Answer :
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
[tex]x = 403[/tex]
b
[tex]x = 436.75[/tex]
Step-by-step explanation:
From the question we are told that
The mean is [tex]\mu = 550[/tex]
The standard deviation is [tex]\sigma = 75[/tex]
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
[tex]P( X < x) = P( \frac{x - \mu }{ \sigma} < \frac{x - 550 }{75 } ) = 0.0250[/tex]
[tex]\frac{X -\mu}{\sigma } = Z (The \ standardized \ value\ of \ X )[/tex]
[tex]P( X < x) = P( Z < z ) = 0.0250[/tex]
Generally the critical value of 0.0250 to the left is
[tex]z = -1.96[/tex]
=> [tex]\frac{x- 550 }{75} = -1.96[/tex]
=> [tex]x = [-1.96 * 75 ]+ 550[/tex]
=> [tex]x = 403[/tex]
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
[tex]P( X < x) = P( \frac{x - \mu }{ \sigma} < \frac{x - 550 }{75 } ) = 0.9345[/tex]
[tex]\frac{X -\mu}{\sigma } = Z (The \ standardized \ value\ of \ X )[/tex]
[tex]P( X < x) = P( Z < z ) = 0.9345[/tex]
Generally the critical value of 0.9345 to the right is
[tex]z = -1.51[/tex]
=> [tex]\frac{x- 550 }{75} = -1.51[/tex]
=> [tex]x = [-1.51 * 75 ]+ 550[/tex]
=> [tex]x = 436.75[/tex]