Question

A survey by KRC Research for U.S. News reported that 40% of people

     plan to spend more on eating out after they retire. Suppose a random sample of 20   

     people are selected and the process follows a binomial distribution, with p = 0.40

a. What is the expected value and standard deviation of the people in the sample who

      plan to spend more on eating out after they retire.

b. What is the probability that 8 or fewer in the sample indicate that they plan to spend

      more on eating out after retirement?

c. What is the probability that at least 9 people (i.e. 9 or more) in the sample indicate

      that they plan to spend more on eating out after retirement?

Answer 1

BIONOMIAL DISTRIBUTION
pmf of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial
I.
mean = np
where
n = total number of repetitions experiment is executed
p = success probability
mean = 20 * 0.4
= 8
II.
variance = npq
where
n = total number of repetitions experiment is executed
p = success probability
q = failure probability
variance = 20 * 0.4 * 0.6
= 4.8
III.
standard deviation = sqrt( variance ) = sqrt(4.8)
=2.1909

a.
the expected value and standard deviation of the people in the sample who
plan to spend more on eating out after they retire.
mean =8 standard deviation=2.1909

b.
the probability that 8 or fewer in the sample indicate that they plan to spend
more on eating out after retirement
P( X < = 8) = P(X=8) + P(X=7) + P(X=6) + P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)
= ( 20 8 ) * 0.4^8 * ( 1- 0.4 ) ^12 + ( 20 7 ) * 0.4^7 * ( 1- 0.4 ) ^13 + ( 20 6 ) * 0.4^6 * ( 1- 0.4 ) ^14 + ( 20 5 ) * 0.4^5 * ( 1- 0.4 ) ^15 + ( 20 4 ) * 0.4^4 * ( 1- 0.4 ) ^16 + ( 20 3 ) * 0.4^3 * ( 1- 0.4 ) ^17 + ( 20 2 ) * 0.4^2 * ( 1- 0.4 ) ^18 + ( 20 1 ) * 0.4^1 * ( 1- 0.4 ) ^19 + ( 20 0 ) * 0.4^0 * ( 1- 0.4 ) ^20
= 0.5956

c.
the probability that at least 9 people (i.e. 9 or more) in the sample indicate
that they plan to spend more on eating out after retirement
P( X < 9) = P(X=8) + P(X=7) + P(X=6) + P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= ( 20 8 ) * 0.4^8 * ( 1- 0.4 ) ^12 + ( 20 7 ) * 0.4^7 * ( 1- 0.4 ) ^13 + ( 20 6 ) * 0.4^6 * ( 1- 0.4 ) ^14 + ( 20 5 ) * 0.4^5 * ( 1- 0.4 ) ^15 + ( 20 4 ) * 0.4^4 * ( 1- 0.4 ) ^16 + ( 20 3 ) * 0.4^3 * ( 1- 0.4 ) ^17 + ( 20 2 ) * 0.4^2 * ( 1- 0.4 ) ^18 + ( 20 1 ) * 0.4^1 * ( 1- 0.4 ) ^19 + ( 20 0 ) * 0.4^0 * ( 1- 0.4 ) ^20
= 0.5956
P( X > = 9 ) = 1 - P( X < 9) = 0.4044