given heights of adult males are normally distributed with a

given heights of adult males are normally distributed with a mean if 69\" and a standard deviation of 2.5\" find the following.

a) the portion of all men over 67\"
b) the portion of all men under 72 \"
c) the portion of men between 67\" and 72\"

Solution

a)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    67      
u = mean =    69      
          
s = standard deviation =    2.5      
          
Thus,          
          
z = (x - u) / s =    -0.8      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   -0.8   ) =    0.788144601 [ANSWER]

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b)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    72      
u = mean =    69      
          
s = standard deviation =    2.5      
          
Thus,          
          
z = (x - u) / s =    1.2      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   1.2   ) =    0.88493033 [ANSWER]

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c)

We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound =    67      
x2 = upper bound =    72      
u = mean =    69      
          
s = standard deviation =    2.5      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -0.8      
z2 = upper z score = (x2 - u) / s =    1.2      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.211855399      
P(z < z2) =    0.88493033      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.673074931   [ANSWER]