#normal distribution mean=80, sd=10
x<-seq(40,120,length(300))
y<-dnorm(x,mean = 80,sd=10)
plot(x,y, type="l",col="red")
lines(x, dnorm(x,mean=80,sd=20),col="blue")

#probability  between 65~75
x2<-seq(65,75,length=200)
y2<-dnorm(x2, mean = 80,sd = 10)
polygon(c(65,x2,75),c(0,y2,0),col="gray")

pnorm(75,mean = 80, sd=10)-pnorm(65,mean = 80,sd=10)

#probability of over 92
1-pnorm(92, mean=80, sd=10)
pnorm(92,mean=80, sd=10, lower.tail = FALSE)


#probability of less than 68
pnorm(68,mean=80,sd=10)

#cutoff that separates the bottom 30%
qnorm(0.3,mean=80,sd=10)

#80th percentile
qnorm(0.8,mean=80,sd=10)


#cutoffs that contain the  middle 60%
qnorm(0.8,mean=80,sd=10)
qnorm(0.2,mean=80,sd=10)