Question

If

np greater than or equals 5np≥5

and

nq greater than or equals 5nq≥5​,

estimate

Upper P left parenthesis fewer than 5 right parenthesisP(fewer than 5)

with

nequals=14

and

pequals=0.4

by using the normal distribution as an approximation to the binomial​ distribution; if

npless than<5

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.

Answer 1

np = 14 * 0.4 = 5.6 >= 10

n ( 1 - p) = 14 * (1 - 0.4) = 8.4 >= 10

Since np >=10 and n( 1 - p) >= 10 , Normal approximation is suitable.

Using Normal Approximation to Binomial
Mean = n * P = ( 14 * 0.4 ) = 5.6
Variance = n * P * Q = ( 14 * 0.4 * 0.6 ) = 3.36
Standard deviation = √(variance) = √(3.36) = 1.833

P(X < x) = P(Z < ( x - mean) / SD )

Using continuity correction,

P(X < 5) = P(X < 4.5)

P ( X < 4.5 ) = P ( Z < 4.5 - 5.6 ) / 1.833 )
= P ( Z < -0.6 )
P ( X < 4.5 ) = 0.2743