P (fewer than 6) Normal Distribution

Question

If np≥5 and nq≥5, estimate P (fewer than 6) with n=14 and p=0.6 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 6) = ______

(round to four decimal places as needed). 

Answer 1

q = 1 - p = 1 - 0.6 = 0.4

Np = (14)(0.6) = 8.4 > 5   <--- this is the mean; the mean is 8.4

Nq = (14)(0.4) = 5.6 > 5

 

So yes, the normal distribution shall approximate the Binomial

 

The standard deviation is sqrt(Npq) = sqrt( 14*0.6*0.4) = 1.833

Prob (X<6) = Prob ( X < 6.5) <---- continuity correction

 

                 = Prob ( z < (6.5-8.4)/1.833 ) 

 

                 = Prob ( z < -1.0365521) = 0.1492 = 14.92%