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](https://tex.z-dn.net/?f=x%20%20%3D%20403)
b
![x = 436.75](https://tex.z-dn.net/?f=x%20%20%3D%20436.75)
Step-by-step explanation:
From the question we are told that
The mean is ![\mu = 550](https://tex.z-dn.net/?f=%5Cmu%20%3D%20550)
The standard deviation is ![\sigma = 75](https://tex.z-dn.net/?f=%5Csigma%20%3D%2075)
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( \frac{x - \mu }{ \sigma} < \frac{x - 550 }{75 } ) = 0.0250](https://tex.z-dn.net/?f=P%28%20X%20%3C%20%20x%29%20%3D%20P%28%20%5Cfrac%7Bx%20-%20%5Cmu%20%20%7D%7B%20%5Csigma%7D%20%20%3C%20%20%5Cfrac%7Bx%20-%20550%20%7D%7B75%20%7D%20%29%20%3D%200.0250)
![\frac{X -\mu}{\sigma } = Z (The \ standardized \ value\ of \ X )](https://tex.z-dn.net/?f=%5Cfrac%7BX%20-%5Cmu%7D%7B%5Csigma%20%7D%20%20%3D%20%20Z%20%28The%20%20%5C%20standardized%20%5C%20%20value%5C%20%20of%20%20%5C%20X%20%29)
![P( X < x) = P( Z < z ) = 0.0250](https://tex.z-dn.net/?f=P%28%20X%20%3C%20%20x%29%20%3D%20P%28%20Z%20%3C%20z%20%29%20%3D%200.0250)
Generally the critical value of 0.0250 to the left is
![z = -1.96](https://tex.z-dn.net/?f=z%20%3D%20-1.96)
=> ![\frac{x- 550 }{75} = -1.96](https://tex.z-dn.net/?f=%5Cfrac%7Bx-%20550%20%7D%7B75%7D%20%3D%20-1.96)
=>
=> ![x = 403](https://tex.z-dn.net/?f=x%20%20%3D%20403)
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( \frac{x - \mu }{ \sigma} < \frac{x - 550 }{75 } ) = 0.9345](https://tex.z-dn.net/?f=P%28%20X%20%3C%20%20x%29%20%3D%20P%28%20%5Cfrac%7Bx%20-%20%5Cmu%20%20%7D%7B%20%5Csigma%7D%20%20%3C%20%20%5Cfrac%7Bx%20-%20550%20%7D%7B75%20%7D%20%29%20%3D%200.9345)
![\frac{X -\mu}{\sigma } = Z (The \ standardized \ value\ of \ X )](https://tex.z-dn.net/?f=%5Cfrac%7BX%20-%5Cmu%7D%7B%5Csigma%20%7D%20%20%3D%20%20Z%20%28The%20%20%5C%20standardized%20%5C%20%20value%5C%20%20of%20%20%5C%20X%20%29)
![P( X < x) = P( Z < z ) = 0.9345](https://tex.z-dn.net/?f=P%28%20X%20%3C%20%20x%29%20%3D%20P%28%20Z%20%3C%20z%20%29%20%3D%200.9345)
Generally the critical value of 0.9345 to the right is
![z = -1.51](https://tex.z-dn.net/?f=z%20%3D%20-1.51)
=> ![\frac{x- 550 }{75} = -1.51](https://tex.z-dn.net/?f=%5Cfrac%7Bx-%20550%20%7D%7B75%7D%20%3D%20-1.51)
=>
=> ![x = 436.75](https://tex.z-dn.net/?f=x%20%20%3D%20436.75)