Question

Assume that x is a binominal random variable with n=100 and p=0.40. Use a normal approximation to find the following:

a) P(x≥38) b) P(x=45) c) P(x>45) d) P(x<45)

Answer 1

Solution:

Given that,

P = 0.40

1 - P = 0.60

n = 100

Here, BIN ( n , P ) that is , BIN (100 , 0.40)

then,

n*p = 100*0.40 = 40 > 5

n(1- P) = 100*0.60 = 60 > 5

According to normal approximation binomial,

X Normal

Mean = = n*P = 40

Standard deviation = =n*p*(1-p) = 100*0.40*0.60 = 24

We using countinuity correction factor

a)

P(X a ) = P(X > a - 0.5)

P(x > 37.5) = 1 - P(x < 37.5)

= 1 - P((x - ) / < (37.5 - 40) / 24)

= 1 - P(z < -0.510)

= 1 - 0.3050

= 0.6950

Probability = 0.6950

b)

P(X = a) = P( a - 0.5 < X < a + 0.5)

P(44.5 < x < 45.5) = P((44.5 - 40)/ 24) < (x - ) /  < (45.5 - 40) / 24) )

= P(0.919 < z < 1.123)

= P(z < 1.123) - P(z < 0.919)

= 0.8693 - 0.8210

= 0.0483

Probability = 0.0483

c)

P(x > a ) = P( X > a + 0.5)

P(x > 45.5) = 1 - P(x <45.5 )

= 1 - P((x - ) / < (45.5-40) / 24)

= 1 - P(z < 1.123)

= 1 - 0.8693

= 0.1307

Probability = 0.1307

d)

P(X < a ) = P(X < a - 0.5)

P(x <44.5 ) = P((44.5 - 40)/ 24) )

= P(z < 0.919)

= 0.8210

Probability = 0.8210