In order to estimate the mean monthly rental of the 300,000 3 ½ room apartments for lower income Housing in New York City a random sample of apartments is chosen and the rents are recorded. The housing authority wishes to be 95% certain that the error in estimating the mean monthly rental is less than $50. How many apartments should be in the sample? The standard deviation of rents is not known.

**APPROACH A: Additional Information**

It is discovered from the housing authority that it is extremely unlikely that the monthly rental of 3 ½ room lower income apartments is outside the $750 to $1750 range. Using this information, determine the sample size. (HINT: Assume that rents are normally distributed and the “extremely unlikely” implies a probability of .05).

**APPROACH B: Double Sampling**

A sample of 10 apartments is drawn and the following rents are obtained:

980, 1230, 1320, 1440, 1180, 1050, 1520, 1630, 880, 920.

1. Estimate the mean rental and the standard deviation of apartment rentals from this data and obtain a 95% confidence interval for the mean.

2. Is this sample mean an adequate estimate of the mean rental of the 3 ½ room lower income apartments in New York City? Discuss.

3. Using the data, estimate the sample size required by the housing authority. How many additional (if any) apartments must be sampled?

4. This method is called double sampling because we sampled the apartments twice, once to estimate the required sample size and a second time, if required, in order to obtain an adequate estimate of the mean rental. Discuss the advantages and disadvantages of this approach.