Question

X is a binomial random variable.

n= 100 p= .4

Use the binomial approach and normal approximation to calculate the follwowing: 1. P(x>=38) 2. P(x=45) 3. P(X>45) 4. P(x <45)

Answer 1

Solution:

Given that,

P = 0.4

1 - P = 0.6

n = 100

Here, BIN ( n , P ) that is , BIN (100 , 0.4)

then,

n*p = 100 * 0.4 = 40 > 5

n(1- P) = 100 * 0.6 = 60 > 5

According to normal approximation binomial,

X Normal

Mean = = n*P = 40

Standard deviation = =n*p*(1-p) = 100 * 0.4 * 0.6= 24

We using continuity correction factor

1)

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 < -0.51)

= 1 - 0.3050   

= 0.6950

Probability = 0.6950

2)

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(0.92 < z < 1.12)

= P(z < 1.12) - P(z < 0.92)

= 0.8686 - 0.8212

Probability = 0.0474

3)

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 < 1.12)

= 1 - 0.8686   

= 0.1314

Probability = 0.1314

4)

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

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

= P(z < 0.92)

Probability = 0.8212