Question

A tax auditor is selecting a sample of 8 tax returns for an audit. If 3 or more of these returns are​ "improper," the entire population of 45 tax returns will be audited. Complete parts​ (a) through​ (e) below.

a. What is the probability that the entire population will be audited if the true number of improper returns in the population is 15​? The probability is? ​(Round to four decimal places as​ needed.)

b. What is the probability that the entire population will be audited if the true number of improper returns in the population is 20​? The probability is?. ​(Round to four decimal places as​ needed.)

c. What is the probability that the entire population will be audited if the true number of improper returns in the population is 5​? The probability is?. ​(Round to four decimal places as​ needed.)

d. What is the probability that the entire population will be audited if the true number of improper returns in the population is 10? The probability is?. ​(Round to four decimal places as​ needed.)

e. Discuss the differences in the​ results, depending on the true number of improper returns in the population. Choose the correct answer below.

A. The probability that the entire group will be audited is not very sensitive to the true number of improper returns in the population. The probability decreases as the true number​ increases, but only slightly.

B. The probability that the entire group will be audited is very sensitive to the true number of improper returns in the population. If the true number is very​ low, the probability is very low. If the true number is very​ high, the probability is very low.

C. The probability that the entire group will be audited is very sensitive to the true number of improper returns in the population. If the true number is very​ low, the probability is very low. If the true number is very​ high, the probability is very high.

D. The probability that the entire group will be audited is not very sensitive to the true number of improper returns in the population. The probability increases as the true number​ increases, but only slightly.

Answer 1

A tax auditor is selecting a sample of 8 tax returns for an audit. If 3 or more of these returns are​ "improper," the entire population of 45 tax returns will be audited. Complete parts​ (a) through​ (e) below.

Hypergeometric distribution used.

P(X=x) = (_N1C_x) (_N-N1C_n-x) (_NC_n)^-1

N=45, n=8

a. What is the probability that the entire population will be audited if the true number of improper returns in the population is 15​? The probability is? ​(Round to four decimal places as​ needed.)

N1=15

P( x ≥ 3) = 0.5419

b. What is the probability that the entire population will be audited if the true number of improper returns in the population is 20​? The probability is?. ​(Round to four decimal places as​ needed.)

N1=20

P( x ≥ 3) = 0.7943

c. What is the probability that the entire population will be audited if the true number of improper returns in the population is 5​? The probability is?. ​(Round to four decimal places as​ needed.)

N1=5

P( x ≥ 3) = 0.0327

d. What is the probability that the entire population will be audited if the true number of improper returns in the population is 10? The probability is?. ​(Round to four decimal places as​ needed.)

N1=10

P( x ≥ 3) = 0.2400

e. Discuss the differences in the​ results, depending on the true number of improper returns in the population. Choose the correct answer below.

A. The probability that the entire group will be audited is not very sensitive to the true number of improper returns in the population. The probability decreases as the true number​ increases, but only slightly.

B. The probability that the entire group will be audited is very sensitive to the true number of improper returns in the population. If the true number is very​ low, the probability is very low. If the true number is very​ high, the probability is very low.

Answer: C. The probability that the entire group will be audited is very sensitive to the true number of improper returns in the population. If the true number is very​ low, the probability is very low. If the true number is very​ high, the probability is very high.

D. The probability that the entire group will be audited is not very sensitive to the true number of improper returns in the population. The probability increases as the true number​ increases, but only slightly.