# Sample size to find a death rate

I need help, please, in working up a sample size. I’m working on a real project to track the incipient famine in Afghanistan by estimating crude death rates over time using survey data gathered from a sample of mosques. Enumerators will visit mosques on a monthly basis and ask the mullah of each how many of his congregants have died since the enumerator’s last visit, and how many congregants there are. The sum of the deaths divided by the sum of the congregation sizes will give me my estimate of the overall population’s crude death rate. I don’t have enough information to segment my sample, or even to make my selection of mosques dependably random. (And there are a thousand other things that could fowl me up, I know, but I’m just looking at the sample size question for now.)

The metro area I’m looking at has an overall population over a million. The crude death rate for Afghanistan was estimated at 13.6 deaths per thousand per year for 2016. To identify a famine, the IPC, a UN affiliated group, uses a threshold rate of 2 deaths per 10,000 per day (=70 deaths per thousand per year).

I’ve tried to use Cochran’s formula to come up with a sample size: Sample size = $$Z^2* p * (1-p) / M^2$$

Where:

Z is the z score of 1.96 for a 95% confidence level,

p is the population proportion. I’m setting it at .001178. I got that by converting the annual crude death rate for Afghanistan in 2016 of 13.6 per thousand to a probability figure for 30 days $$(0.001178 = 13.6 / 1000 / 360 * 30)$$ .

M is the ‘confidence interval’ or 'margin of error', within which I want my estimate to fall 95% of the time. I’m putting it at .10 for 10%.

So I do the math, but my answer seems a little low, even when I round it up to 1.

$$1.962^2 * 0.001178 * (1-0.001178) / 0.1^2= 0.428934049$$

I have tried an alternative approach of writing a hundred thousand rows on a spreadsheet and assigning “life” or “death” randomly to each, based on that tiny daily crude death rate of 0.001178, and then randomly sampling 10,000 rows. I run the sample repeatedly and get terribly erratic results despite the huge sample size.

I realize that friend Cochran had a better grasp of the subject than I, but I can’t see where I went wrong. Can anyone think what I’m misunderstanding, please, or steer me to a better approach?

In this formula $$M$$ is the absolute error and not the relative error. Your crude estimate of $$p = 0.001178$$ means that an error margin of $$0.1$$ is near meaningless. $$M= 0.0001$$ is a better choice which gets you a sample size of 452931. Obviously this can be made higher or lower depending on what data you have available.