Adam and Jenny Mersereau have operated a small collection company in North Carolina for 4 years.Throughout this time they have worked as agents for various companies in the area doing debt collection work.For example, they are currently under contract with a local medical clinic to collect past due accounts.As with most of their contracts, Merserau Collections gets a percentage of all money collected.This has worked out well over the years. Armed with only a cursory knowledge of accounting, the Mersereaus have earned a nice supplement to their income.

However, Adam and Jenny are now facing a major decision that involves significant risk and significant reward as well.Bed Bath & Beyond recently declared bankruptcy.The local BB&B store has over 12,000 receivable accounts that had balances that were over 90 days late.The bankruptcy judge handling the case has ordered the delinquent accounts to be sold. Adam and Jenny have the opportunity to purchase these accounts at a fraction of their face value.If they do, then any money they collect will be theirs.Adam has figured that if they can collect on 30% or more of the accounts, they will make a sizeable profit.However, if the collection rate is a lot less than this, they will be in big trouble themselves.The judge has agreed to allow the Mersereaus the opportunity to test their collection process on a sample of 50 of BB&B's accounts.Adam and Jenny have decided that if they are able to collect on 15 or more of these, they will take the deal.If they are successful on 14 or fewer, they will walk away from the opportunity.

Adam poured himself another Boodles, grabbed his hand calculator, and tried to figure out to turn it on.While he's trying to figure it out, help him answer the following questions independently of each other.

a) Suppose the true collection rate for the 12,000 accounts is actually 20%.What is the probability that Adam and Jenny, based on their sample, will take the deal?What is the probability that they will walk away from the opportunity?

b) Suppose instead that the true collection rate for the 12,000 accounts is actually 30%.What is the probability that Adam and Jenny, based on their sample, will take the deal?What is the probability that they will walk away from the opportunity?

c) Suppose now that Adam and Jenny have been able to collect on 20 of the 50 accounts.Develop a 95% confidence interval for the true proportion of accounts they can collect on.

d) Suppose now that Adam and Jenny have been able to collect on 20 of the 50 accounts. They have been thinking further about the opportunity and don't really want to proceed unless they are very confident that they will make money.They thus want to test:

H₀: = 0.30 versus H_a: > 0.30. Test at the 5 percent level of significance.Show all relevant calculations, and explain how you reached your conclusion.Draw a picture to illustrate your test.What is the prob-value of the test?Explain your conclusion in terms of the statement of the problem.

e) Suppose now that Adam and Jenny have been able to collect on 20 of the 50 accounts, but Adam and Jenny are not satisfied with the results from the previous question. How large a sample do they need to ask the judge to take in order for a reasonable 95% confidence interval of their estimate of the true collection rate to be of the form "plus or minus 5%"?