Question

54% of all Americans are home owners. If 48 Americans are randomly selected, find the probability that

a. Exactly 25 of them are are home owners.
b. At most 24 of them are are home owners.
c. At least 28 of them are home owners.
d. Between 24 and 29 (including 24 and 29) of them are home owners.  

Answer 1

P(an American is a home owner), p = 0.54

q = 1 - p = 0.46

n = 48

By normal approximation to binomial, P(X < A) = P(Z < (A - mean)/standard deviation)

Mean = np

= 48 x 0.54

= 25.92

Standard deviation =

=

= 3.453

a. P(exactly 25) = P(24.5 < X < 25.5)

= P(X < 25.5) - P(X < 24.5)

= P(Z < (25.5 - 25.92)/3.453) - P(Z < (24.5 - 25.92)/3.453)

= P(Z < -0.12) - P(Z < -0.41)

= 0.5478 - 0.3409

= 0.2069

b. P(at most 24) = P(X < 24.5)

= 0.3409

c. P(at least 28) = 1 - P(X < 27.5)

= 1 - P(Z < (27.5 - 25.92)/3.453)

= 1 - P(Z < 0.46)

= 1 - 0.6772

= 0.3228

d. P(between 24 and 29, inclusive) = P(23.5 < X < 29.5)

= P(X < 29.5) - P(X < 23.5)

= P(Z < (29.5 - 25.92)/3.453) - P(Z < (23.5 - 25.92)/3.453)

= P(Z < 1.04) - P(Z < -0.70)

= 0.8508 - 0.2420

= 0.6088