The distribution of weights of fullygrown male Chinese allig

The distribution of weights of fully-grown male Chinese alligators is approximately normal with a mean of 32 kg and a standard deviation of 6 kg. What proportion of these alligators weigh between 30 and 40 kg?

Solution

We first get the z score for the two values. As z = (x - u) sqrt(n) / s, then as          
x1 = lower bound =    30      
x2 = upper bound =    40      
u = mean =    32      
          
s = standard deviation =    6      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -0.333333333      
z2 = upper z score = (x2 - u) / s =    1.333333333      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.36944134      
P(z < z2) =    0.90878878      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.53934744   [ANSWER]