Statistics

a) You and your team have built a chess computer. To invEstigate how well it performs, you organise a tournament with l randomly selected grandmasters {a title given to expert chass players by the world chess organization FIDE], where your computer will play each of the grandmasters once, and it is racorded whether the game is a win for the computEr, a win for the grandmaster, or a draw. For each win against a grandmaster, the computer scores 1 point, and for each draw the computer scores half a point. At the end of the tournament, the chess computer has scored ED points. Problem Sheet 3 Mathematics for AI i} Identify the population, sample and sample size. iiJ Identify a reasonable parameter and parameter space. iii] Using the results of the tournament, give an estimate of the parameter. Explain whether this is an unbiased estimate. Subsequently, you and your team go away to improve the chess computer. After a while, you organise another tournament, where the computer will again play It'll] randomly selected grandmasters. Two members of your team make claims about the new version of the computer. Alice claims that the new version will perform better against grandmasters than the previous version. Bob claims that the new version will score an average of at least [1.3" points in games against a grandmaster. ivJ Give the null and alternative hypotheses for both Alice and Bob's claims. v) Using the concept of the critical value, explain how Alice's claim can be tested. vi] Using the concept of the p-value, explain how Bob's claim can be tested. Based on the results of the second tournament, you decide there is sufficient evidence for both Alice and Bob's claims to be accepted. However, you later discover that both your tournaments occurred at the same time as the Candidates Tournament {one of the most important chess tournaments}, and so, each time, your random selection of grandmasters excluded the best l'lt. vii} 1With reference to Type I andfor Type II errors, explain how this discovery might change your attitude to each of Alice and Bob's claims