Answer:
(a) 0.00605
(b) 0.0403
(c) 0.9536
(d) 0.98809
Step-by-step explanation:
We are given that 40% of first-round appeals were successful (The Wall Street Journal, October 22, 2012) and suppose ten first-round appeals have just been received by a Medicare appeals office.
This situation can be represented through Binomial distribution as;
where, n = number of trials (samples) taken = 10
r = number of success
p = probability of success which in our question is % of first-round
appeals that were successful, i.e.; 40%
So, here X ~ 
(a) Probability that none of the appeals will be successful = P(X = 0)
P(X = 0) = 
= 
= 0.00605
(b) Probability that exactly one of the appeals will be successful = P(X = 1)
P(X = 1) = 
= 
= 0.0403
(c) Probability that at least two of the appeals will be successful = P(X>=2)
P(X >= 2) = 1 - P(X = 0) - P(X = 1)
= 1 - 
= 1 - 0.00605 - 0.0403 = 0.9536
(d) Probability that more than half of the appeals will be successful = P(X > 0.5)
For this probability we will convert our distribution into normal such that;
X ~ N(
)
and standard normal z has distribution as;
Z = 
~ N(0,1)
P(X > 0.5) = P( > 
) = P(Z > -2.26) = P(Z < 2.26) = 0.98809
