Using R for Introductory Statistics : John Verzani

Using R for introductory statistics 222For illustration, the p-value for the two-sided alternative can be computed as follows:> k = max(obs, n − obs)> k[1] 12> 2*(1 − pbinom(k−1 , n, 1/2))[1] 0.035168.4.2 The signed-rank testThe signed-rank test is an improvement to the sign test when the population is symmetric,but not close enough to normal to use a t-test. Assume H 0 : median =m. If X i are from acontinuous distribution with density f() that is symmetric about m, then not only is X iequally likely to be on either side of m, but the distance from m is independent of theside. Thus, if we rank all the data by its distance to m, the sum corresponding to thevalues larger than m may be viewed as a random sample of a certain size from thenumbers 1 through n. The distribution of this sum can be characterized, so the sum of theranks can be an effective test statistic.The Wilcoxon signed-rank test for the medianIf the data, X 1 , X 2 , …, X n , is an i.i.d. sample from a continuous, symmetric distribution,then a significance test of the hypothesesH 0 : the median=m, H A : medianm, or median≠mcan be performed with test statisticUnder H 0 , the distribution of T can be calculated. Large values of T support thealternative hypothsis H A : median>m.In R, the function wilcox.test() performs the test aswilcox.text(x, mu=..., alt="two.sided”)The data is contained in x, the null hypothesis is specified by the argument mu=, andthe alternative is specified with the argument alt=. This argument takes a value of “less”,“greater”, or “two. sided” (the default value). If desired, the distribution of T is given bythe function psignrank().A typical application of the signed-rank test is to use it after transforming the data tomake it look like the parent distribution is symmetric.■ Example 8.7: Number of recruits In salmon populations, there is a relationshipbetween the number of spawners and the subsequent number of “recruits” that theyproduce. A common model involves two parameters, which describe how many recruitsthere are when there are few spawners and how many there are when there are manyspawners. The data set salmon. rate (UsingR) contains simulated data on one of the