Solved: Use R program : Given that X has a normal distribution with mean 18 and standard deviation 2.5

Given that X has a normal distribution with mean 18 and standard deviation 2.5, find

a. $P(X < 15)$;
b. the value k such that $P(X < k) = 0.2236$;
c. the value k such that $P(X > k) = 0.1814$;
d. $P(17 < X < 21)$.

Solution

Given that X has a normal distribution with mean 18 and standard deviation 2.5. That is $\mu=18$ and $\sigma=2.5$.

Define mean and sd of given normal distribution

mu<-18
s<-2.5

a. To calculate $P(X < 15)$ use the pnorm() function as follows:

p1<-pnorm(15,mu,s)
## Print the value of p1
p1

[1] 0.1150697

b. To find the value k such that $P(X < k) = 0.2236$, use qnorm() function as follows:

x1<-qnorm(0.2236,mu,s)
## print the value of x1
x1

[1] 16.09977

c. To find the value k such that $P(X > k) = 0.1814$, use qnorm() function with lower.tail=FLASE as follows:

x2<-qnorm(0.1814,mu,s,lower.tail = FALSE)
## print the value of x1
x2

[1] 20.27511

d. To find $P(17 < X < 21)=P(X < 21)-P(X < 17)$, use pnorm() function as follows:

p2<-pnorm(21,mu,s)-pnorm(17,mu,s)
## print the required probability
p2

[1] 0.5403521