Question

72% of all Americans live in cities with population greater than 100,000 people. If 46 Americans are randomly selected, find the probability that

a. Exactly 31 of them live in cities with population greater than 100,000 people.  
b. At most 34 of them live in cities with population greater than 100,000 people.
c. At least 30 of them live in cities with population greater than 100,000 people.  
d. Between 27 and 34 (including 27 and 34) of them live in cities with population greater than 100,000 people.

Answer 1

Using Normal Approximation to Binomial
Mean = n * P = ( 46 * 0.72 ) = 33.12
Variance = n * P * Q = ( 46 * 0.72 * 0.28 ) = 9.2736
Standard deviation = √(variance) = √(9.2736) = 3.0453

Part a)

P ( X = 31 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 31 - 0.5 < X < 31 + 0.5 ) = P ( 30.5 < X < 31.5 )

X ~ N ( µ = 33.12 , σ = 3.0453 )
P ( 30.5 < X < 31.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 30.5 - 33.12 ) / 3.0453
Z = -0.86
Z = ( 31.5 - 33.12 ) / 3.0453
Z = -0.53
P ( -0.86 < Z < -0.53 )
P ( 30.5 < X < 31.5 ) = P ( Z < -0.53 ) - P ( Z < -0.86 )
P ( 30.5 < X < 31.5 ) = 0.2981 - 0.1949
P ( 30.5 < X < 31.5 ) = 0.1032

Part b)

P ( X ≤ 34 )
Using continuity correction
P ( X < n + 0.5 ) = P ( X < 34 + 0.5 ) = P ( X < 34.5 )

X ~ N ( µ = 33.12 , σ = 3.0453 )
P ( X < 34.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 34.5 - 33.12 ) / 3.0453
Z = 0.45
P ( ( X - µ ) / σ ) < ( 34.5 - 33.12 ) / 3.0453 )
P ( X < 34.5 ) = P ( Z < 0.45 )
P ( X < 34.5 ) = 0.6736

Part c)

P ( X ≥ 30 )
Using continuity correction
P ( X > n - 0.5 ) = P ( X > 30 - 0.5 ) =P ( X > 29.5 )
X ~ N ( µ = 33.12 , σ = 3.0453 )
P ( X > 29.5 ) = 1 - P ( X < 29.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 29.5 - 33.12 ) / 3.0453
Z = -1.19
P ( ( X - µ ) / σ ) > ( 29.5 - 33.12 ) / 3.0453 )
P ( Z > -1.19 )
P ( X > 29.5 ) = 1 - P ( Z < -1.19 )
P ( X > 29.5 ) = 1 - 0.117
P ( X > 29.5 ) = 0.883

Part d)

P ( 27 <= X <= 34 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 27 - 0.5 < X < 34 + 0.5 ) = P ( 26.5 < X < 34.5 )
X ~ N ( µ = 33.12 , σ = 3.0453 )
P ( 26.5 < X < 34.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 26.5 - 33.12 ) / 3.0453
Z = -2.17
Z = ( 34.5 - 33.12 ) / 3.0453
Z = 0.45
P ( -2.17 < Z < 0.45 )
P ( 26.5 < X < 34.5 ) = P ( Z < 0.45 ) - P ( Z < -2.17 )
P ( 26.5 < X < 34.5 ) = 0.6736 - 0.015
P ( 26.5 < X < 34.5 ) = 0.6586