Question

62% of all Americans are home owners. If 44 Americans are randomly selected, find the probability that

a. Exactly 25 of them are are home owners.  ____
b. At most 26 of them are are home owners. ___
c. At least 28 of them are home owners.  _____
d. Between 24 and 28 (including 24 and 28) of them are home owners.  ___

Answer 1

Since n = 44 > 30, we use Normal approximation

Mean = n * P = ( 44 * 0.62 ) = 27.28
Variance = n * P * Q = ( 44 * 0.62 * 0.38 ) = 10.3664
Standard deviation = = 3.2197

Part a)

P ( X = 25 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 25 - 0.5 < X < 25 + 0.5 ) = P ( 24.5 < X < 25.5 )

P ( 24.5 < X < 25.5 )
Standardizing the value

Z = ( 24.5 - 27.28 ) / 3.2197
Z = -0.86
Z = ( 25.5 - 27.28 ) / 3.2197
Z = -0.55
P ( -0.86 < Z < -0.55 )
P ( 24.5 < X < 25.5 ) = P ( Z < -0.55 ) - P ( Z < -0.86 )
P ( 24.5 < X < 25.5 ) = 0.2902 - 0.1939
P ( 24.5 < X < 25.5 ) = 0.0962

Part b)

P ( X <= 26 )
Using continuity correction
P ( X < n + 0.5 ) = P ( X < 26 + 0.5 ) = P ( X < 26.5 )

P ( X < 26.5 )
Standardizing the value

Z = ( 26.5 - 27.28 ) / 3.2197
Z = -0.24

P ( X < 26.5 ) = P ( Z < -0.24 )
P ( X < 26.5 ) = 0.4052

Part c)

P ( X >= 28 )
Using continuity correction
P ( X > n - 0.5 ) = P ( X > 28 - 0.5 ) =P ( X > 27.5 )

P ( X > 27.5 ) = 1 - P ( X < 27.5 )
Standardizing the value

Z = ( 27.5 - 27.28 ) / 3.2197
Z = 0.07

P ( Z > 0.07 )
P ( X > 27.5 ) = 1 - P ( Z < 0.07 )
P ( X > 27.5 ) = 1 - 0.5279
P ( X > 27.5 ) = 0.4721

Part d)

P ( 24 <= X <= 28 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 24 - 0.5 < X < 28 + 0.5 ) = P ( 23.5 < X < 28.5 )

P ( 23.5 < X < 28.5 )
Standardizing the value

Z = ( 23.5 - 27.28 ) / 3.2197
Z = -1.17
Z = ( 28.5 - 27.28 ) / 3.2197
Z = 0.38
P ( -1.17 < Z < 0.38 )
P ( 23.5 < X < 28.5 ) = P ( Z < 0.38 ) - P ( Z < -1.17 )
P ( 23.5 < X < 28.5 ) = 0.6476 - 0.1202
P ( 23.5 < X < 28.5 ) = 0.5274