Question

If np ≥ 5 and nq ≥ 5​, estimate P(fewer than 3) with

n =14 and p = 0.4 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. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

A. P(fewer than 3)=

or

B. the normal approximation is not suitable

Answer 1

Condition check for Normal Approximation to Binomial
n * P >= 5 = 14 * 0.4 = 5.6
n * (1 - P ) >= 5 = 14 * ( 1 - 0.4 ) = 8.4

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 < 3 )
Using continuity correction
P ( X < n - 0.5 ) = P ( X < 3 - 0.5 ) = P ( X < 2.5 )

X ~ N ( µ = 5.6 , σ = 1.833 )
P ( X < 2.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 2.5 - 5.6 ) / 1.833
Z = -1.69
P ( ( X - µ ) / σ ) < ( 2.5 - 5.6 ) / 1.833 )
P ( X < 2.5 ) = P ( Z < -1.69 )
P ( X < 2.5 ) = 0.0455

A. P(fewer than 3) = 0.0455