I am working on a very simple statistics problem, but I am unsure of two parts of the problem due to unfamiliar wording.
The question is as follows:
Suppose there is a 30% probability that someone has had a flu shot this year. You select 10 people at random.
(a) What is the probability that at least half of these people have had the flu shot?
(b) Find the expected value and standard deviation of the number of those 10 who have received a flu shot.
(c) Find the expected value and standard deviation of the proportion of those 10 who have received a flu shot.
I've completed part (a), simply using $P(x)=\frac{n!} {x!(n-x)!}P^{x}(1-P)^{(n-x)}$ for $P(5), P(6), P(7), P(8), P(9), $and $P(10)$, and then summing the results. The result was $P(5\leq X\leq 10)=0.15042$.
For parts (b) and (c) I am stuck due to not knowing the meaning of the words number and proportion in their context within the question.
My textbook gives the following definitions for the derived mean and variance of a binomial probability distribution.
Let $X$ be the number of successes in $n$ independent trials, each with probability of success $P$. Then $X$ follows a binomial distribution with mean and variance:
$\mu =E(X)=nP$
and
$\sigma_{X}^{2}=E[(X-\mu_{X})^{2}]=nP(1-P)$
In terms of the question I am answering I would have $P=0.3$ and $n=10$. But there is clearly some other meaning. Thanks for any help.