# The exact upper tail binomial probability for Example 1.1.2
 # Note that pbinom is the CDF for the binomial distribution

> 1 -pbinom(25, 40, .5)
[1] 0.04034523

 # Normal approximation with continuity correction

>  1 -pnorm(1.74,0,1)
[1] 0.04092951

 # Normal approximation without continuity correction

>  1 -pnorm(1.90,0,1)
[1] 0.02871656


 # ------------------------------------------------------------------

 tab111 <- c(72.1, 72.8, 72.9, 73.3, 73.3, 73.3, 73.9, 74.0, 74.2, 74.2,
74.3, 74.6, 74.7, 75.0, 75.1, 75.1, 75.2, 75.3, 75.3, 75.3,
75.4, 76.1, 76.5, 76.5, 76.6, 76.9, 77.1, 77.2, 77.4, 77.4,
77.7, 78.0, 78.3, 78.6, 78.8, 78.9, 79.7, 80.3, 80.5, 81.0)

 # wilcox.test computes a nonparametric confidence interval for the median,
 # but it not exactly the same as what we get, it is slightly shorter in length

wilcox.test(tab111,conf.int=TRUE,conf.level=0.95,exact=TRUE)


 # -------------------------------------------------------------------

 # one way to plot the empirical cdf                                   

 tab121 <- c(7,  11,  15,  16,  20,  22,  24,  25,  29,  33,  
 34,  37,  41,  42,  49,  57,  66,  71,  84,  90) 
   

 plot(ecdf(tab121),main="ECDF from Table 1.2.1")