Find the value of xthat yields the probability shown where X

Find the value of x*that yields the probability shown, where X is a normally distributed random variable X with mean 54 and standard deviation 12.

P(X<x*)=0.0900

P(X<x*)=0.6500

Solution

A)

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.09      
          
Then, using table or technology,          
          
z =    -1.340755034      
          
As x = u + z * s,          
          
where          
          
u = mean =    54      
z = the critical z score =    -1.340755034      
s = standard deviation =    12      
          
Then          
          
x* = critical value =    37.9109396   [ANSWER]  

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B)

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.65      
          
Then, using table or technology,          
          
z =    0.385320466      
          
As x = u + z * s,          
          
where          
          
u = mean =    54      
z = the critical z score =    0.385320466      
s = standard deviation =    12      
          
Then          
          
x* = critical value =    58.6238456   [ANSWER]