Math 207—Confidence Interval for a Population Mean

Example (Motivation for the Method)

A large bank wants to estimate the average amount of money owed by all its delinquent debtors (i.e., people more than two months behind on payments). It is known from past experience that the standard deviation of the amounts is $90. [Reality Check: Yes, everything is online now, so it’s seems the bank should know the average amount owed, but let’s assume they are still in the “dark ages” of paper. Yes, it seems silly that we magically know the population standard deviation when we don’t know the mean, but suspend reality for just a few minutes and then we’ll deal with the unknown-standard-deviation situation.]

The bank takes a random sample of 100 of its delinquent accounts and finds . How can we use this sample mean to estimate the population mean? The Central Limit Theorem to the rescue! (Let’s start with a 95% confidence interval.)

Example Continued

Now find a 99% confidence interval for the mean amount of money owed by its delinquent debtors. (Is this interval wider or narrower than the 95% interval?)

Example Continued Even More

Now suppose a random sample of 200 accounts is taken, and the sample mean is $233.28. Again, find a 99% confidence interval for the mean amount of money owed by its delinquent debtors. (Is this interval wider or narrower than the 99% interval based on 100 accounts?)