Using R for Introductory Statistics : John Verzani

Describing populations 149variance 1. A key property of the normal distribution is that for any normal randomvariable the z-score, (X−µ)/σ, is a standard normal. This says that areas are determined byz-scores. In Figure 5.5 the two shaded areas are the same, as each represents the area tothe left of 1.5 standard deviations above the mean.We can verify this with the “p” function:> pnorm(1.5, mean=0,sd=1)[1] 0.9332> pnorm(4.75, mean=4,sd=1/2) # same z-score asabove[1] 0.9332Figure 5.5 Two normal densities: thestandard normal, f(x|0,1), andf(x|4,1/2). For each, the shaded areacorresponds to a z-score of 3/2 orless.It is useful to know some areas for the normal distribution based on z-scores. Forexample, the IQR is the range of the middle 50%. We can find this for the standardnormal by breaking the total area into quarters.> qnorm(c(.25,.5,.75))[1] −0.6745 0.0000 0.6745We use qnorm() to specify the area we want. The mean and standard deviation are takenfrom the defaults of 0 and 1. For any normal random variable, this says the IQR is about1.35σ.How much area is no more than one standard deviation from the mean? We usepnorm() to find this:> pnorm(1)−pnorm(−1)[1] 0.6827