Problem 8. About 4% of the scientists in the National Academy of Sciences believ
ID: 3052262 • Letter: P
Question
Problem 8. About 4% of the scientists in the National Academy of Sciences believe in PSI (parapsychology such as psychokinesis, telepathy, and astral projection). At a convention 183 members show up. Let X be the number of scientists at the convention who believe in PSI. You may assume for the following questions that X is binomially distributed with parameters n = 183 and p = 0.04. (a) How many scientists at the convention do you expect do believe in PSI? (b) What are the chances that none of the scientists at the convention believes in PSI? (c) What are the chances that at least 4 scientists at the convention believe in PSI? (d) What is the standard deviation of X?Explanation / Answer
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
a.
mean = np
where
n = total number of repetitions experiment is excueted
p = success probability
mean = 183 * 0.04
= 7.32
b.
P( X = 0 ) = ( 183 0 ) * ( 0.04^0) * ( 1 - 0.04 )^183
= 0.00057
c.
P( X < 4) = P(X=3) + P(X=2) + P(X=1) + P(X=0)
= ( 183 3 ) * 0.04^3 * ( 1- 0.04 ) ^180 + ( 183 2 ) * 0.04^2 * ( 1- 0.04 ) ^181 + ( 183 1 ) * 0.04^1 * ( 1- 0.04 ) ^182 + ( 183 0 ) * 0.04^0 * ( 1- 0.04 ) ^183
= 0.062789
P( X > = 4 ) = 1 - P( X < 4) = 0.937211
d.
variance = npq
where
n = total number of repetitions experiment is excueted
p = success probability
q = failure probability
variance = 183 * 0.04 * 0.96
= 7.0272
standard deviation = sqrt( variance ) = sqrt(7.0272)
=2.650887