Question

If np greater than or equals 5 and nq greater than or equals 5​, estimate Upper P left parenthesis fewer than 6 right parenthesis with nequals14 and pequals0.5 by using the normal distribution as an approximation to the binomial​ distribution; if npless than5 or nqless than​5, then state that the normal approximation is not suitable. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. A. Upper P left parenthesis fewer than 6 right parenthesisequals nothing ​(Round to four decimal places as​ needed.) B. The normal approximation is not suitable.

Answer 1

Solution:

Given p = 0.5 and n = 14.

q = 1-p = 1-0.5 = 0.5

np = 14(0.5) = 7

nq = 14(0.5)= 7
since np and nq both are greater than 5, we can use the normal distribution as an approximation to the binomial​ distribution.
mean μ = np = 14(0.5) = 7
sd = sqrt(npq) = sqrt(14*0.5*0.5) = 1.871

P(x < 6) = P(x < 6-0.5) = P(x < 5.5) (Using continuity correction factor)
P(x < 5.5) = P(z < 5.5- 7/1.871) = P(z < -0.80)
= 1-P (Z<0.8 ) = 1−0.7881 = 0.2119