Question

3. One thing that often surprises San Diego newcomers is how present the U.S. military is here. As a statistician at DQ Industries, your boss has tasked you with finding the proportion of San Diego jobs that are connected to the military (this includes jobs at bases, contractors, military R & D, etc.). You are required to draw a large-enough sample so that the sample proportion will be within 1.5% of the true value 90% of the time. a. Suppose you have no information about this proportion and that it costs $5 to contact each person in your sample. What is the least amount of money you can spend to meet your requirements? b. Your boss is horrified by the cost estimate in part a. You decide to do some Google searching to get an estimate for the proportion. At www.sdmac.org, you see that 1 in 5 jobs in San Diego is linked to the military sector. Since this is a pro-military group, you figure this number can serve as an upper bound on the true proportion. What is the new minimal cost estimate?

Answer 1

a.

Least amount of money can be spent, when we have the minimum sample size to have margin of error 1.5% with 90% confidence interval.

Margin of error = 1.5% = 0.015
z value for 90% confidence interval is 1.645
As, we do not know the estimates of p, we take conservative approach to have p = 0.5 which will give the required sample size to have margin of error 1.5% with 90% confidence interval.

Minimum sample size = (z / E)² * p(1-p) = (1.645 / 0.015)² * 0.5 * (1 - 0.5) = 3007 (Rounded to nearest integer)

Least amount of money = 3007 * $5 = $15035

b.

Assuming that the upper limit of true proportion is 1/5 = 0.2

So, the true proportion would be estimated as difference between upper limit and margin of error = 0.2 - 0.015 = 0.185

Now,

Minimum sample size = (z / E)² * p(1-p) = (1.645 / 0.015)² * 0.185 * (1 - 0.185) = 1813 (Rounded to nearest integer)

New minimal cost estimate = 1813 * $5 = $9065