Question

55% of all Americans are home owners. If 40 Americans are randomly selected, find the probability that

a. Exactly 21 of them are are home owners.____
b. At most 21 of them are are home owners.____
c. At least 22 of them are home owners.____
d. Between 19 and 23 (including 19 and 23) of them are home owners.____

Answer 1

Mean = n * P = ( 40 * 0.55 ) = 22
Variance = n * P * Q = ( 40 * 0.55 * 0.45 ) = 9.9
Standard deviation = = 3.1464

Part a)

P ( X = 21 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 21 - 0.5 < X < 21 + 0.5 ) = P ( 20.5 < X < 21.5 )

P ( 20.5 < X < 21.5 )
Standardizing the value

Z = ( 20.5 - 22 ) / 3.1464
Z = -0.48
Z = ( 21.5 - 22 ) / 3.1464
Z = -0.16
P ( -0.48 < Z < -0.16 )
P ( 20.5 < X < 21.5 ) = P ( Z < -0.16 ) - P ( Z < -0.48 )
P ( 20.5 < X < 21.5 ) = 0.4369 - 0.3168
P ( 20.5 < X < 21.5 ) = 0.1201

Part b)

P ( X <= 21 )
Using continuity correction
P ( X < n + 0.5 ) = P ( X < 21 + 0.5 ) = P ( X < 21.5 )

P ( X < 21.5 )
Standardizing the value

Z = ( 21.5 - 22 ) / 3.1464
Z = -0.16

P ( X < 21.5 ) = P ( Z < -0.16 )
P ( X < 21.5 ) = 0.4364

part c)

P ( X >= 22 )
Using continuity correction
P ( X > n - 0.5 ) = P ( X > 22 - 0.5 ) =P ( X > 21.5 )

P ( X > 21.5 ) = 1 - P ( X < 21.5 )
Standardizing the value

Z = ( 21.5 - 22 ) / 3.1464
Z = -0.16

P ( Z > -0.16 )
P ( X > 21.5 ) = 1 - P ( Z < -0.16 )
P ( X > 21.5 ) = 1 - 0.4364
P ( X > 21.5 ) = 0.5636

Part d)

P ( 19 <= X <= 23 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 19 - 0.5 < X < 23 + 0.5 ) = P ( 18.5 < X < 23.5 )

P ( 18.5 < X < 23.5 )
Standardizing the value

Z = ( 18.5 - 22 ) / 3.1464
Z = -1.11
Z = ( 23.5 - 22 ) / 3.1464
Z = 0.48
P ( -1.11 < Z < 0.48 )
P ( 18.5 < X < 23.5 ) = P ( Z < 0.48 ) - P ( Z < -1.11 )
P ( 18.5 < X < 23.5 ) = 0.6832 - 0.133
P ( 18.5 < X < 23.5 ) = 0.5502