picked particular cities for maintenance bases, cargo handling
centers, offices, and terminals.
Staffing Tab – Hire the pilots and staff to operate the airplanes. Make an educated guess with
the staffing members. You will come back to fix them later (without being penalized financially).
You have to staff an airplane before you can schedule it. The “Actual Number” employed will
stay zero until the simulation is run.
Explain who you hired and why.
If students hired a staffing manager they will have recommendations as to the number of staff to
hire and the salary for each staff member. Students should explain if they keep these
recommendations or made changes and why.
Service Levels Tab – Create a New Flight Profile. This is where you designate what services
each class will get. Save the profile. You can make more than one profile.
Explain the profile(s) you set up and why. What effect do you believe your profile(s) will have on
your profit/loss?
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Students may decide to make up profiles based on the distance of the particular routes. Students
may also decide to make profiles based on the planes that travel to and from particular
locations. Students need to explain what effect the different profiles may have on the earnings of
the airline.
Day 2
Building an Airline (Part 2)
Analyst Tab (if you hired any) – Look over the analyst information which includes a list of the
airports with the highest demand and click on the City Pair Data button for more demand
information. This tab is full of useful information for scheduling.
It is helpful for students to look through all of the information provided under the analyst tab and
where they can use it in future decisions.
Scheduling Tab – Select the aircraft you want to schedule. Choose the destination and click
Complete Scheduling. Select the times for A and B checks (the table at the top of the screen
specifies the time requirements for A and B checks). The maintenance checks work only if the
aircraft is arriving at one of your maintenance bases and if there is enough time for a particular
maintenance check. Generally, a plane needs about 2 A checks and 1 B check every week. You
need to select Apply Maintenance Checks and then press Complete Scheduling three more times.
Then proceed to assign prices and luggage fees. If you want to keep the prices that are originally
there while adding the service profile to a flight, click View next to the flight and then at the
bottom of the screen select the service level profile. Select the flight and choose Apply to
Selected. Finish the scheduling of that airplane by pressing Complete.
Explain what you did and why. Did the size of your airplanes affect the destination they went?
How? Did the distance to destination affect the service level for the flight? How?
Scheduling takes a lot of time. Students need to look at which airlines fly to the same cities they
fly to and which cities are unique to their particular airline. Students also need to consider the
distance between cities and the length of time they are at a maintenance base for A and B checks.
Students need to look at what other airlines charge for different routes and makes decisions on
prices accordingly. Going to the analyst tab throughout this process and taking notes is helpful.
Aircraft Tab – Go to configuration. Create a new configuration. You can make more than one.
(The attached Airplane Types sheet can help you with figuring out the approximate sizes of the
planes.) To make a cargo only airplane, save the configuration screen without anything checked.
Then assign those configurations to the airplanes. (You may have to press the Back button at the
top of the screen before assigning configurations to the airplanes.) If you get an error, it is
because you have too many seats to fit that particular airplane.
Explain what you did and why.
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There are a number of reasons for creating particular configurations. Students should give
reasons for why they created each configuration. Also, students do not have to use each
configuration created.
Aircraft Tab – Go to cargo infrastructure. Buy x-rays machines, containers, and loading
equipment as needed. If you hired an analyst, suggested numbers should appear.
Did you have recommendations? Did you follow the recommendations? Why? If you did not have
recommendations, how did you choose the number of items to purchase?
Answers will vary.
Staffing Tab – Go to Staff Salaries. Make sure your staffing numbers are reasonable for the
number of airplanes owned. If you hired an analyst, suggested staffing numbers should appear.
How did you decide what was reasonable for staffing your planes?
Students may wish to go to the analyst tab to see if they meet union requirements. Students
should discuss the decisions they made concerning staffing and salaries.
Advertising Tab – Fill out how much you want to spend in each category for advertising and the
percentages. If you hired an advertising manager, you can click on the suggestions button so you
will not have to mentally make the percentages sum to 100%. Also, set the commission
percentage.
How did you allocate your advertising dollars? Why?
It is helpful to look at the effectiveness of different categories in the advertising campaign. The
analyst tab will also let students know if they are not using their advertising dollars wisely.
Airfares Tab – Search the average prices your competitors have on flights between any two
destinations. You can also change your prices on tickets in order to have competitive pricing.
Analyst Tab – If you hired an analyst, this will give you any overall recommendations on your
airline.
Aircraft Tab – Go to Maintenance. Schedule C and D checks if needed. An aircraft will be
grounded if it has been 7500 hours since the last C check or if it has flown 22,000 hours. D
checks reset flown hours to zero. Every quarter, an airplane flies about 800 hours (if it flies about
9 hours per day).
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Explain what you did and why.
Students will need to look at the total hours flown and estimate when C and D maintenance
checks are needed. Based on this information, students need to explain why they did or did not
decide to schedule a particular maintenance check.
When you have completed these steps, feel free to play around with the other functions of this
program (connecting flights, code shares, bonds, etc.). Completing the above steps makes sure
you have a functioning airline.
A simulation will be run following this time period and at the end of the maintenance time for
each of the following seven days.
End of class days 3, 4, 5, 6, 7, 8, and 9. The teacher may choose to continue this process until
the Probability section of this unit is complete.
Maintaining an Airline
Complete the Profitability Chart for the most recent simulation.
If your airline went bankrupt during the latest simulation run, you will have to start from scratch
again. You will start just as you did at the beginning of the simulation. If your airline made it
through the latest simulation run, continue with the following instructions for maintaining an
airline.
Aircraft Tab – Go to maintenance. Schedule C and D checks if needed. An aircraft will be
grounded if it has been 7500 hours since the last C check or if it has flown 22,000 hours. D
checks reset flown hours to zero. Every quarter, an airplane flies about 800 hours (if it flies about
9 hours per day).
Explain what you did and why.
Students should discuss all maintenance changes, addition, they made to their airline and why.
Analyst Tab – If you hired an analyst this will give you any overall recommendations for your
airline. Adjust you settings as needed.
What changes did you make and why? Did you purchase additional airplanes? Why? Did you
add additional routes? Where and Why?
Answers will vary. Teachers should monitor the reasonableness of the changes and voice
recommendations if needed. This is particularly important if a group of students goes bankrupt
during a simulation period.
Airfares Tab – Search the average prices of your competitors for any flights between two
destinations. You can change prices on tickets to have competitive pricing.
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Did you make any changes? What did you change? Why?
Answers will vary. Teachers should monitor the reasonableness of the changes and voice
recommendations if needed. This is particularly important if a group of students goes bankrupt
during a simulation period.
Three-letter Airport Codes
MRY
LAX
LAS
PHX
SLC
ABQ
DEN
DFW
Monterey
Los Angeles
Las Vegas
Phoenix
Salt Lake City
Albuquerque
Denver
Dallas-Fort Worth
Type
Engine
Max
Capacity (all
seats in
economy)
Max
Distance
(km)
Max
Distance
(miles)
Fuel
Efficiency
(l/km)
Min Runway
length (m)
MSP
ORD
MDW
ATL
BWI
LGA
JFK
Minneapolis-St. Paul
Chicago-O’Hare
Chicago-Midway
Atlanta
Baltimore
New York-Laguardia
New York-Kennedy
Airplane Types
777200ER
CRJ700
RR
Boeing Regional
RollsGeneral
Royce
Electric
717200HG
W
Boeing
RollsRoyce
767400ER
GE
Boeing
General
Electric
117
375
440
3815
10420
2371
CRJ900
EMB
190LR
ERJ145LR
Regional
General
Electric
Regional
General
Electric
Regional
RollsRoyce
70
90
98
50
11037
3124
2774
4260
2955
6475
6858
1941
1724
2647
1836
3.2
8.7
11.9
2
2.3
2.5
1.5
1570
2600
2600
1560
1870
1800
1245
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The teacher may decide to allow other aircraft according to the needs of the students and the
restrictions the teacher desires to impose.
End of Airport Simulation Instructions
________________________________________________________________________
Profitability Chart:
Iteration
1
2
3
4
5
6
7
Beginning Balance
Ending Balance
Profit/Loss
Percent
Profit/Loss
This section should take the two days following the simulation activity. This may be adjusted
depending on the duration of the Probability section of the unit.
Day 1
III – Explain Activity
Materials: Profitability Chart from the Explore Activity, Median-Median Line Activity, Least
Squares Regression Line Activity, Sum of Squared Error Activity.
Activity Type: Small group, Large group/Class
Timetable: This task will take two days following the simulation (Explore) activity.
Instructions:
1. Students will plot the data from the Profitability Chart and discuss the correlation coefficient
of the data and the linearity of the data.
2. Using the plot from #1, students will draw a possible line of best fit and find the equation of
their line (“Eyeballing” a line of best fit).
3. Using the data found in the Profitability Chart and the Median-Median Line Activity, students
will generate a median-median line of best fit.
4. Using the data found in the Profitability Chart and the Least Squares Regression Line (LSRL)
Activity, students will generate a LSRL for the data.
5. Using the Sum of Squared Error (SSE) Activity, students will compute the SSE for each line
and use this measure to compare the two lines.
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Tasks:
“Eyeballing” a line of best fit:
1. Create data points using the form (iteration, percent profit/loss).
( _____, _______ ),
( _____, _______ ),
( _____, _______ ),
( _____, _______),
( _____, _______ ),
( _____, _______ ),
( _____, _______),
( _____, _______)
2. Plot the points on a Coordinate plane where the x-axis
covers the domain [0, highest number of iterations used]
and the range is the range of the profit/loss in your
Profitability Chart.
Answers will vary.
3. Draw a line that you believe comes closest to hitting all
of your data points.
Answers will vary.
4. Choose two points on your line and find the equation of the line that goes through those two
points. You may use any form to find your equation but please write you final equation in y = mx
+ b form to be consistent.
Answers will vary but should match the data from #3. Students should list the points they use and
show their work for finding the linear equation.
Median-Median Line Activity:
1. Create data points using the form (iteration, percent profit/loss).
( _____, _______ ),
( _____, _______ ),
( _____, _______ ),
( _____, _______),
( _____, _______ ),
( _____, _______ ),
( _____, _______),
( _____, _______)
Answers should match those from #1 in the previous section.
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2. Plot the points on a Coordinate plane where the x-axis
covers the domain [0, highest number of iterations used]
and the range is the range of the profit/loss in your
Profitability Chart.
The graph, at this point, should match the one in the previous section.
3. Order your data by iteration and break the set up into three equal parts. (If it is not possible to
have three equal parts, make sure the first and third parts are equal and the middle part contains
+/- one data point when compared to the first and third parts.)
Answers will vary.
4. Find the point (median X, median Y) for each of the three parts of your data. This will yield
points M1, M2, and M3, which are the corresponding (median X, median Y) for each of your
three subsets of data.
Answers will vary.
5. Find the equation of the line, in y = mx + b form, passing through points M1 and M3. Think of
the y-intercept of this line as (b1).
Answers will vary.
6. Using the slope from the equation in #5 and the point M2, find the equation of the line, in y =
mx + b form, parallel to the line in #5 and passing through the point M2. Think of the y-intercept
of this line as (b2).
Answers will vary.
7. If (b1) > (b2) then let (b3) = (b1) - |(b1) - (b2)|/3, if (b2) > (b1) then let (b3) = (b1) + |(b1) –
(b2)|/3. The Median-Median Line is the line with the equation yˆ  mx  b3 , where m is the same
slope as found in #5 and #6 above and b3 is the y-intercept that is 1/3 above or below b1,
depending on the position of b2.
Answers will vary.
Least Squares Regression Line (LSRL) Activity:
1. Create data points using the form (iteration, percent profit/loss).
( _____, _______ ),
( _____, _______ ),
( _____, _______ ),
( _____, _______),
( _____, _______ ),
( _____, _______ ),
( _____, _______),
( _____, _______)
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Answers should match #1 in both of the previous sections.
2. Plot the points on a Coordinate plane where the x-axis
covers the domain [0, highest number of iterations used]
and the range is the range of the profit/loss in your
Profitability Chart.
The graph should match the ones in the previous sections,
#2.
3. Using the TI-83, TI-84, or TI-nSpire, go to STAT, EDIT, and place the x-values in L1 and the
corresponding y-values in L2, pressing enter after each value. Go to STAT, CALC, and then go
down to LinReg (a+bx), it should be #8 on your list, press ENTER, and press ENTER again (the
calculator will default to L1, L2). The calculator will then give you the values for a (the yintercept) and b (the slope) of your LSRL. If you have diagnostics turned on the calculator will
also give you values for r and r-squared. Your teacher may not allow the use of graphing
calculators since some states will not allow their use on End of Course Tests. In this case, your
teacher will provide an alternate method for finding the LSRL.
LSRL: ŷ  _____________________________
Answers will vary. The teacher may decide to give each group their particular LSRL.
Day 2
Sum of Squared Error (SSE) Activity:
Use your data from the Profitability Chart and the appropriate regression equation to complete
the following charts used to find the Sum of Squared Error.
1.
Sum of Squared Error for the line created by “eyeballing”
( y  yˆ )
Iteration (x) Profit/Loss (y) Predicted value ( ŷ )
1
2
3
4
5
6
7
The sum of the rows is the Sum of Squared Error
( y  yˆ ) 2
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2.
Sum of Squared Error for the Median-Median Line
( y  yˆ )
Iteration (x) Profit/Loss (y) Predicted value ( ŷ )
1
2
3
4
5
6
7
The sum of the rows is the Sum of Squared Error
3.
Sum of Squared Error for the LSRL
( y  yˆ )
Iteration (x) Profit/Loss (y) Predicted value ( ŷ )
1
2
3
4
5
6
7
The sum of the rows is the Sum of Squared Error
( y  yˆ ) 2
( y  yˆ ) 2
All answers for this section should vary.
4. Compare the SSE for each regression line. The regression line with the lowest SSE is the
better regression line. This process works for comparing any regression lines for a given set of
data. Which of your lines is the best fit line?
The SSE for the LSRL should be lowest.
5. Discuss your results with other groups in the class. What do you discover?
This provides teachers the opportunity to explain the meaning of the LSRL in terms of SSE.
6. Explain why you think your discovery in #4 is important.
This should open the class up to a discussion of regression lines and curves in general. This
discussion will be extended in the next activity.
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7. If you were to continue this process what would you predict your percent profit/loss for the
10th iteration as compared to the 9th iteration?
This process is called extrapolation because
you are predicting for values outside the domain. Would you consider extrapolation good or bad?
Why?
Aanswers may vary as students do not have a concept of extrapolation yet.
8. Predictions for values inside the domain is called interpolation. Which would be better,
extrapolation or interpolation?
Why?
Answers may vary. Following this activity the teacher needs to have a class discussion of
interpolation and extrapolation.
The teacher may choose to use this as a full-class activity instead of small group/large group.
This activity should take 1 day including the wrap-up discussion.
IV – Extend Activity
Materials: Profitability Chart from the Explore Activity, Scatter Plot and Sum of Squared Error
for the LSRL in the Explain Activity component, Sum of Squared Error Activity for this
component.
Activity Type: Small group, Large group/Class
Timetable: This task will take one day following the Explain component.
Instructions:
1. Students will graph the LSRL on the scatter plot of the data from the Explain Activity
component.
2. Students will create a quadratic regression of the data from the Explore Activity.
3. Using the Sum of Squared Error (SSE) Activity, students will compute the SSE for the LSRL
and the quadratic regression model.
4. The teacher will present additional modeling equations and their use in modeling real
phenomena.
Tasks:
1. Copy your original data into the following table and create a scatterplot of the data.
Profit/Loss
Iteration (x)
Percentage
1
2
3
4
5
6
7
17
2. Graph your LSRL on the scatterplot.
The answers to #1 and #2 should follow from the previous
work.
3. (a) How close does your line come to hitting all of the data points?
Answer will vary.
(b) What measure quantitatively supports your argument in part (a)?
SSE.
4. Is there a better model than a linear one? ____________ Using the following procedure, create
a quadratic model of the data. Using the TI-83, TI-84, or TI-nSpire, go to STAT, EDIT, and
place the x-values in L1 and the corresponding y-values in L2, pressing enter after each value.
Go to STAT, CALC, and then go down to QuadReg, it should be #5 on your list, press ENTER,
and press ENTER again (the calculator will default to L1, L2). The calculator will then give you
the values for a, b, and c of your Quadratic Regression model. Your teacher may not allow the
use of graphing calculators since some states will not allow their use on End of Course Tests. In
this case, your teacher will provide an alternate method for finding the Quadratic Regression
model or the model itself.
QuadReg: ŷ 
The teacher may need to provide the quadratic regressions if students are not using graphing
calculators.
5. Graph your quadratic regression equation on the scatterplot in question #1. Did it come closer
to hitting the data points than the LSRL?
Answers will vary and should provide the opportunity to discuss quadratic regressions and the
shape of the graph that should be expected.
6. Defend your answer in question #5 by finding the SSE for your Quadratic Regression model
and comparing it to the SSE for your LSRL.
Sum of Squared Error for the LSRL
Iteration (x) Profit/Loss (y) Predicted value ( ŷ )
1
2
3
( y  yˆ )
( y  yˆ ) 2
18
4
5
6
7
The sum of the rows is the Sum of Squared Error
Sum of Squared Error for the Quadratic Model
( y  yˆ )
Iteration (x) Profit/Loss (y) Predicted value ( ŷ )
1
2
3
4
5
6
7
The sum of the rows is the Sum of Squared Error
( y  yˆ ) 2
Which is the better model and why?
A discussion of the SSE should be provided. Answers will vary as to which is the better fit
depending on the actual data.
7. Notice that there are other regression models available on your calculator. Explain when each
of the following models would be used.
(a) LinReg:
(b) QuadReg:
(c) CubicReg:
(d) LnReg:
(e) ExpReg:
(f) Logistic:
19
The teacher should graph the parent graphs of each of the above regression equations and
explain the general shape of the data that would be useful in determining which regression
model should be used. For this course we will be using only linear and quadratic regression
models.
Summative evaluation day for the regression part of this unit. This activity can be used in
conjunction with the remaining summative activities for the unit.
V – Evaluate Activity
Materials: GRASP based Summative Evaluation Task with scoring guide..
Activity Type: Individual
Timetable: This task will take one day at the end of the unit.
Instructions:
1. Students will plot the Average Temperatures for McDonough from 1998 to 2007 from the
data in the provided chart.
2. Showing all necessary work, students will determine each of the following regression models:
a. Median-Median line
b. Least Squares Regression Line
c. Quadratic Regression
3. Using the method of SSE and showing all work, students will determine which of the above
regression models is most appropriate for the data and defend their answer.
4. Students will determine whether average temperatures are decreasing, staying the same, or
increasing and defend their determination based on the findings of their regression model.
5. Students will predict the average temperature in McDonough for August 1, 1990, August 1,
1997, August 1, 2005, and August 1, 2009. Students will then determine which of their
predictions they would consider most accurate.
6. Provided the actual data for August 1, 1990, August 1, 1997, August 1, 2005, and August 1,
2009, students will discuss the relationship of the real data to the predicted results and
whether adding this data prior to making their models would have an effect on the model.
7. Students will write a short paper discussing their findings.
Tasks:
20
Summative Assessment for Regression Part of Data Analysis and
Probability
The total number of points that can be
___________________________
achieved in a particular situation are found
in square brackets [ points ] beside the
____________
situation.
Student Name
Date _______________
Period
The Guidelines for this performance assessment are:
Real-world Goal: The goal is to take a defendable position on the concept of Climate Change.
Real-world Role: Your role as a data analyst is to organize and analyze the provided data to
demonstrate evidence for or against the idea of an increase in temperature over time.
Real-world Audience: Your target audience is the average person who may or may not
understand the process of analyzing data to demonstrate evidence for or against climate change.
Real-world Situation: You are provided average daily temperatures for the same date over
multiple years for a given location. You must organize and analyze this information and make a
judgment on its implications regarding climate change.
Real-world Products and Performances: You are provided a task that will be used to defend
your conclusions concerning climate change. The final part of the task is to communicate with
your audience the conclusion you draw by writing a short paper stating your position on climate
change based on the data provided and the defense you have for your position.
Standards: As you work through the task, you will acquire points that sum to a final grade for
the overall task.
Data: The following data, gleaned from
www.wunderground.com, is the average daily temperatures for
McDonough, GA on August 1 of the given year in the form date temperature.
1998 – 78o, 1999 – 86o, 2000 – 78o, 2001 – 78o, 2002 – 82o, 2003
– 78o, 2004 – 84o, 2006 – 84o, 2007 – 83o.
1. Organize the data in the chart at the right
with X being the number of years since 1990
and Y being the average temperature on
August 1 of the year. [ 5 pts. ]
X
8
9
10
11
12
13
14
16
17
Y
78
86
78
78
82
78
84
84
83
21
2. Plot the points from your
chart as a scatterplot on the
graph below. Make sure you
label each axis. [ 5 pts. ]
3. Showing all work, for the data in #1, find the (a) Median-Median Line, (b) Least Squares
Regression Line, and (c) the Quadratic Regression Equation. Graph these regression equations
on the graph in #2. [ 30 pts. total ]
(a) Median-Median Line
(Predicted Temperature) = 69.4286 + 0.8571 (number of years since 1990)
(b) LSRL
(Predicted Temperature) = 76.2794 + 0.4044 (number of years since 1990)
(c) Quadratic Regression Equation
(Predicted Temperature) = 0.0751x2 – 1.4796x + 87.4508,
where x is the number of years since 1990.
22
4. Showing all work, determine the SSE for each of the regression equations in #3. Based on the
SSE and the graphs in #2, discuss which is the best regression equation and defend your answer.
[ 30 points total ]
(a) SSE for Median-Median Line
Sum of Squared Error for the Median-Median Line
( y  yˆ )
Predicted temp ( ŷ )
year (x)
temp (y)
8
78
76.2854
1.7146
9
86
77.1425
8.8575
10
78
77.9996
0.0004
11
78
78.8567
-0.8567
12
82
79.7138
2.2862
13
78
80.5709
-2.5709
14
84
81.4280
2.572
16
84
83.1422
0.8578
17
83
83.9993
-0.9993
The sum of the rows is the Sum of Squared Error
(b) SSE for LSRL
Sum of Squared Error for the LSRL
( y  yˆ )
Predicted temp ( ŷ )
year (x)
temp (y)
8
78
79.5146
-1.5146
9
86
79.9190
6.0810
10
78
80.3234
-2.3234
11
78
80.7278
-2.7278
12
82
81.1322
0.8678
13
78
81.5366
-3.5366
14
84
81.9410
2.0590
16
84
82.7498
1.2502
17
83
83.1542
-0.1542
The sum of the rows is the Sum of Squared Error
(c) SSE for Quadratic Regression Equation
Sum of Squared Error for the Quadratic Regression
( y  yˆ )
Predicted temp ( ŷ )
year (x)
temp (y)
8
78
80.4204
-2.4204
9
86
80.2175
5.7825
10
78
80.1648
-2.1648
11
78
80.2623
-2.2623
12
82
80.5100
1.4900
13
78
80.9079
-2.9079
14
84
81.4560
2.5440
( y  yˆ ) 2
2.9399
78.4553
0.0000
0.7339
5.2267
6.6095
6.6152
0.7358
0.9986
102.3149
( y  yˆ ) 2
2.2940
36.9786
5.3982
7.4409
0.7531
12.5075
4.2395
1.5630
0.0238
71.1986
( y  yˆ ) 2
5.8583
33.4373
4.6864
5.1180
2.2201
8.4559
6.4719
23
16
17
84
83.0028
0.9972
0.9944
83
84.0015
-1.0015
1.0060
The sum of the rows is the Sum of Squared Error
68.2483
5. Based on your findings at this point, are the average temperatures on August 1 in McDonough,
GA increasing, staying the same, or decreasing. Defend your answer.
[ 5 points ]
Answers may vary but should be based on the findings from problems 2 and 3. The solutions
from problems 2 and 3 should be a part of the discussion for full credit.
6. Using the regression equation you found to be best in #4, predict the average temperature for
the following dates and explain why it is or is not a good prediction.
[ 5 points total ]
August 1, 1990 - __________
87.4508o
extrapolation
August 1, 1997 - __________
80.7755o
extrapolation
August 1, 2005 - __________
82.1633o
interpolation
August 1, 2009 - __________
86.4638o
extrapolation
7. At this point in the task, see your teacher to get the actual average temperatures in
McDonough, GA on the above dates. Discuss the relationship between your predicted values and
the real data. If you had added these data points to the original data set, would it have an effect
on the regression equation you found in #4? If so, what would that effect have been? [ 5 points ]
Answers will vary. Some students will discuss the graph some will input the points and discuss
the change in the regression equation. Grade according to the completeness of the discussion.
24
8. Based on the results of your findings, take a position on climate change and defend that
position using the information and understanding you have developed during this task. Write a
short paper describing your position and it defense in such a way that a person who does not
understand data analysis and has not seen your work in problems #1 through #7 will understand
what you are discussing. [ 15 points ]
Answers will vary. Teachers will need to determine if the defense follows the information found
in throughout the task.
Resource data for use in the Evaluation Activity. This data was downloaded from the website
www.wunderground.com.
Year
1990
1991
1992
1993
1994
1995
1996
Average
Temperature on
August 1 in
McDonough,
GA
78o
78o
75o
82o
76o
81o
78o
Year
1997
1998
1999
2000
2001
2002
2003
Average
Temperature on
August 1 in
McDonough,
GA
69o
78o
86o
78o
78o
82o
78o
Year
2004
2005
2006
2007
2008
2009
Average
Temperature on
August 1 in
McDonough,
GA
84o
77o
84o
83o
80o
81o
25
26
Task 2: INTRODUCTION
On-time departure is an essential part of a successful transportation network. This
problem is set up to evaluate delayed departures in the aviation ind ustry. By
looking at Delta's flights departing out of Atlanta's Hartsfield-Jackson Airport
(ATL) on Saturday, September 13, 2008 various delay parameters can be evaluated.
This data can lead to improving the efficiency of the aviation network. This topic
was chosen since it has a wide-spread interest. Most people traveling have
important schedules that they need to or want to meet. Thus, this topic should
peak the interest of many of the high school students who are learning probability
and statistics with it.
Specifically, the problem is going to be set up as if particular individuals, Tim and
Brittany, are traveling out of Atlanta on a Delta flight. The problem is set up to use
two sets of data. The first set of data is from the Bureau of Transportation
Statistics, and simply offers on-time arrival performance for the entire aviation
network (1). The second set of data contains all the Delta flights that flew out of
ATL on Saturday, September 13, their scheduled departure time, their actual
departure time, and whether or not the flight arrived to Atlanta late. This data has
been simplified to include the flight number and relative departure time and how
late it originally arrived in Atlanta. This data set can be found in the
appendix. It was acquired from FlightStats.com (2).
WEB RESOURCES
http:// www.transtats.bts.gov/OT_Delay/OT_DelayCausetasp
http://www.flightstats.com/go/FlightStatus/flightStatusByAirport.do
27
FIVE ESSENTIAL QUESTIONS AND THE STANDARDS THEY MEET
Using the Accelerated Mathematics 1 track, the following questions were
developed and expanded upon:
1. What are the various causes of delayed flights?
a. Probability calculations:
i.What is the probability that a flight departs early?
ii.What is the probability that a flight departs on-time?
iii.What is the probability that a flight is delayed?
iv.What is the probability that a flight is cancelled?
v.What is the probability that a flight is early or on-time?
b. Are Delta flights out of Atlanta delayed more or less often than the
national average?
c. Conditional Probability: Given that their flight is delayed, what is the
probability that the cause of delay is air carrier delay?
d. What are the changes that both flights are delayed?
e. What's the probability that the first flight is delayed due to weather and the
second is delayed due to security?
[MA1D2]
2. Should someone departing out of Altanta's Hartsfield-Jackson airport except to
leave on time?
a. What percentage of Delta flights out of Atlanta's Hartsfield-Jackson airport depart early,
on-time (+ 15 minutes), or late?
b. What is the average delay time? What is the most common delay time?
c. What is the expected value of departure time of a Delta Flight out of ATL compared to its
scheduled departure time?
[MA1D2, MA2D1(b)]
3. Should someone departing out of any given airport except to leave on time?
a. How does Delta's delay time out of ATL compare to that of the entire population?
[MA2D1 (a)]
4. Does the amount of departure delay dependent on the significant of the late
arrival of the aircraft?
a. Plot the departure delay against arrival delay of all the flights with delayed
departures that arrived to the airport late.
b. Determine the regression line for predicting departure time from arrival time
and display it on the graph .
c. What does the low R2 value mean mathematically and in the context of how
departure time relates to the arrival time of each aircraft?
[MA1D5]
5. Although, most travelers only consider delay, a fair amount of times airplanes actually leave
ahead of schedule. Is this ahead of schedule departure significant?
a. Find the probability that a flight is not late.
b. Find the probability that a flight leaves 15 minutes late at the latest.
c. Find the probability that the flight leaves over 45 minutes late.
d. Find the probability that a flight leaves between 10 and 30 minutes late
[MA2D3]
28
Task
Tim and Brittany are planning their honeymoon. They have decided to fly out of
Atlanta's Hartsfield-Jackson Airport (ATL) on Tuesday, September 15, 2009. They have
acquired delay information for flights departing out of Atlanta's Hartsfield-Jackson
Airport (ATL) on the Tuesday a year prior to their scheduled trip, assuming it will mimic
similar behavior for their trip in 2009. Brittany and Tim want your help analyzing the data
to determine whether or not they will have an on-time departure to their honeymoon.
First, fill out the percentages of early flights, on-time flights, delayed flights, and
cancelled flights in the chart below.
Early
On-Time
Delayed
Cancelled
# of Flights
603
223
81
3
% of Flights
.6626
.2451
.089
.0033
Remember that probability is the number that meet a condition divided by the total in the
# correct
sample space. P( x) 
total
1.
A flight is considered early if it leaves before its scheduled departure time.
What is the probability that Tim and Brittany's flight departs out of ATL early?
P(early departure) = 603 / (603+223+81+3) = .6626
2.
An on-time departure is any flight that leaves between its scheduled departure time
and fifteen minutes late. What is the probability that Tim and Brittany's flight departs out of
ATL on-time?
P(on-time departure) = 223 / (603+223+81+3) = .2451
3.
A delayed flight is any flight that departs 15 minutes after its schedule departure time.
What is the probability that Tim and Brittany's flight has a delayed departure?
P(delayed departure) = 81 / (603+223+81+3) = .089
4.
What's the probability that Tim and Brittany's flight is cancelled?
P(delayed departure) = 3 / (603+223+81+3) = .0033
29
5.
Tim and Brittany always arrive early to the airport. What are the chances that their flight
departs either early or on-time?
P(early or on-time departure) = (603+223) / (603+223+81+3) = .9077
Brittany and Tim decided to compare Delta's flight departure statistics to the national on -time
arrival performance. Data acquired from the Bureau of Transportation Statistics can be seen
below:
National On-Time Arrival Performance (%)
Diverted, 0.3
Cancelled, 1.697
Aircraft Arriving Late. 7.17
Security, 0.05
National Aviation System
Delay. 7.78
Weather Delay, 1.01
Air Carrier Delay, 6.3
6.
Are Delta flights out of ATL on this particular day delayed more often or less often
than the national average?
On this particular day, 8.9% of Delta's flights were delayed. This is significantly less than the
National
On-Time Arrival Performance delay percentage of 22.31% as found by slimming air carrier delay,
weather
delay, national aviation system delay, security delay, and delay caused by aircrafts arriving late.
7.
Given that their flight is delayed, what is the probability that the cause of delay is air
carrier delay?
P(air carrier delay | flight delayed)=6.3 / (6.3+1.01+7.78+.05+7.17) = .2824
30
Assume Tim and Brittany have to transfer flights on their way to their honeymoon destination:
8.
What are the chances that both flights are delayed? (Hint: use Atlanta delay statistics for
the first leg and the national performance for the second leg, and assume delays are
independent)
Probability of both flights delayed = .089 * .2231 = .0199
9.
What's the probability that the first flight is delayed due to weather and the second is
delayed due to security?
P(flight 1 weather delay and flight 2 security delay) = .0101*.0005 = .00000505
31
Task 3: Correlation and its meaning
As observed above in the National On-Time Arrival Performance, 7.17% of flights are
delayed due to the aircraft arriving late. Tim and Brittany have speculated that, on average, the
later the flight arrives to ATL, the longer the delay will be. Departure time for flights with
previous legs arriving late into ATL can be seen below. This data was obtained from
www.flightstatus.com.
Flight
Late Arrival
Departure
DL 802
21
12
DL 2016
52
49
DL 6773
28
15
DL 1548
17
31
DL 724
16
12
DL 4516
85
96
DL 1272
20
-2
DL 4422
29
14
DL 4798
37
49
DL 6515
18
0
DL 6493
21
8
DL 879
46
47
DL 1447
33
-2
DL 4416
22
25
DL 363
57
60
DL 411
17
9
DL 1551
25
37
DL 4427
15
-3
DL 1619
19
-1
DL 1592
40
-1
DL 836
27
14
32
DL 727
16
-3
DL 19
36
31
DL 1679
35
5
DL 4740
36
44
DL 4715
15
-2
DL 1541
18
-5
DL 763
31
4
DL 857
37
-3
DL 756
27
-7
DL 881
18
-2
DL 1268
23
11
DL 4594
21
8
DL 1717
27
-8
DL 1635
18
22
DL 5880
26
-3
DL 4294
45
35
DL 1041
64
79
DL 745
29
37
DL 6492
27
5
DL 1577
26
8
DL 6622
101
99
DL 4801
20
3
DL 4025
17
8
DL 705
30
30
DL 1682
15
29
DL 2010
20
-5
DL 1622
27
32
33
DL 6908
15
10
DL 865
21
-3
DL 971
24
40
DL 4230
26
23
DL 6284
46
70
DL 4447
32
31
DL 1770
32
-3
DL 6322
98
111
DL 6448
79
85
DL 46
78
128
DL 973
24
26
DL 6456
138
150
DL 34
55
70
DL 1566
30
-1
DL 4612
15
15
DL 5859
20
-6
DL 947
23
26
DL 4343
66
63
DL 4544
63
75
DL 2024
19
-6
DL 50
19
15
DL 4910
22
17
DL 349
28
29
DL 317
17
5
DL 1681
31
35
DL 6345
38
45
DL 1145
16
-5
34
DL 341
26
-2
DL 3674
16
10
DL 5905
26
45
DL 1033
34
50
DL 4695
17
21
DL 76
17
24
DL 63
18
23
DL 180
42
79
DL 4658
29
30
DL 1555
60
59
DL 114
19
DL 4450
103
125
DL 719
41
33
DL 211
19
DL 4579
41
50
DL 6434
39
47
DL 8517
33
72
DL 4845
86
94
DL 6631
47
70
DL 1766
33
41
DL 4309
46
56
DL 147
42
40
DL 897
22
22
5
5
Brittany and Tim speculate that there is a linear relationship between how delayed a flight
arrives to ATL and how late it departs from ATL. They want you to follow the steps below to
predict a straight line relationship between arrival time and departure time of flights.
1.
By hand or using technology, draw a scatterplot of the data: Graph the data from the table,
arrival time (x) and departure time (y).
The teacher will need to provide graph paper or allow students to use a graphing calculator and
explain how to use it to graph this information.
35
2.
Determine the regression line for predicting departure time from arrival time and display
it and the r -value on the graph.
The teacher may decide to provide students with the regression equation (LSRL) and the
corresponding r-value.
Predicted departure time = 13.571+ 0.3371 arrival time
r = 0.2834
3.
What does the low r-value represent mathematically?
The data is not very linear, and thus the line is probability not a good fit for the data. However, the
line still represents the average y values for each x value input for the sample space.
4.
What does the low r-value mean in the context of how departure time relates to the arrival
time of each aircraft?
There's large variation in departure time for each arrival time of the aircrafts. Although there's a
positive slope for the best fit line, the linear relationship between x and y is not as strong as we may
have expected.
5.
Assume that the specific flight Tim and Brittany are planning to take arrives to ATL 35
minutes late. Using the linear regression equation, on average how late will their flight depart out
of ATL?
Predicted departure time = 13.571+ 0.3371 arrival time = 13.571 + 0.3371 *3 5 = 25.3695 minutes