Using R for Introductory Statistics : John Verzani

Using R for introductory statistics 240> cols =c("blue","brown","green","orange","red","yellow","purple")> prob = c(1,1,1,1,2,2,2) # ratio of colors> bagfull.mms=sample(cols,30,replace=TRUE, prob=prob)> table(bagfull.mms)bagfull.mmsblue brown green orange purple red yellow2 3 1 3 6 10 5A formula for the multinomial distribution is similar to that for the binomial distributionexcept that more factors are involved, as there are more categories to choose from. Thedistribution can be specified as follows:As an example, consider the voter survey. Suppose we expected the percentages to be35% Republican, 35% Democrat, and 30% undecided. What is the probability in a surveyof 100 that we see 35, 40, and 25 respectively? It isThis is found with> choose(100,30)*choose(70,40) * .35^35 * .35^40 *.30^25[1] 0.008794(We skip the last coefficient, as for any j.) This small value is the probability ofthe observed value, but it is not a p-value. A p-value also includes the probability ofseeing more extreme values than the observed one. We still need to specify what thatmeans.9.1.2 Pearson’s χ 2 statisticTrying to use the multinomial distribution directly to answer a problem about the p-valueis difficult, as the variables Y i are correlated. If one is large the others are more likely tobe small, so the meaning of “extreme” in calculating a pvalue is not immediately clear.As an alternative, the problem of testing whether a given set of probabilities could haveproduced the data is done as before: by comparing the observed value with the expectedvalue and then normalizing to get something with a known distribution.Each Y i is a random variable telling us how many of the n choices were in category i.If we focus on a single i, then Y i is seen to be Binomial(n, p i ). Again, the Y i are notindependent but correlated, as one large one implies that the others are more likelysmaller. However, we know that the expected number of Y i is np i . Based on this, a goodstatistic might be