Question

Suppose the probability that a person has type O- blood is 0.05. Consider a group of 200 people.

a. What is the probability that exactly 3 people of the people in the group have type O- blood? (3 points)

b. Show that this binomial distribution satisfies both conditions needed to use the normal approxi- mation to the binomial. (2 points)

c. Using the normal approximation to the binomial, find the probability that at least 13 people in the group have type O- blood. (5 points)

Answer 1

Part a)

P(X = 3 ) =

Part b)

For Normal approximation to Binomial, need to satisfy the condition

np and nq > 5

np = 200 * 0.05 = 10 > 5

nq = 200 * 0.95 = 190 > 5

Hence both condition is satisfied, we can use Normal approximation.

Part c)

Mean = n * P = ( 200 * 0.05 ) = 10
Variance = n * P * Q = ( 200 * 0.05 * 0.95 ) = 9.5
Standard deviation = = 3.0822

P ( X >= 13 )
Using continuity correction
P ( X > n - 0.5 ) = P ( X > 13 - 0.5 ) =P ( X > 12.5 )

P ( X > 12.5 ) = 1 - P ( X < 12.5 )
Standardizing the value

Z = ( 12.5 - 10 ) / 3.0822
Z = 0.81

P ( Z > 0.81 )
P ( X > 12.5 ) = 1 - P ( Z < 0.81 )
P ( X > 12.5 ) = 1 - 0.791
P ( X > 12.5 ) = 0.2090