Question

Suppose that x has a binomial distribution with n = 50 and p = 0.6, so that μ = np = 30 and σ = np(1 − p) = 3.4641. Approximate the following probabilities using the normal approximation with the continuity correction. (Hint: 25 < x < 39 is the same as 26 ≤ x ≤ 38. Round your answers to four decimal places.)

(a) P(x = 30)

(b) P(x = 25) (

c) P(x ≤ 25)

(d) P(25 ≤ x ≤ 39)

(e) P(25 < x < 39)

Answer 1

x has a binomial distribution with
n = 50
p = .6,
so that
μ = np = 30
σ = np(1 − p) = 3.4641.
(a)
P(x = 30) = P [((x-mean)/sd) = ((30-30)/3.4641)]
= P (z = 0)
= 0.5

(b)
P(x = 25) = P [((x-mean)/sd) = ((25-30)/3.4641)]
= P (z = -5/3.46)
= -1.445

(c) As we need to find P(x ≤ 25) we need to include 25. So we can consider an upper bound of 25.5.

P(x ≤ 25) = P [((x-mean)/sd)  = (25.5-30)/3.4641]

= -1.299

(d) As we need to find P(25 ≤ x ≤ 39) we need to include 25 and 39. So we can consider an upper bound of 38.5 and a lower bound of 24.5.

P(25 ≤ x ≤ 39) = P [((x-mean)/sd)  = (24.5-30)/3.4641] -  P [((x-mean)/sd)  = (38.5-30)/3.4641]

= 2.4537 - -1.5877

= 4.0414.

(e)

As we need to find P(25 ≤ x ≤ 39) we need to include 25 and 39. So we can consider an upper bound of 37.5 and a lower bound of 25.5.

P(25 ≤ x ≤ 39) = P [((x-mean)/sd)  = (25.5-30)/3.4641] -  P [((x-mean)/sd)  = (37.5-30)/3.4641]

= 2.165 - -1.299

= 3.464.