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;
![P(X=r)= \binom{n}{r}p^{r}(1-p)^{n-r} ; x = 0,1,2,3,....](https://tex.z-dn.net/?f=P%28X%3Dr%29%3D%20%5Cbinom%7Bn%7D%7Br%7Dp%5E%7Br%7D%281-p%29%5E%7Bn-r%7D%20%3B%20x%20%3D%200%2C1%2C2%2C3%2C....)
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