The length X of a fish from a particular mountain lake in Id

The length, X, of a fish from a particular mountain lake in Idaho is normally distributed with =7.7 inches and =1.6inches.

(a) Find the probability that the length of a chosen fish was greater than 8.7 inches:  .

(b) Find the probability that the length of a chosen fish was between 5.7 and 8.7 inches:

Solution

a)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    8.7      
u = mean =    7.7      
          
s = standard deviation =    1.6      
          
Thus,          
          
z = (x - u) / s =    0.625      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   0.625   ) =    0.265985529 [answer]

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b)

We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound =    5.7      
x2 = upper bound =    8.7      
u = mean =    7.7      
          
s = standard deviation =    1.6      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -1.25      
z2 = upper z score = (x2 - u) / s =    0.625      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.105649774      
P(z < z2) =    0.734014471      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.628364697   [answer]