Stats 100A, Spring 99

Stats 100A, Spring 99 Quiz 6

NAME:

From the LA Times, Thursday, May 27, 1999:

"Rare coins, jewelry and keepsakes taken from safe deposit boxes of about 2,000

"lost" Californians will be auctioned today and Friday, but a random check by The

Times has raised questions about the adequacy of efforts to find the owners.

In a 24-hour period immediately before the massive sale, The Times was able to

find and contact one-quarter of the owners of so-called abandoned property from a sample

list of 24 individuals supplied by the state controller's office."

Background information: There are laws that require that property kept in safe deposit boxes be turned over to the state if the owners of the boxes are "lost".

Suppose the 24 individuals from the sample are a random sample.

1. What's the population?

The population consists of the 2000 "lost" Californians.

2. What's the estimate of the percentage of the population who could be found and contacted?

The Times estimates that 25% of the population is not really lost, assuming that their sample of 24 "lost" Californians is representative of the population.

3. What's the standard error for this estimate?

The number of "lost" people who are found, X, is a binomial random variable. So phat = X/24

represents our estimate of p, the true proportion of lost Californians who could be found. The SD

of X is sqrt(np(1-p)) estimated as sqrt(n*phat * (1 - phat)) = sqrt(24*.25*.75) = 2.1213. But the SD (or SE) of the proportion is SD(phat) = sqrt(.25*.75/24) = 0.0883883 or 8.8%. (This isn't rocket science: if 2.12 out of 24 people are the SD, then the proportion of people this represents is 2.12/24 = .0883883).

4. Use the normal approximation to find an approximate 95% confidence interval for the percent of the population of "lost" Californians who could actually be found. (Hint: z(.05) = 1.645, z(.025) =1.96, where P(Z > z(alpha)) = alpha.)

A 95% CI is estimate +- z(.025)*SE

.25 +- 1.96*.0883883

.25 +- 0.173241

(0.0767589, 0.423241)

So we're 95% confident that the true percentage of lost Californians who could be found again is between 7.7% and 42.3%. (This is a wide interval, but then our sample size was only 24.)

5. This is probably not a good approximation. Why?