Question

The chances of a tax return being audited are about 19 in 1 000 if an income is less than​ $100,000 and 36 in 1 000 if an income is​ $100,000 or more.

Complete parts a through e.

a. What is the probability that a taxpayer with income less than​ $100,000 will be​ audited? With income of​ $100,000 or​ more? Upper P left parenthesis taxpayer with income less than $ 100 comma 000 is audited right ​(Type an integer or a​ decimal.) What is the probability that a taxpayer with income of​ $100,000 or more will be​ audited? Upper P left parenthesis taxpayer with income of $ 100 comma 000 or higher is audited ​(Type an integer or a​ decimal.)

b. If five taxpayers with incomes under​ $100,000 are randomly​ selected, what is the probability that exactly one will be​ audited? That more than one will be​ audited? Upper P left parenthesis x equals 1 right parenthesisequals nothing ​(Round to four decimal places as​ needed.) What is the probability that more than one will be​ audited? Upper P left parenthesis x greater than 1 right parenthesisequals nothing ​(Round to four decimal places as​ needed.)

c. Repeat part b assuming that five taxpayers with incomes of​ $100,000 or more are randomly selected. Upper P left parenthesis x equals 1 right parenthesisequals nothing ​(Round to four decimal places as​ needed.) What is the probability that more than one will be​ audited? Upper P left parenthesis x greater than 1 right parenthesisequals nothing ​(Round to four decimal places as​ needed.)

d. If two taxpayers with incomes under​ $100,000 are randomly selected and two with incomes more than​ $100,000 are randomly​ selected, what is the probability that none of these taxpayers will be​ audited? ​P(none of the taxpayers will be ​audited)equals nothing ​(Round to four decimal places as​ needed.)

e. What assumptions did you have to make in order to answer these​ questions? A. We must assume that the variables are binomial random variables. We must assume that the trials are identical and dependent. B. We must assume that the variables are random variables. We must assume that the trials are​ identical, and the probability of success varies from trial to trial. C. We must assume that the variables are binomial random variables. We must assume that the trials are​ identical, the probability of success varies from trial to​ trial, and that the trials are dependent. D. We must assume that the variables are binomial random variables. We must assume that the trials are​ identical, the probability of success is the same from trial to​ trial, and that the trials are independent. Click to select your answer(s).

Answer 1

a) The chances of a tax return being audited are about 19 in 1 000 if an income is less than​ $100,000. This is same as  the probability that a taxpayer with income less than​ $100,000 will be​ audited is 19/1000=0.019

ans:

The chances of a tax return being audited are about 36 in 1 000 if an income is​ $100,000 or more. This is same as the probability that a taxpayer with income of​ $100,000 or more will be​ audited is 36/1000=0.036

ans:

b)  five taxpayers with incomes under​ $100,000 are randomly​ selected. Let X be the number of tax payers out of 5 selected who are audited. We know that  the probability that a taxpayer with income under​ $100,000 will be​ audited is =0.019 and this probability remains the same for all the 5 tax payers (with incomes under​ $100,000) that are selected. Let us call this probability the success probability p=0.019 and n=5 as the number of trials.

We can say that X has a Binomial distribution with parameters, number of trials n=5 and success probability p=0.019.

The Binomial probability of X=x are audited out of 5 randomly selected taxpayers with incomes under​ $100,000 is

The probability that exactly one will be​ audited is same as the probability that X=1

ans: The probability that exactly one will be​ audited is

The probability that more than one will be​ audited is same as the probability that X>1

ans: The probability that more than one will be​ audited is

c)  five taxpayers with incomes of​ $100,000 or more are randomly​ selected. Let X be the number of tax payers out of 5 selected who are audited. We know that  the probability that a taxpayer with income of $100,000 or more will be​ audited is =0.036 and this probability remains the same for all the 5 tax payers (with incomes of​ $100,000 or more) that are selected. Let us call this probability the success probability p=0.036 and n=5 as the number of trials.

We can say that X has a Binomial distribution with parameters, number of trials n=5 and success probability p=0.036.

The Binomial probability of X=x are audited out of 5 randomly selected taxpayers with incomes of ​ $100,000 or more is

The probability that exactly one will be​ audited is same as the probability that X=1

ans: The probability that exactly one will be​ audited is

The probability that more than one will be​ audited is same as the probability that X>1

ans: The probability that more than one will be​ audited is

d)  two taxpayers with incomes under​ $100,000 are randomly selected and two with incomes more than​ $100,000 are randomly​ selected.

Let X be the number of tax payers out of 2 randomly selected taxpayers with incomes under​ $100,000 who are audited.

X has a Binomial distribution with parameters, number of trials n=2 and success probability p=0.019

Let Y be the number of tax payers out of 2 randomly selected taxpayers with incomes more than​ $100,000 who are audited.

Y has a Binomial distribution with parameters, number of trials n=2 and success probability p=0.036

X and Y are independent random variables as selection of two taxpayers with incomes under​ $100,000 is independent of selection of  two with incomes more than​ $100,000.

Hence we can say that the joint probability that X=x taxpayers with incomes under​ $100,000 are audited and Y=y with incomes more than​ $100,000 are audited (out of the 2 each selected) is

The probability that none of these taxpayers will be​ audited is same as the probability that X=0 and Y=0 are audited.

ans: The probability that none of these taxpayers will be​ audited is  ​P(none of the taxpayers will be ​audited) = 0.8943

e) We needed that the probability of success remains constant through out the 5 trails and each trial is independent of others.

ans: D. We must assume that the variables are binomial random variables. We must assume that the trials are​ identical, the probability of success is the same from trial to​ trial, and that the trials are independent.