Question

If np greater than or equals 5(np≥5) and nq greater than or equals 5(nq≥5​), estimate P(fewer than 4) with nequals=13 and p equals=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.

Answer 1

np = 13 * 0.5 = 6.5 >= 5

nq = 13 * 0.5 = 6.5 >= 5

Using Normal Approximation to Binomial
Mean = n * P = ( 13 * 0.5 ) = 6.5
Variance = n * P * Q = ( 13 * 0.5 * 0.5 ) = 3.25
Standard deviation = √(variance) = √(3.25) = 1.8028

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

P ( X < 4 ) = ?
Using continuity correction
P ( X < n - 0.5 ) = P ( X < 4 - 0.5 ) = P ( X < 3.5 )

Standardizing the value
Z = ( X - µ ) / σ
Z = ( 3.5 - 6.5 ) / 1.8028
Z = -1.66
P ( ( X - µ ) / σ ) < ( 3.5 - 6.5 ) / 1.8028 )
P ( X < 3.5 ) = P ( Z < -1.66 )
P ( X < 3.5 ) = 0.0485