In a study by Peter D. Hart Research Associates for the Nasdaq Stock Market, it
ID: 3170734 • Letter: I
Question
In a study by Peter D. Hart Research Associates for the Nasdaq Stock Market, it was determined that 20% of all stock investors are retired people. In addition, 40% of all adults invest in mutual funds. Suppose a random sample of 25 stock investors is taken.
a) Suppose a random sample of 20 adults is taken. What is the probability that exactly seven adults invested in mutual funds?
P(x = 7)=
b) Suppose a random sample of 20 adults is taken. What is the probability that fewer than five adults invested in mutual funds?
P(x < 5)=
c)Suppose a random sample of 20 adults is taken. What is the probability that exactly two adults invested in mutual funds?
P(x = 2)=
d)Suppose a random sample of 20 adults is taken. What is the probability that 11 or more adults invested in mutual funds?
P(x 11)=
Explanation / Answer
Binomial 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
a.
P( X = 7 ) = ( 20 7 ) * ( 0.4^7) * ( 1 - 0.4 )^13
= 0.1659
b.
P( X < 5) = P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)
= ( 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
= 0.051
c.
P( X = 2 ) = ( 20 2 ) * ( 0.4^2) * ( 1 - 0.4 )^18
= 0.0031
d.
P( X < 11) = P(X=10) + 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) + P9X=0)
= ( 20 10 ) * 0.4^10 * ( 1- 0.4 ) ^10 + ( 20 9 ) * 0.4^9 * ( 1- 0.4 ) ^11 + ( 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
= 0.8725
P( X > = 11 ) = 1 - P( X < 11) = 0.1275