
Andy C. answered 07/18/18
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X' is Binomial N=38, p=0.25 ---> q=0.75
mean = Np = 38*0.25 = 9.5 > 5
So 5 < 9.5 = 38*0.25 < 38*0.75 = Nq
So the normal distribution can be used to approximate the binomial.
The mean is , as stated, Np=9.5
the standard deviation is sqrt(Npq) = sqrt( 38*0.25*.75) = 2.669269563007827802703....
Prob (X' < 8) = Prob(X < 8.5) <--- continuity correction
= Prob (z < (8.5 - 9.5)/2.669269....) =
= Prob ( z < -0.37463) = 0.3538 = 35.38%