© 21St Century Math Projects

Home , AFC North, NFC North, Super Bowl 50

Project Title: Gridiron Glory

Standard Focus: Data Analysis & Probability Time Range: 3-5 Days (can be modified)

Supplies: Graphing Calculators could be helpful (w/ regression functions), Paper Footballs, Long Tables (maybe borrow the cafeteria?!)

Topics of Focus:

- Measures of Central Tendency

- Graphical Displays

- Correlations

- Expected Value

- Data Collection

Benchmarks :

Interpreting Categorical 1. Represent data with plots on the real number line (dot plots, histograms, and S-ID and Quantitative Data box plots).

2. Use statistics appropriate to the shape of the data distribution to compare Interpreting Categorical S-ID center (median, mean) and spread (interquartile range, standard deviation) of two and Quantitative Data or more different data sets.

Interpreting Categorical 3. Interpret differences in shape, center, and spread in the context of the data S-ID and Quantitative Data sets, accounting for possible effects of extreme data points (outliers).

Interpreting Categorical 6. Represent data on two quantitative variables on a scatter plot, and describe S-ID and Quantitative Data how the variables are related.

Interpreting Categorical 8. Compute (using technology) and interpret the correlation coefficient of a S-ID and Quantitative Data linear fit.

Making Inferences and 2. Decide if a specified model is consistent with results from a given data- S-IC Justifying Conclusions generating process, e.g., using simulation.

Using Probability to 2. (+) Calculate the expected value of a random variable; interpret it as the mean S-MD Make Decisions of the probability distribution.

3. (+) Develop a probability distribution for a random variable defined for a Using Probability to S-MD sample space in which theoretical probabilities can be calculated; find the Make Decisions expected value.

Using Probability to S-MD 5b. Evaluate and compare strategies on the basis of expected values. Make Decisions

Procedures:

A) Task 1: (p 4 – 6) Provide students with “Bragging Rights”. This assignment is designed more toward middle school students, but can be used for high school review. Students will analyze Super Bowl data to determine which divisions have historically had the most success. They will compute Winning Percentage and construct a bar graph and a pie graph. Provide students with copies of the Super Bowl Result table.

B) Task 2: (p 7-8) Complete “Super Sunday LX”. This assignment is geared more toward students with algebra skills. Students will use historical television data to create lines of best to predict the future. This assignment would work best on TI graphing calculators. This also shows the limits of linear models because the Commercial Cost linear model has serious limitation. Provide students with copies of the Historical Super Bowl Television Data table.

C) Task 3: (p 9-13) Complete “Super Sunday Predictions”. This is an advanced statistics assignment that will require calculations of Mean, Standard Deviation, Variance and beginning level usage of a T-Test. Students will analyze Points For and Points Against of Super Bowl Winners and Losers to see if there is a significant difference between the means (this is what the T-Test is for). If there is a significant difference, then that could be used to predict the winner of the next Super Bowl. Find out!

D) Task 4: (p 14-15) Complete “Training Camp”. This is a preliminary assignment to the grand finale a Paper Football Championship Tournament! Students will measure their Paper Football skills with a variety of tasks. You will need a lot of flat table space. The cafeteria or library may be needed depending on your classroom. Students will need to work in groups of 3 or 4 to collect the data. 1 will be the player, 1 will be the aid (and goalposts) and 1 will record the data. Rules for Paper Football are provided in this PDF. With the results of the assignment, students will need to calculate their expected value. At the end they will use these four expected values to calculate their “Paper Football Rating”.

Once all of the Paper Football Ratings are calculated, you should rank the students from highest to lowest. For the tournament, I would take either the top 8 or top 16 students and set up a tournament bracket (Round of 8 bracket provided). The students not playing can either be referees, coaches or they can play each other on the side.

E) Task 5: (p 16 – 19) Students will participate in a Paper Football Championship Tournament. You can go as crazy with this as you’d like. If you want students to have team names and coaches, go for it. If you want a prize or a trophy, go for it. They will get into this. Each game should have two students playing the roles of referee and statisticians. These students should receive the “Drive Log” and mark attempts for further statistical analysis and keep score.

The essential question with the Paper Football Championship is does the Expected Value “Paper Football Ratings” predict who the winner of the tournament will be. This is ultimately theoretical versus experimental probability. Nonetheless, it will be fun.

Every NFL team has a rivalry. Many of these rivalries stem from being in the same division and playing each other more often than any other teams. Some of these rivalries go back 50 years (Bears vs. Packers) and others are relatively new (Ravens vs. Steelers), but nonetheless they are all competing for the ultimate prize -- a Super Bowl Championship. Every team may feel their division is difficult, but which one has had the most historical success in the Big Game?

Use the Super Bowl Results table to tabulate the correct data below. Then, use the Super Bowl Win data and total appearances to calculate a team’s Winning Percentage. You will use these results to answer follow-up questions (Hint: There have been 47 Super Bowls. If you total the wins and losses, these should add to 47 each.)

Super Super Super Super Winning Winning Bowl Bowl Bowl Bowl AFC Pct. NFC Pct Wins Losses Wins Losses East East Buffalo Bills Dallas Cowboys Miami Dolphins New York Giants New England Patriots Philadelphia Eagles New York Jets Washington Redskins Total Total North North Baltimore Ravens Chicago Bears Cincinnati Bengals Detroit Lions Cleveland Browns Green Bay Packers Pittsburgh Steelers Minnesota Vikings Total Total South South Houston Texans Atlanta Falcons Indianapolis Colts* Carolina Panthers Jacksonville Jaguars New Orleans Saints Tennessee Titans Tampa Bay Buccaneers Total Total West West Denver Broncos Arizona Cardinals Kansas City Chiefs San Francisco 49ers Oakland Raiders** Seattle Seahawks San Diego Chargers St. Louis Rams*** Total Total Footnotes: *Formerly the Baltimore Colts **Also the Los Angeles Raiders ***Formerly the St. Louis Rams

Use the data create graphs and determine answers to the following questions.

1. Construct a Double Bar Graph to display the Wins & Super Bowl Appearances of the eight divisions.

2. Construct a Pie Graph for Super Bowl Wins by division.

3. Which division has the most Super Bowl Wins? Which has the least?

4. Which division boasts the highest winning percentage in the Super Bowl?

5. In your opinion, which division has achieved the most success in the Super Bowl? Use evidence from the data to support your claim.

Super Bowl Results

Super Bowl Winning Team Score Losing Team I Green Bay Packers 35 – 10 Kansas City Chiefs II Green Bay Packers 33 – 14 Oakland Raiders III New York Jets 16 – 7 Baltimore Colts IV Kansas City Chiefs 23 – 7 Minnesota Vikings V Baltimore Colts 16 – 13 Dallas Cowboys VI Dallas Cowboys 24 – 3 Miami Dolphins VII Miami Dolphins 14 – 7 Washington Redskins VIII Miami Dolphins 24 – 7 Minnesota Vikings IX Pittsburgh Steelers 16 – 6 Minnesota Vikings X Pittsburgh Steelers 21 – 17 Dallas Cowboys XI Oakland Raiders 32 – 14 Minnesota Vikings XII Dallas Cowboys 27 – 10 Denver Broncos XIII Pittsburgh Steelers 35 – 31 Dallas Cowboys XIV Pittsburgh Steelers 31 – 19 Los Angeles Rams XV Oakland Raiders 27 – 10 Philadelphia Eagles XVI San Francisco 49ers 26 – 21 Cincinnati Bengals XVII Washington Redskins 27 – 17 Miami Dolphins XVIII Los Angeles Raiders 38 – 9 Washington Redskins XIX San Francisco 49ers 38 – 16 Miami Dolphins XX Chicago Bears 46 – 10 New England Patriots XXI New York Giants 39 – 20 Denver Broncos XXII Washington Redskins 42 – 10 Denver Broncos XXIII San Francisco 49ers 20 – 16 Cincinnati Bengals XXIV San Francisco 49ers 55 – 10 Denver Broncos XXV New York Giants 20 – 19 Buffalo Bills XXVI Washington Redskins 37 – 24 Buffalo Bills XXVII Dallas Cowboys 52 – 17 Buffalo Bills XXVIII Dallas Cowboys 30 – 13 Buffalo Bills XXIX San Francisco 49ers 49 – 26 San Diego Chargers XXX Dallas Cowboys 27 – 17 Pittsburgh Steelers XXXI Green Bay Packers 35 – 21 New England Patriots XXXII Denver Broncos 31 – 24 Green Bay Packers XXXIII Denver Broncos 34 – 19 Atlanta Falcons XXXIV St. Louis Rams 23 – 16 Tennessee Titans XXXV Baltimore Ravens 34 – 7 New York Giants XXXVI New England Patriots 20 – 17 St. Louis Rams XXXVII Tampa Bay Buccaneers 48 – 21 Oakland Raiders XXXVIII New England Patriots 32 – 29 Carolina Panthers XXXIX New England Patriots 24 – 21 Philadelphia Eagles XL Pittsburgh Steelers 21 – 10 Seattle Seahawks XLI Indianapolis Colts 29 – 17 Chicago Bears XLII New York Giants 17 – 14 New England Patriots XLIII Pittsburgh Steelers 27 -23 Arizona Cardinals XLIV New Orleans Saints 31 – 17 Indianapolis Colts XLV Green Bay Packers 31 – 25 Pittsburgh Steelers XLVI New York Giants 21 – 17 New England Patriots XLVII Baltimore Ravens 34 – 31 San Francisco 49ers

Every year the Super Bowl typically tops the highest rated television events and it is usually by a wide margin. Just like everything else, this has been the product of years of growth and increased interest. Now, people wonder out loud if the Monday following Super Bowl Sunday should be a national holiday. As Viewers rise, the costs of commercial advertisement grows as well. For Super Bowl XLVII, the cost of a 30-second television ad reached an all-time high of 4 million dollars. While we can reasonably predict the viewership and cost of adverting of the Super Bowl L (50), what about Super Sunday LX?

In this assignment, you will use linear regression to calculate mathematical models that you can use to predict the future. To create the models in a TI Calculator, program L1 with the Super Bowl Number, L2 with the Cost of the 30 Second Ad and L3 with the Average Number of Homes Watching.

Average Number of Cost of 30 Second Ad ($ Adjusted for Inflation) Homes Watching (in thousands)

Line of Best Fit

Describe the Slope

in Words

Use the models to determine make the following predictions.

Average Number of Cost of 30 Second Ad ($ Adjusted for Inflation) Homes Watching (in thousands) Super Bowl 50

Super Bowl 55

Super Bowl 60

1. Critique the models. Do you think they will offer accurate predictions of the future?

2. What are the limitations of the models? What could be done to improve them?

Historical Super Bowl Television Data

Cost of 30 Average Number Super Winning Team Losing Team Second Ad of Homes Bowl ($ Adjusted for Watching Inflation) (in thousands) I Green Bay Packers Kansas City Chiefs 282,857 12,410 II Green Bay Packers Oakland Raiders 349,043 20,610 III New York Jets Baltimore Colts 337,102 20,520 IV Kansas City Chiefs Minnesota Vikings 452,196 23,050 V Baltimore Colts Dallas Cowboys 399,891 23,980 VI Dallas Cowboys Miami Dolphins 462,793 27,450 VII Miami Dolphins Washington Redskins 445,825 27,670 VIII Miami Dolphins Minnesota Vikings 469,953 27,540 IX Pittsburgh Steelers Minnesota Vikings 447,369 29,040 X Pittsburgh Steelers Dallas Cowboys 434,855 29,440 XI Oakland Raiders Minnesota Vikings 463,983 31,610 XII Dallas Cowboys Denver Broncos 558,897 34,410 XIII Pittsburgh Steelers Dallas Cowboys 573,191 35,090 XIV Pittsburgh Steelers Los Angeles Rams 606,024 35,330 XV Oakland Raiders Philadelphia Eagles 801,762 34,540 XVI San Francisco 49ers Cincinnati Bengals 755,235 40,020 XVII Washington Redskins Miami Dolphins 903,369 40,480 XVIII Los Angeles Raiders Washington Redskins 796,704 38,880 XIX San Francisco 49ers Miami Dolphins 1,097,518 39,390 XX Chicago Bears New England Patriots 1,128,799 41,490 XXI New York Giants Denver Broncos 1,188,058 40,030 XXII Washington Redskins Denver Broncos 1,226,421 37,120 XXIII San Francisco 49ers Cincinnati Bengals 1,224,466 39,320 XXIV San Francisco 49ers Denver Broncos 1,204,723 35,920 XXV New York Giants Buffalo Bills 1,321,227 39,010 XXVI Washington Redskins Buffalo Bills 1,362,780 37,120 XXVII Dallas Cowboys Buffalo Bills 1,323,170 41,990 XXVIII Dallas Cowboys Buffalo Bills 1,366,026 42,860 XXIX San Francisco 49ers San Diego Chargers 1,697,374 39,400 XXX Dallas Cowboys Pittsburgh Steelers 1,555,505 44,145 XXXI Green Bay Packers New England Patriots 1,681,786 42,000 XXXII Denver Broncos Green Bay Packers 1,793,992 43,630 XXXIII Denver Broncos Atlanta Falcons 2,160,278 39,992 XXXIV St. Louis Rams Tennessee Titans 1,436,892 43,618 XXXV Baltimore Ravens New York Giants 2,667,260 41,270 XXXVI New England Patriots St. Louis Rams 2,375,675 42,664 XXXVII Tampa Bay Buccaneers Oakland Raiders 2,567,260 43,433 XXXVIII New England Patriots Carolina Panthers 2,619,723 44,908 XXXIX New England Patriots Philadelphia Eagles 2,764,227 45,081 XL Pittsburgh Steelers Seattle Seahawks 2,789,422 45,867 XLI Indianapolis Colts Chicago Bears 2,820,660 47,505 XLII New York Giants New England Patriots 2,820,839 48,665 XLIII Pittsburgh Steelers Arizona Cardinals 2,830,911 48,139 XLIV New Orleans Saints Indianapolis Colts 2,578,913 51,728 XLV Green Bay Packers Pittsburgh Steelers 3,000,000 53,282 XLVI New York Giants New England Patriots 3,500,000 53,910 XLVII Baltimore Ravens San Francisco 49ers 4,000,000 52,998

Using data can an accurate prediction of the Super Bowl Winner be made? Can regular season data predict Super Bowl victories? Does defense win championships or does offense make the difference? In this assignment, these questions will be investigated.

Since the modern 16 game schedule was put into place 28 years ago, this assignment will analyze data of the last 28 Super Bowls. The most often used measurement of Offense and Defense are the statistics Points For and Points Against. The total number of points a team scores in a season is their Points For and the total number of points that they allow is Points Against. Is there a difference between the averages of regular season Points For or Points Against with Super Bowl Winners versus the Losers? We will compare the means of the two groups to see if there is a statistically significant difference.

Use this table of data to calculate sample means, variances and standard deviations.

Super Points For Points For Winning Team Losing Team Bowl (x1) (x2) XX Chicago Bears 456 New England Patriots 362 XXI New York Giants 371 Denver Broncos 378 XXII Washington Redskins 379 Denver Broncos 379 XXIII San Francisco 49ers 369 Cincinnati Bengals 448 XXIV San Francisco 49ers 442 Denver Broncos 362 XXV New York Giants 335 Buffalo Bills 428 XXVI Washington Redskins 485 Buffalo Bills 458 XXVII Dallas Cowboys 409 Buffalo Bills 381 XXVIII Dallas Cowboys 376 Buffalo Bills 329 XXIX San Francisco 49ers 505 San Diego Chargers 381 XXX Dallas Cowboys 435 Pittsburgh Steelers 407 XXXI Green Bay Packers 456 New England Patriots 418 XXXII Denver Broncos 472 Green Bay Packers 422 XXXIII Denver Broncos 501 Atlanta Falcons 442 XXXIV St. Louis Rams 526 Tennessee Titans 392 XXXV Baltimore Ravens 333 New York Giants 328 XXXVI New England Patriots 371 St. Louis Rams 503 XXXVII Tampa Bay Buccaneers 346 Oakland Raiders 450 XXXVIII New England Patriots 348 Carolina Panthers 325 XXXIX New England Patriots 437 Philadelphia Eagles 386 XL Pittsburgh Steelers 389 Seattle Seahawks 452 XLI Indianapolis Colts 427 Chicago Bears 427 XLII New York Giants 373 New England Patriots 589 XLIII Pittsburgh Steelers 347 Arizona Cardinals 427 XLIV New Orleans Saints 510 Indianapolis Colts 416 XLV Green Bay Packers 388 Pittsburgh Steelers 375 XLVI New York Giants 394 New England Patriots 513 XLVII Baltimore Ravens 398 San Francisco 49ers 397 (number of subjects in sample 1) n1 (number of subjects in sample 2) n2 (mean of sample 1) 푥̅̅1̅ (mean of sample 2) 푥̅̅2̅ 2 2 (variance in sample 1) 휎1 (variance in sample 2) 휎2 (standard deviation in sample 1) 휎1 (standard deviation in sample 2) 휎2

Use the data from the study to answer the following questions.

1. What are the means of the two groups? Do you believe there is a significant difference?

2. What are the standard deviations of the two groups? What are the range of values within one standard deviation of the mean? Do these seem statistically different?

In order to prove if a difference is actually statistically significant (instead of happening by chance), a t-test must be performed to compare the means. Before using a t-test, writing a null hypothesis is necessary. In this case, the following is the null hypothesis (the case where there is no difference!)

Null Hypothesis: The mean scores of Points For is statistically the same for the Super Bowl Winners and Super Bowl Losers.

The goal of most research is to REJECT THE NULL. This would mean that there IS a statistical difference in the two groups.

In this case to reject the null, the value of the T-Test must fall into the critical areas of the T-Distribution. From referencing a T-Table the critical value for this case is 2.048 for a test with 27 degrees of freedom and a critical value of α =0.05.

T-Test for Independent Samples Calculate the t-value of this data set.

푥̅1 − 푥̅2 푡 = (휎 )2 (휎 )2 √ 1 2 푛 + 푛 1 2 3. After comparing the t-value to the critical values, is there a statistically significant difference between the means of the Regular Season Points Scored by the Super Bowl Winners & the Super Bowl Losers?

4. Is this a good predictor of who will win the Super Bowl?

Super Points Points Winning Team Losing Team Bowl Against (x1) Against (x2) XX Chicago Bears 198 New England Patriots 290 XXI New York Giants 236 Denver Broncos 327 XXII Washington Redskins 285 Denver Broncos 288 XXIII San Francisco 49ers 294 Cincinnati Bengals 329 XXIV San Francisco 49ers 253 Denver Broncos 226 XXV New York Giants 211 Buffalo Bills 263 XXVI Washington Redskins 224 Buffalo Bills 318 XXVII Dallas Cowboys 243 Buffalo Bills 283 XXVIII Dallas Cowboys 229 Buffalo Bills 242 XXIX San Francisco 49ers 296 San Diego Chargers 306 XXX Dallas Cowboys 291 Pittsburgh Steelers 327 XXXI Green Bay Packers 210 New England Patriots 313 XXXII Denver Broncos 287 Green Bay Packers 282 XXXIII Denver Broncos 309 Atlanta Falcons 289 XXXIV St. Louis Rams 242 Tennessee Titans 324 XXXV Baltimore Ravens 165 New York Giants 246 XXXVI New England Patriots 272 St. Louis Rams 273 XXXVII Tampa Bay Buccaneers 196 Oakland Raiders 304 XXXVIII New England Patriots 238 Carolina Panthers 304 XXXIX New England Patriots 260 Philadelphia Eagles 260 XL Pittsburgh Steelers 258 Seattle Seahawks 271 XLI Indianapolis Colts 360 Chicago Bears 255 XLII New York Giants 351 New England Patriots 274 XLIII Pittsburgh Steelers 223 Arizona Cardinals 426 XLIV New Orleans Saints 341 Indianapolis Colts 307 XLV Green Bay Packers 240 Pittsburgh Steelers 232 XLVI New York Giants 400 New England Patriots 342 XLVII Baltimore Ravens 344 San Francisco 49ers 273 (number of subjects in sample 1) n1 (number of subjects in sample 2) n2 (mean of sample 1) 푥̅̅1̅ (mean of sample 2) 푥̅̅2̅ 2 2 (variance in sample 1) 휎1 (variance in sample 2) 휎2 (standard deviation in sample 1) 휎1 (standard deviation in sample 2) 휎2

Use the data from the study to answer the following questions.

5. What are the means of the two groups? Do you believe there is a significant difference?

6. What are the standard deviations of the two groups? What are the range of values within one standard deviation of the mean? Do these seem statistically different?

Write the Null Hypothesis for this Data Set:

T-Test for Independent Samples Calculate the t-value of this data set.

푥̅1 − 푥̅2 푡 = (휎 )2 (휎 )2 √ 1 2 푛 + 푛 1 2 7. After comparing the t-value to the critical values, is there a statistically significant difference between the means of the Regular Season Points Allowed by the Super Bowl Winners & the Super Bowl Losers?

8. Is this a good predictor of who will win the Super Bowl? Is it better than Points For?

9. What factors might affect the outcome of this study? How could it be improved?

10. Points For and Points Against are only two of thousands of statistical measures kept by the NFL, what other measurements do you think could have a stronger predictor of a Super Bowl Winner?

Number Winning Team PF PA Score Losing Team PF PA XX Chicago Bears 456 198 46 – 10 New England Patriots 362 290 XXI New York Giants 371 236 39 – 20 Denver Broncos 378 327 XXII Washington Redskins 379 285 42 – 10 Denver Broncos 379 288 XXIII San Francisco 49ers 369 294 20 – 16 Cincinnati Bengals 448 329 XXIV San Francisco 49ers 442 253 55 – 10 Denver Broncos 362 226 XXV New York Giants 335 211 20 – 19 Buffalo Bills 428 263 XXVI Washington Redskins 485 224 37 – 24 Buffalo Bills 458 318 XXVII Dallas Cowboys 409 243 52 – 17 Buffalo Bills 381 283 XXVIII Dallas Cowboys 376 229 30 – 13 Buffalo Bills 329 242 XXIX San Francisco 49ers 505 296 49 – 26 San Diego Chargers 381 306 XXX Dallas Cowboys 435 291 27 – 17 Pittsburgh Steelers 407 327 XXXI Green Bay Packers 456 210 35 – 21 New England Patriots 418 313 XXXII Denver Broncos 472 287 31 – 24 Green Bay Packers 422 282 XXXIII Denver Broncos 501 309 34 – 19 Atlanta Falcons 442 289 XXXIV St. Louis Rams 526 242 23 – 16 Tennessee Titans 392 324 XXXV Baltimore Ravens 333 165 34 – 7 New York Giants 328 246 XXXVI New England Patriots 371 272 20 – 17 St. Louis Rams 503 273 XXXVII Tampa Bay Buccaneers 346 196 48 – 21 Oakland Raiders 450 304 XXXVIII New England Patriots 348 238 32 – 29 Carolina Panthers 325 304 XXXIX New England Patriots 437 260 24 – 21 Philadelphia Eagles 386 260 XL Pittsburgh Steelers 389 258 21 – 10 Seattle Seahawks 452 271 XLI Indianapolis Colts 427 360 29 – 17 Chicago Bears 427 255 XLII New York Giants 373 351 17 – 14 New England Patriots 589 274 XLIII Pittsburgh Steelers 347 223 27 -23 Arizona Cardinals 427 426 XLIV New Orleans Saints 510 341 31 – 17 Indianapolis Colts 416 307 XLV Green Bay Packers 388 240 31 – 25 Pittsburgh Steelers 375 232 XLVI New York Giants 394 400 21 – 17 New England Patriots 513 342 XLVII Baltimore Ravens 398 344 34 – 31 San Francisco 49ers 397 273

In preparation for the hotly anticipated Paper Football Championship, it is time to show the skills necessary to compete at the highest level. Rather than go through the motions of a complete regular season, qualifying for a spot in the playoffs will come down to a series of skill competition and the expected value of each player. It’s time to bring your A-Game or else you could find yourself holding a clipboard recording stats tomorrow.

In this activity, it will be helpful to do it in groups of three. One will be the player, one will be the aid and the other will be the recorder.

The kicking game is a critical component of Paper Football. In this exercise, from one end of the table to the other you will attempt 20 kicks. To form the goal posts, your partner needs to places both elbows on the table, with their thumbs touching together. Then they extend their index fingers to form the uprights.

In the table, use tally marks to record the number of attempts, the number made and the number missed. Use the results to calculate the percent made, the percent missed and the expected value of a single kick.

Made Missed Attempts Percent Made Percent Missed (3 pts) (0 pts)

Expected Value of a Field Goal attempt: E(x) = (percent made)(3) + (percent missed)(0)

For the rest of the challenges you will push or flick a Paper Football from different distances in an attempt to score a Touchdown. A Touchdown is scored when a Paper Football stops with part of it hanging over the edge of the table (without falling off).

There are two types of acceptable ways to move a Paper Football -- a “flick” or a “bump”. A flick will use one finger to strike the Paper Football forward. A bump will use TWO FINGERTIPS to bump the side of the Paper Football.

Use the diagram as a reference for where to start the footballs. You will make 20 attempts from each distance. Touchdowns will count as 6 points. Touchback (attempts that fall off the table) will count as -1 point.

Touchdowns Touchbacks Percent Percent Attempts (6 pts) (-1 pts) Touchdowns Touchbacks

Expected Value of a Red Zone attempt: E(x) = (percent TD)(6) + (percent touchbacks)(-1)

Touchdowns Touchbacks Percent Percent Attempts (6 pts) (-1 pts) Touchdowns Touchbacks

Expected Value of a From the 50 attempt:

Touchdowns Touchbacks Percent Percent Attempts (6 pts) (-1 pts) Touchdowns Touchbacks

Expected Value of a From the 50 attempt:

(the Paper Football Rating will be used to determine playoff seeding)

Expected Value Expected Value Expected Value Expected Value Paper Football Rating + from the Red + + from a Hail = of Field Goal from the 50 (PFR) Zone Mary

+ + + =

Although Paper Football is played international with a variety of rules, in order to maintain the integrity of the Paper Football Championship, a standard must be set.

1. Players use Rock-Paper-Scissors to see who will “Kickoff” to start the game.

2. Place the paper football on the palm of your hand. With an upward stroke, flip the paper football into the air. Where it lands on the field is where gameplay begins.

1. There are only two types of acceptable ways to move a Paper Football -- a “flick” or a “bump”. A flick will use one finger to strike the Paper Football forward. A bump will use TWO FINGERTIPS to bump the side of the Paper Football. No prolonged contact is allowed or a penalty will be called and the opponent will get a penalty kick.

2. Players take turns, one move each, back and forth until someone scores. If the ball goes off the table it is a “Touchback” and the other player gets to place the ball at the 50 yard line. Players must count their Touchbacks because after a player has three Touchbacks, their opponent will get to attempt a Field Goal. After this field goal attempt, the Touchbacks of both players will be reset to zero.

1. To get a Touchdown, you must flick/bump the ball and have the ball stop with part of it hanging over the edge of the table (without falling off). A Touchdown is 6 points. If a Touchdown is disputed, take a pencil and run in vertically against the table. If it contact the Paper Football, it is a touchdown.

2. After a Touchdown, the player can kick an Extra Point (see Kicking) for 1 point OR attempt a 2-point conversion. On a 2-point conversion, the ball is placed at the 50 yard line (the middle of the table) and the player has one flick or bump to score another Touchdown.

1. Forming the goal post: The opponent places both elbows on the table, with their thumbs touching together. Then they extend their index fingers to form the uprights.

2. Field-goal kicker: Balancing the football vertically with your index finger, flick the football with your other hand to launch it in the air from their side of the table (approximately ¼ of the way). If the ball sails between the two goals posts, the try is good. A Field Goal is 3 points. An Extra Point is 1 point.

The first player to 30 points is declared the winner. No crying.

Use this sheet to record the results of your Paper Football Playoff Games. The overarching question that we will attempt to answer is…

“Will players with a higher expected value perform the best in the tournament?”

a drive. Mark T for Touchdown, X for Touchback o or for Make approximate marks on the game board from where your player attempts a Miss

Thank you for being my Math Friend!

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Use the Super Bowl Results table to tabulate the correct data in the table below. Then, use the Super Bowl Win data and total appearances to calculate a team’s Winning Percentage. You will use these results to answer follow-up questions (Hint: There have been 47 Super Bowls. If you total the wins and losses, these should add to 47 each.)

Super Super Super Super Winning Winning Bowl Bowl Bowl Bowl AFC Pct. NFC Pct Wins Losses Wins Losses East East Buffalo Bills 0 4 .000 Dallas Cowboys 5 3 .625 Miami Dolphins 2 3 .400 New York Giants 4 1 .800 New England Patriots 3 4 .429 Philadelphia Eagles 0 2 .000 New York Jets 1 0 1.000 Washington Redskins 3 2 .600 Total 6 11 .353 Total 12 8 .600 North North Baltimore Ravens 2 0 1.000 Chicago Bears 1 1 .500 Cincinnati Bengals 0 2 .000 Detroit Lions 0 0 -- Cleveland Browns 0 0 -- Green Bay Packers 4 1 .800 Pittsburgh Steelers 6 2 .750 Minnesota Vikings 0 4 .000 Total 8 4 .667 Total 5 6 .455 South South Houston Texans 0 0 -- Atlanta Falcons 0 1 .000 Indianapolis Colts* 2 2 .500 Carolina Panthers 0 1 .000 Jacksonville Jaguars 0 0 -- New Orleans Saints 1 0 1.000 Tennessee Titans 0 1 .000 Tampa Bay Buccaneers 1 0 1.000 Total 2 3 .400 Total 2 2 .500 West West Denver Broncos 2 4 .333 Arizona Cardinals 0 1 .000 Kansas City Chiefs 1 1 .500 San Francisco 49ers 5 1 .833 Oakland Raiders** 3 2 .600 Seattle Seahawks 0 1 .000 San Diego Chargers 0 1 .000 St. Louis Rams*** 1 2 .333 Total 6 8 .429 Total 6 5 .545 Footnotes: *Formerly the Baltimore Colts **Also the Los Angeles Raiders ***Formerly the St. Louis Rams

Use the data create graphs and determine answers to the following questions.

1. Construct a Double Bar Graph to display the Wins & Super Bowl Appearances of the eight divisions.

Super Bowl Wins & Appearances by Division 25

0 AFC East AFC North AFC South AFC West NFC East NFC North NFC South NFC West

Super Bowl Wins Appearances

2. Construct a Pie Graph for Super Bowl Wins by division. Super Bowl Wins NFC West AFC East 13% 13% NFC South 4% NFC North AFC North 11% 17%

NFC East AFC South 25% 4% AFC West 13%

3. Which division has the most Super Bowl Wins? Which has the least?

The NFC East has the most wins with 12. The NFC South and AFC South both have the least with 2.

4. Which division boasts the highest winning percentage in the Super Bowl?

The AFC North has the highest winning percentage.

5. In your opinion, which division has achieved the most success in the Super Bowl? Use evidence from the data to support your claim.

Answers will vary. Strong arguments can be made for the NFC East and the AFC North. Since the AFC North is heavily influenced by the Steelers, I’d prefer the NFC East.

Every year the Super Bowl typically tops the highest rated television events and it is usually by a wide margin. Just like everything else, this has been the product of years of growth and increased interest. Now people wonder out loud if the Monday following Super Bowl Sunday should be a national holiday. As Viewers rise, the costs of commercial advertisement grows as well. For Super Bowl XLVII, the cost of a 30-second television ad reached an all-time high of 4 million dollars. While we can reasonably predict the viewership and cost of adverting of the Super Bowl L (50), what about Super Sunday LX?

Average Number of Cost of 30 Second Ad Homes Watching ($ Adjusted for Inflation) (in thousands)

Line of Best Fit Y =69436x - 184645 Y = 623.39x + 23306

Describe the Slope Every year the cost of a 30-second add Every year the average number of homes in Words increases by $69,436. watching increases by 623,000 people

Use the models to determine make the following predictions.

Average Number of Cost of 30 Second Ad ($ Adjusted for Inflation) Homes Watching (in thousands) Super Bowl 50 3,287,155 54,475.5

Super Bowl 55 3,634,355 57,592.45

Super Bowl 60 3,981,515 60,709.4

1. Critique the models. Do you think they will offer accurate predictions of the future?

The Average Number of Homes is likely accurate, the Cost of the Ads is really off because of the rapid growth in recent years. This can really help students understand the value of other types of models.

2. What are the limitations of the models? What could be done to improve them?

The rapid growth distorts the Advertisement model either using less and more recent years could improve the model. That or using an exponential model would make it more accurate. © 21st Century Math Projects

Based on statistical data can a prediction of the Super Bowl Winner be made? Can regular season data predict Super Bowl victories? Does defense win championships or does offense make the difference? In this assignment, these questions will be investigated.

Since the modern 16 game schedule was put into place 28 years ago the assignment will analyze data of the last 28 Super Bowls. The most often used measurement of Offense and Defense are the statistics Points For and Points Against. The total number of points a team scores in a season is their Points For and the total number of points that they allow is Points Against. Is there a difference between the averages of regular season Points For or Points Against with Super Bowl Winners versus the Losers? We will compare the means of the two groups to see if there is a statistically significant difference.

Use this table of data to calculate sample means, variances and standard deviations.

Super Points For Points For Winning Team Losing Team Bowl (x1) (x2) XX Chicago Bears 456 New England Patriots 362 XXI New York Giants 371 Denver Broncos 378 XXII Washington Redskins 379 Denver Broncos 379 XXIII San Francisco 49ers 369 Cincinnati Bengals 448 XXIV San Francisco 49ers 442 Denver Broncos 362 XXV New York Giants 335 Buffalo Bills 428 XXVI Washington Redskins 485 Buffalo Bills 458 XXVII Dallas Cowboys 409 Buffalo Bills 381 XXVIII Dallas Cowboys 376 Buffalo Bills 329 XXIX San Francisco 49ers 505 San Diego Chargers 381 XXX Dallas Cowboys 435 Pittsburgh Steelers 407 XXXI Green Bay Packers 456 New England Patriots 418 XXXII Denver Broncos 472 Green Bay Packers 422 XXXIII Denver Broncos 501 Atlanta Falcons 442 XXXIV St. Louis Rams 526 Tennessee Titans 392 XXXV Baltimore Ravens 333 New York Giants 328 XXXVI New England Patriots 371 St. Louis Rams 503 XXXVII Tampa Bay Buccaneers 346 Oakland Raiders 450 XXXVIII New England Patriots 348 Carolina Panthers 325 XXXIX New England Patriots 437 Philadelphia Eagles 386 XL Pittsburgh Steelers 389 Seattle Seahawks 452 XLI Indianapolis Colts 427 Chicago Bears 427 XLII New York Giants 373 New England Patriots 589 XLIII Pittsburgh Steelers 347 Arizona Cardinals 427 XLIV New Orleans Saints 510 Indianapolis Colts 416 XLV Green Bay Packers 388 Pittsburgh Steelers 375 XLVI New York Giants 394 New England Patriots 513 XLVII Baltimore Ravens 398 San Francisco 49ers 397 (number of subjects in sample 1) n1 28 (number of subjects in sample 2) n2 28 (mean of sample 1) 푥̅̅1̅ 413.5 (mean of sample 2) 푥̅̅2̅ 413.39 2 3206.61 2 3262.38 (variance in sample 1) 휎1 (variance in sample 2) 휎2 (standard deviation in sample 1) 휎1 56.63 (standard deviation in sample 2) 휎2 57.11

Use the data from the study to answer the following questions.

1. What are the means of the two groups? Do you believe there is a significant difference?

These stats happen to be shockingly the same.

2. What are the standard deviations of the two groups? What are the range of values within one standard deviation of the mean? Do these seem statistically different?

That means that 68% of all Super Bowl Winning Teams would score 356.87 and 470.13. 68% of the Super Bowl Losing Teams would score between 356.28 and 470.5. These are very similar.

In order to prove if a difference is actually statistically significant (instead of happening by chance). A t-test must be performed to compare the means. Before using a t-test, writing a null hypothesis is necessary. In this case, the following is the null hypothesis (the case where there is no difference!)

Null Hypothesis: The mean scores of Points For is statistically the same for the Super Bowl Winners and Super Bowl Losers.

The goal of most research is to REJECT THE NULL. This would mean that there IS a statistical difference in the two groups.

T-Test for Independent Samples Calculate the t-value of this data set.

0.9945 푥̅1 − 푥̅2 푡 = (휎 )2 (휎 )2 √ 1 + 2 푛 푛 1 2 3. After comparing the t-value to the critical values, is there a statistically significant difference between the means of the Regular Season Points Scored by the Super Bowl Winners & the Super Bowl Losers?

This does not lie in the critical area and it is not close.

4. Is this a good predictor of who will win the Super Bowl? No, it is not.

Super Points Points Winning Team Losing Team Bowl Against (x1) Against (x2) XX Chicago Bears 198 New England Patriots 290 XXI New York Giants 236 Denver Broncos 327 XXII Washington Redskins 285 Denver Broncos 288 XXIII San Francisco 49ers 294 Cincinnati Bengals 329 XXIV San Francisco 49ers 253 Denver Broncos 226 XXV New York Giants 211 Buffalo Bills 263 XXVI Washington Redskins 224 Buffalo Bills 318 XXVII Dallas Cowboys 243 Buffalo Bills 283 XXVIII Dallas Cowboys 229 Buffalo Bills 242 XXIX San Francisco 49ers 296 San Diego Chargers 306 XXX Dallas Cowboys 291 Pittsburgh Steelers 327 XXXI Green Bay Packers 210 New England Patriots 313 XXXII Denver Broncos 287 Green Bay Packers 282 XXXIII Denver Broncos 309 Atlanta Falcons 289 XXXIV St. Louis Rams 242 Tennessee Titans 324 XXXV Baltimore Ravens 165 New York Giants 246 XXXVI New England Patriots 272 St. Louis Rams 273 XXXVII Tampa Bay Buccaneers 196 Oakland Raiders 304 XXXVIII New England Patriots 238 Carolina Panthers 304 XXXIX New England Patriots 260 Philadelphia Eagles 260 XL Pittsburgh Steelers 258 Seattle Seahawks 271 XLI Indianapolis Colts 360 Chicago Bears 255 XLII New York Giants 351 New England Patriots 274 XLIII Pittsburgh Steelers 223 Arizona Cardinals 426 XLIV New Orleans Saints 341 Indianapolis Colts 307 XLV Green Bay Packers 240 Pittsburgh Steelers 232 XLVI New York Giants 400 New England Patriots 342 XLVII Baltimore Ravens 344 San Francisco 49ers 273 (number of subjects in sample 1) n1 28 (number of subjects in sample 2) n2 28 (mean of sample 1) 푥̅̅1̅ 266.286 (mean of sample 2) 푥̅̅2̅ 291.93 2 3047.20 2 1593.85 (variance in sample 1) 휎1 (variance in sample 2) 휎2 (standard deviation in sample 1) 휎1 55.20 (standard deviation in sample 2) 휎2 39.92

Use the data from the study to answer the following questions.

5. What are the means of the two groups? Do you believe there is a significant difference?

The Super Bowl winners allowed 266.286 points a year while the losers allowed 291.93. This seems to be a significant difference.

6. What are the standard deviations of the two groups? What are the range of values within one standard deviation of the mean? Do these seem statistically different?

68% of the Super Bowl Winners would allow between 211.08 and 321.49 points whereas 68% of the losers would allow 252.01 and 331.85. Other observations will vary.

Write the Null Hypothesis for this Data Set:

There is no difference between the amount of regular season points that are allowed by the Super Bowl Winner and the Super Bowl Loser.

T-Test for Independent Samples Calculate the t-value of this data set. 0.056 푥̅1 − 푥̅2 푡 = (휎 )2 (휎 )2 √ 1 2 푛 + 푛 1 2 7. After comparing the t-value to the critical values, is there a statistically significant difference between the means of the Regular Season Points Allowed by the Super Bowl Winners & the Super Bowl Losers?

No, it is not significantly different. The value very close to 0 indicates that it is likely due to chance.

8. Is this a good predictor of who will win the Super Bowl? Is it better than Points For?

No. It actually proves to be a worse predictor than Points For because more credit is given to randomness.

9. What factors might affect the outcome of this study? How could it be improved?

The change in offensive football in the last 10 years will greatly affect the means. Perhaps looking at shorter amounts of years, but this would also affect the statistical significance.

10. Points For and Points Against are only two of thousands of statistical measures kept by the NFL, what other measurements do you think could have a stronger predictor of a Super Bowl Winner?

Answers will vary. Professional Statisticians do not have any stat that has proven to be a reliable predictor.