A continuous random variable X has a normal distribution with mean 50.5. The probability that X takes a value less than 54 is 0.76. Use this information and the symmetry of the density function to find the probability that X takes a value greater than 47. Sketch the density curve with relevant regions shaded to illustrate the computation.

Answer:

For a left tailed area of 0.76, the z score is, by table/technology,

z = 0.706302563

Hence, the standard deviation is

sigma = (X-u)/z = (54-50.5)/(0.706302563) = 4.955383406

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    47      
u = mean =    50.5      
          
s = standard deviation =    4.955383406      
          
Thus,          
          
z = (x - u) / s =    -0.706302563      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   -0.706302563   ) =    0.76 [ANSWER]

PFA