I am unsure of how to solve for this The Better Business Bur

I am unsure of how to solve for this.

The Better Business Bureau settles 85% of complaints it receives involving new car dealers. Suppose a sample of 90 complaints involving new car dealers is selected. Find the probability that Better Business Bureau settles less than 83 of these complaints.

Solution

I am attaching two possible solutions:

a) If you use binomial distribution for this,

Note that P(fewer than x) = P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    90      
p = the probability of a success =    0.85      
x = our critical value of successes =    83      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   82   ) =    0.969158509
          
Which is also          
          
P(fewer than   83   ) =    0.969158509 [ANSWER]

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B)

If you use normal approximation for this,

u = n p = 90*0.85 = 76.5
s = sqrt(n p (1-p)) = sqrt(90*0.85*(1-0.85)) = 3.387476937

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    82.5      
u = mean =    76.5      
          
s = standard deviation =    3.387476937      
          
Thus,          
          
z = (x - u) / s =    1.771229771      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   1.771229771   ) =    0.96173875 [ANSWER]