Question

77% of all Americans live in cities with population greater than 100,000 people. If 37 Americans are randomly selected, find the probability that

a. Exactly 30 of them live in cities with population greater than 100,000 people.

b. At most 30 of them live in cities with population greater than 100,000 people.

c. At least 26 of them live in cities with population greater than 100,000 people.

d. Between 26 and 31 (including 26 and 31) of them live in cities with population greater than 100,000 people.

Answer 1

Mean = n * P = ( 37 * 0.77 ) = 28.49
Variance = n * P * Q = ( 37 * 0.77 * 0.23 ) = 6.5527
Standard deviation = = 2.5598

Part a)
P ( X = 30 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 30 - 0.5 < X < 30 + 0.5 ) = P ( 29.5 < X < 30.5 )

P ( 29.5 < X < 30.5 )
Standardizing the value

Z = ( 29.5 - 28.49 ) / 2.5598
Z = 0.39
Z = ( 30.5 - 28.49 ) / 2.5598
Z = 0.79
P ( 0.39 < Z < 0.79 )

P ( 29.5 < X < 30.5 ) = 0.7838 - 0.6534
P ( 29.5 < X < 30.5 ) = 0.1304

Part b)

P ( X <= 30 )
Using continuity correction
P ( X < n + 0.5 ) = P ( X < 30 + 0.5 ) = P ( X < 30.5 )

P ( X < 30.5 )
Standardizing the value

Z = ( 30.5 - 28.49 ) / 2.5598
Z = 0.79

P ( X < 30.5 ) = P ( Z < 0.79 )
P ( X < 30.5 ) = 0.7852

Part c)

P ( X >= 26 )
Using continuity correction
P ( X > n - 0.5 ) = P ( X > 26 - 0.5 ) =P ( X > 25.5 )

P ( X > 25.5 ) = 1 - P ( X < 25.5 )
Standardizing the value

Z = ( 25.5 - 28.49 ) / 2.5598
Z = -1.17

P ( Z > -1.17 )
P ( X > 25.5 ) = 1 - P ( Z < -1.17 )
P ( X > 25.5 ) = 1 - 0.121
P ( X > 25.5 ) = 0.879

Part d)

P ( 26 <= X <= 31 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 26 - 0.5 < X < 31 + 0.5 ) = P ( 25.5 < X < 31.5 )

P ( 25.5 < X < 31.5 )
Standardizing the value

Z = ( 25.5 - 28.49 ) / 2.5598
Z = -1.17
Z = ( 31.5 - 28.49 ) / 2.5598
Z = 1.18
P ( -1.17 < Z < 1.18 )
P ( 25.5 < X < 31.5 ) = P ( Z < 1.18 ) - P ( Z < -1.17 )
P ( 25.5 < X < 31.5 ) = 0.8802 - 0.1214
P ( 25.5 < X < 31.5 ) = 0.7588