Using R for Introductory Statistics : John Verzani

Significance tests 221Sign test for the medianAssume X 1 , X 2 , …, X n are from a continuous distribution with positive density. Asignificance test of the hypothesesH 0 :median=m, H A :medianm, or median≠m,can be performed with test statisticT=the number of X i with X i >m.If the data has values equal to m, then delete those values from the data set. Under H 0 ,T has a Binomial(n, 1/2) distribution. Large values of T support the alternative that themedian is greater than M; small values of T support the alternative that the median issmaller than M. For two-sided alternatives, large or small values of T support H A . The p-value is calculated byIn R, the test statistic can be computed using sum(). The p-values are found usingpbinom(k). However, as P(T≥k)=1−P(T≤k−1), the p-value is is found with1−pbinom(k−1, n, 1/2).■ Example 8.6: Length of cell-phone calls Suppose a cell-phone bill contains this datafor the number of minutes per call:2 1 3 3 3 3 1 3 16 2 2 12 20 31Is this consistent with an assumption that the median call length is 5 minutes, or does itindicate that the median length is less than 5?The hypothesis test isH 0 :the median =5, H A :the median < 5.The data is clearly nonnormal, so a t-test is inappropriate. A sign test can be used. Here,small values of T support the alternative.> calls = c(2, 1, 3, 3, 3, 3, 1, 3, 16, 2, 2, 12, 20,3, 1)> obs = sum(calls > 5) # find observed value ofT> n = length(calls)> n − obs[1] 12> 1 − pbinom(11,n,1/2) # we want P(T >= 12)[1] 0.01758We get a p-value of 0.0176, which leads us to believe that the median is less than 5.