Question

2. Suppose that the probability that a grant proposal is awarded by a funding agency is 0.3. (a) If, for a particular year, there are 100 proposals submitted to that agency, what is the probability that at most 20 proposals are awarded? Consider any necessary approximation.

Answer 1

Solution:

Given that,

P = 0.3

1 - P = 0.7

n = 100

Here, BIN ( n , P ) that is , BIN (100 , 0.3)

then,

n*p = 100 * 0.3 = 30 > 5

n(1- P) = 100 * 0.7 = 70 > 5

According to normal approximation binomial,

X Normal

Mean = = n*P = 100 * 0.3 = 30

Standard deviation = =n*p*(1-p) = 100*0.3*0.7= 21

We using continuity correction factor

P( X a ) = P(X < a + 0.5)

P(x < 20.5) = P((x - ) / < (20.5 - 30 ) / 21)

= P(z < -2.07)

Probability = 0.0192

The probability that at most 20 proposals are awarded is 0.0192