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`