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

 $x = 436.75$

Step-by-step 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

     $P( X < x) = P( \frac{x - \mu }{ \sigma} < \frac{x - 550 }{75 } ) = 0.0250$

$\frac{X -\mu}{\sigma } = Z (The \ standardized \ value\ of \ X )$

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

Generally the critical value of  0.0250 to the left  is  

       $z = -1.96$

=>    $\frac{x- 550 }{75} = -1.96$

=>    $x = [-1.96 * 75 ]+ 550$      

=>    $x = 403$

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

        $P( X < x) = P( \frac{x - \mu }{ \sigma} < \frac{x - 550 }{75 } ) = 0.9345$

$\frac{X -\mu}{\sigma } = Z (The \ standardized \ value\ of \ X )$

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

Generally the critical value of  0.9345 to the right  is  

       $z = -1.51$

=>    $\frac{x- 550 }{75} = -1.51$

=>    $x = [-1.51 * 75 ]+ 550$      

=>    $x = 436.75$