Normal distribution and binomial distribution

Question

If np(greater than or equal to) and nq(greater than or equal to) 5, estimate P (fewer than 7) with n=14 and p=0.5 by using the normal distribution as an approximation to the binomial distribution; if np<5 or nq<5, then state that the normal approximation is not suitable. 

 

P (fewer than 7) = 

          OR 

The normal approximation is not suitable 

 

Answer 1

Hello Kate...

did some good fishin' at the Crooked Horse out there in Kattanning...

 

 

since p=0.5, the normal distribution shall give a fair estimation to the binomial.

 

mean U = np = (14)(0.5) = 7 >5 which further supports the approximation, as is nq = 14(1-1/2) = (14)(1/2)=7

 

stddev s =sqrt( npq) = sqrt((14)(0.5)(1-0.5)) = sqrt((14)(0.5)(0.5)) = sqrt(7/2) = sqrt(3.5) = 1.8708286934....

 

Prob ( X<7) = Prob ( x' < 6.5 ) <--- continuity correction

                  = Prob( z < (6.5-7)/1.8708286934... ) <---- z-score

                  = Prob (- 0.267261242....) 

                  0.3936 <--- per the normal distribution table

 

 

The EXACT ANSWER 0.3952.. is given in the following table; 

The error is less than 1/2% = 0.005, or the 3rd decimal place

 

BINOMIAL Probabilities N=14, p=1/2,  q = 1-1/2 = 1/2

k      (N choose k)       p^k             (1-p)^(N-k)              Prob(X=k)
0             1                  1                 6.10352E-05          6.10352E-05 <--- essentially zero
1            14                0.5                0.00012207            0.000854492
2            91               0.25               0.000244141           0.005554199
3           364             0.125              0.000488281            0.022216797
4            1001          0.0625             0.000976563          0.061096191
5           2002           0.03125           0.001953125          0.122192383
6           3003          0.015625           0.00390625           0.183288574
                                                                         TOTAL 0.395263672