Question

Let be a continuous random variable that follows a normal distribution with a mean of and a standard deviation of . Find the value of so that the area under the normal curve to the right of is . Round your answer to two decimal places.

Answer 1

Complete Question

Let x be a continuous random variable that follows a normal distribution with a mean of 550 and a standard deviation of 75.

a

Find the value of x so that the area under the normal curve to the left of x is .0250.

b

Find the value of x so that the area under the normal curve to the right ot x is .9345.

Answer:

a

`x = 403`

b

`x = 436.75`

Explanation:

From the question we are told that

The mean is
`\mu = 550`

The standard deviation is
`\sigma = 75`

Generally the value of x so that the area under the normal curve to the left of x is 0.0250 is mathematically represented as

`P( X < x) = P( (x - \mu )/( \sigma) < (x - 550 )/(75 ) ) = 0.0250`

`(X -\mu)/(\sigma )  =  Z (The  \ standardized \  value\  of  \ X )`

`P( X < x) = P( Z < z ) = 0.0250`

Generally the critical value of 0.0250 to the left is

`z = -1.96`

=>
`(x- 550 )/(75) = -1.96`

=>
`x = [-1.96 * 75 ]+ 550`

=>
`x = 403`

Generally the value of x so that the area under the normal curve to the right of x is 0.9345 is mathematically represented as

`P( X < x) = P( (x - \mu )/( \sigma) < (x - 550 )/(75 ) ) = 0.9345`

`(X -\mu)/(\sigma )  =  Z (The  \ standardized \  value\  of  \ X )`

`P( X < x) = P( Z < z ) = 0.9345`

Generally the critical value of 0.9345 to the right is

`z = -1.51`

=>
`(x- 550 )/(75) = -1.51`

=>
`x = [-1.51 * 75 ]+ 550`

=>
`x = 436.75`