Question: topic iv confidence interval amp hypothesis testing during the european...
Topic IV: Confidence Interval & Hypothesis Testing
During the European football championships in 2008, and the football World Cup in 2010, an octopus called Paul living at an aquarium in Oberhausen, Germany, was used to predict the outcome of football matches, mostly involving the German national football team To obtain Pauls' predictions, his keepers at the aquarium would present him with two boxes of food before each match. Each box was covered in the flag of the two nations that were participating, and the box that Paul chose to feed from first determined which nation he predicted would win. Paul was asked to predict the outcome of 14 matches, 12 of which involved Germany. He correctly predicted the outcomes of 12 matches, only incorrectly guessing that Germany would beat Croatia in the Euro 2008 group stage, and that Germany would beat Spain in the Euro 2008 final. Some people claimed he was an 'animal oracle':
Answer the questions below regarding the claim that Paul the Octopus is an oracle. Provide working, reasoning or explanations that you have used, as appropriate.
1. Calculate an estimate of Paul's success rate at predicting football matches. Calculate a 95% confidence interval for this estimate, and summarise/describe your results appropriately. Show working as required. (3 Marks)
2. Using hypothesis testing, test the hypothesis that Paul just 'got lucky' and was randomly guessing the outcomes of the football matches. Write down explicitly the hypothesis that you are testing, and then calculate a p -value using the approximate approach for testing a Bernoulli population. What does this p -value suggest? (4 Marks)
3. Paul has a sister Emily. The aquarium also wants to know whether Emily has the same clairvoyant ability similar to that of Paul's. Thus, they also conduct another similar experiment on Emily. Out of 14 matches, Emily correctly guessed 8 of them. Test the hypothesis that Emily has the same abilities as Paul's (i.e. difference between the two Bernoulli population), calculate the appropriate p-value for the null hypothesis, and give a conclusion on the hypothesis as well. (5 Marks)