CONTEXT:Paul the Octopus correctly predicted the winner of 8 out of 8 games played in the 2010 World Cup. His success brought him lots of fame, including being covered by various news outlets writing about Paul’s “psychic” abilities. In Germany, soccer fans loved watching as Paul the Octopus was presented with two boxes in his tank, each containing a mussel and decorated with the country’s flag of a soccer team. Whichever box Paul chose to eat his mussel from was considered his “prediction” for the winner of the soccer game. Key question to investigate: Could Paul’s success rate have happened by chance? In other words, is Paul’s success rate of 8 out of 8 unusual enough that a model based on random chance is reasonable, or should this model be rejected in favor of a different model to explain his prediction of successes? Our null model (or hypothesis) is that the probability of choosing either box is 0.5. The alternative model (or hypothesis) is that the probability is something other than 0.5 for choosing the left or right box. Designing a Method to Simulate Paul’s Predictions. Paul did his initial predicting in 2010. To test a hypothesis that Paul’s prediction is not likely to have happened by chance, we would ideally want Paul to repeat his experiment and see how many predictions he got correct. Since this is not possible, we can instead build a simulation of Paul’s prediction process. In order to simulate Paul’s predictions, we have to conceive of Paul's individual predictions as repeatable actions. In each prediction, Paul does the following: Paul swims towards two boxes, each with a country flag in it representing the two teams. He chooses one box to eat from, indicating his prediction for who will win the game. The teams play the game and one team wins. The winner either matches or does not match Paul's prediction (e.g., Paul’s prediction was either correct or not). Since Paul made 8 predictions, the string of 8 predictions that he made also needs to be conceived as repeatable. That is, we need to do a set of 8 predictions over and over again. Design two different methods to simulate the context of Paul's 8 predictions using physical (hands-on) devices. You can use whatever devices you want. Describe in enough detail that a student of yours would clearly understand the mapping of the real-world scenario and the simulated process. Be sure to also describe the event of interest that you will record. Recall that our initial problem had us questioning whether Paul’s success could have happened simply by chance. Method 1: Method 2: Consider the following ways some students have thought about simulate the scenario of Paul predicting the winner of a set of 8 games. Some responses are viable ways of modeling the scenario and simulating the prediction process, while others are incomplete or contain flaws. for the followin sample response (A), tell whether the student gives a viable and complete way to model the situation, and explain why or why not. Student A) “The repeatable process is selecting the winning team of the matches. Therefore, there are two possible outcomes, which are Paul correctly predicts the winning team or Paul incorrectly predicts the winning team. To investigate if Paul is psychic or not, we should first assume that he has an equally likely chance of being right or wrong. I would have bag of an even number of marbles. There would be two different colored marbles, let’s say blue and red. Choosing a blue marble from the bag would represent Paul correctly choosing the winner of the match. Choosing a red marble from the bag would represent Paul incorrectly choosing the winner of the match. It would be important to shake the bag up to ensure that the marbles are mixed as well as make sure that the person selecting the marble cannot see into the bag. With the marble simulation, I would choose 8 marbles, one at a time, with replacement. The 8 selected marbles would represent one trial of Paul's selection since we want to see how likely it is for us to choose 8 blue marbles in a row.”

Question

CONTEXT:Paul the Octopus correctly predicted the winner of 8 out of 8 games played in the 2010 World Cup. His success brought him lots of fame, including being covered by various news outlets writing about Paul’s “psychic” abilities.

In Germany, soccer fans loved watching as Paul the Octopus was presented with two boxes in his tank, each containing a mussel and decorated with the country’s flag of a soccer team. Whichever box Paul chose to eat his mussel from was considered his “prediction” for the winner of the soccer game.

Key question to investigate: Could Paul’s success rate have happened by chance? In other words, is Paul’s success rate of 8 out of 8 unusual enough that a model based on random chance is reasonable, or should this model be rejected in favor of a different model to explain his prediction of successes?

Our null model (or hypothesis) is that the probability of choosing either box is 0.5.

The alternative model (or hypothesis) is that the probability is something other than 0.5 for choosing the left or right box.

Designing a Method to Simulate Paul’s Predictions.

Paul did his initial predicting in 2010. To test a hypothesis that Paul’s prediction is not likely to have happened by chance, we would ideally want Paul to repeat his experiment and see how many predictions he got correct. Since this is not possible, we can instead build a simulation of Paul’s prediction process. In order to simulate Paul’s predictions, we have to conceive of Paul's individual predictions as repeatable actions. In each prediction, Paul does the following:

Paul swims towards two boxes, each with a country flag in it representing the two teams. He chooses one box to eat from, indicating his prediction for who will win the game. The teams play the game and one team wins. The winner either matches or does not match Paul's prediction (e.g., Paul’s prediction was either correct or not).

Since Paul made 8 predictions, the string of 8 predictions that he made also needs to be conceived as repeatable. That is, we need to do a set of 8 predictions over and over again.

Design two different methods to simulate the context of Paul's 8 predictions using physical (hands-on) devices. You can use whatever devices you want. Describe in enough detail that a student of yours would clearly understand the mapping of the real-world scenario and the simulated process. Be sure to also describe the event of interest that you will record. Recall that our initial problem had us questioning whether Paul’s success could have happened simply by chance.

Method 1:

Method 2:

Consider the following ways some students have thought about simulate the scenario of Paul predicting the winner of a set of 8 games. Some responses are viable ways of modeling the scenario and simulating the prediction process, while others are incomplete or contain flaws.

for the followin sample response (A), tell whether the student gives a viable and complete way to model the situation, and explain why or why not.

Student A) “The repeatable process is selecting the winning team of the matches. Therefore, there are two possible outcomes, which are Paul correctly predicts the winning team or Paul incorrectly predicts the winning team. To investigate if Paul is psychic or not, we should first assume that he has an equally likely chance of being right or wrong. I would have bag of an even number of marbles. There would be two different colored marbles, let’s say blue and red. Choosing a blue marble from the bag would represent Paul correctly choosing the winner of the match. Choosing a red marble from the bag would represent Paul incorrectly choosing the winner of the match. It would be important to shake the bag up to ensure that the marbles are mixed as well as make sure that the person selecting the marble cannot see into the bag. With the marble simulation, I would choose 8 marbles, one at a time, with replacement. The 8 selected marbles would represent one trial of Paul's selection since we want to see how likely it is for us to choose 8 blue marbles in a row.”