Final answer:
The probability that the entire population of 45 tax returns will be audited when 20 are improper is calculated using the hypergeometric distribution by summing up the probabilities for finding 3 or more improper returns in a sample of 9.
Step-by-step explanation:
To calculate the probability that the entire population of 45 tax returns will be audited given that the true number of improper returns in the population is 20, we need to consider a sampling distribution. Since we are sampling without replacement from a finite population, the hypergeometric distribution is appropriate to model this scenario. The formula for the hypergeometric distribution is:
P(X=k) = [ (C(20, k)) * (C(25, 9-k)) ] / C(45, 9)
Where C(n, k) represents the combination of selecting k items from a total of n without regard to order. To find the probability of auditing the entire population, we're interested in the probability of finding 3 or more improper returns in our sample of 9.
Thus, we need to sum the probabilities for k = 3, 4, ..., 9:
P(X>=3) = ∑ P(X=k) from k=3 to k=9
This involves calculating the probability for each individual value of k and then summing them up.
However, since calculating this by hand may be quite tedious, we may use statistical software or a hypergeometric probability calculator to find the result more efficiently.