The University of California, Berkeley (Cal) and StanfordUniversity are athletic archrivals in the Pacific 10 conference.Stanford fans claim Stanford's basketball team is better than Cal'steam; Cal fans challenge this assertion. In 2004, StanfordUniversity's basketball team went nearly undefeated within the Pac10. Stanford's record, and those of Cal and the other eight teamsin the conference, are listed in In all, there were 89 games playedamong the Pac 10 teams in the season. Stanford won 17 of the 18games it played; Cal won 9 of 18. We would like to use these datato test the Stanford fans' claim that Stanford's team is betterthan Cal's. That is, we would like to determine whether thedifference between the two teams' performance reasonably could beattributed to chance, if the Stanford and Cal teams in fact haveequal skill.
To test the hypothesis, we shall make a number of simplifyingassumptions. First of all, we shall ignore the fact that some ofthe games were played between Stanford and Cal: we shall pretendthat all the games were played against other teams in theconference. One strong version of the hypothesis that the two teamshave equal skill is that the outcomes of the games would have beenthe same had the two teams swapped schedules. That is, suppose thatwhen Washington played Stanford on a particular day, Stanford won.Under this strong hypothesis, had Washington played Cal that dayinstead of Stanford, Cal would have won.
A weaker version of the hypothesis is that the outcome ofStanford's games is determined by independent draws from a 0-1 boxthat has a fraction pC of tickets labeled \"1\"(Stanford wins the game if the ticket drawn is labeled \"1\"), thatthe outcome of Berkeley's games is determined similarly, byindependent draws from a 0-1 box with a fractionpS of tickets labeled \"1,\" and thatpS = pC. This model hassome shortcomings. (For instance, when Berkeley and Stanford playeach other, the independence assumption breaks down, and thefraction of tickets labeled \"1\" would need to be 50%. Also, itseems unreasonable to think that the chance of winning does notdepend on the opponent. We could refine the model, but that wouldrequire knowing more details about who played whom, and theoutcome.)
Nonetheless, this model does shed some light on how surprisingthe records would be if the teams were, in some sense, equallyskilled. This box model version allows us to use Fisher's Exacttest for independent samples, considering \"treatment\" to be playingagainst Stanford, and \"control\" to be playing against Cal, andconditioning on the total number of wins by both teams (26).
Q1) On the assumption that the null hypothesis is true, thebootstrap estimate of the standard error of the sample percentageof games won by Stanford is ?
Q2) On the assumption that the null hypothesis is true, thebootstrap estimate of the standard error of the sample percentageof games won by Cal is ?
Q3) The approximate P-value for z test againstthe two-sided alternative that the Stanford and Berkeleyteams have different skills is ?
Note :*The z-score for the difference in samplepercentages is 2.97685*
