Question

The chance of an IRS audit for a tax return with over $25,000 in income is about 2% per year. Suppose that 100 people with tax returns over $25,000 are randomly picked. We are interested in the number of people audited in one year. Use a Poisson distribution to answer the following questions.

a. The mean number of people audited is 2

b. The mean number of people audited is 4

c. The mean number of people audited is 5

d. The mean number of people audited is 1

Answer 1

Final Answers:

a. P(X = 2) ≈ 0.2707

b. P(X = 4) ≈ 0.1954

c. P(X = 5) ≈ 0.1755

d. P(X = 1) ≈ 0.1839

Step-by-step explanation:

The Poisson distribution calculates the probability of a specific number of events occurring in a fixed interval of time or space, given the average rate of occurrence and assuming events happen independently. The formula to compute the probability for a Poisson distribution is P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the average rate of occurrence.

For these scenarios, each mean number of people audited serves as λ. Using the Poisson probability formula, we can calculate the probabilities:

a. For a mean of 2 people audited (λ = 2), the probability of exactly 2 people being audited is around 0.2707.

b. With a mean of 4 people audited (λ = 4), the probability of exactly 4 individuals being audited is approximately 0.1954.

c. At a mean of 5 individuals audited (λ = 5), the probability of exactly 5 people being audited is roughly 0.1755.

d. Finally, for a mean of 1 person audited (λ = 1), the probability of exactly 1 person being audited is about 0.1839.

These probabilities demonstrate the likelihood of specific numbers of individuals being audited among the randomly selected 100 people with tax returns over $25,000 in one year.