# Approximation for the sampling error of the number of positive cases in a Bernoulli trial

Reading the book "Energy for Future Presidents" I found a way of approximating the binomial proportion sampling error which I never saw before, and I would like to know if my derivation is correct.

## Notes, page 307, Chapter 1, Note 1:

If 4400 cancers are expected, ordinary statistical fluctuations are plus or minus 66 (the square root of 4400) for one standard deviation [..]

In the main text, we get the additional information that the whole population consists of 22000 individuals, and that the number 4400 is a rate/frequency:

[..] the entire population [..] about 22000 people [..] Even without the accident, the cancer rate is about 20% of the population, or 4400 cancers

This seems to me a textbook example of estimating a probability from an iid Bernoulli sample. The sampling error is (using the Wald confidence interval for simplicity, even though it's not a great choice):

$$\bar{\sigma}=\sqrt{\frac{\hat{p}(1-\hat{p})}{N}}=\sqrt{\frac{0.2(1-0.2)}{22000}}\approx2.7\cdot 10^{-3}$$

Thus a 1-standard deviation confidence interval for the number of cancers would be

$$4400\pm\bar{\sigma}N=4400\pm 59$$

The number is close to the 66 quoted by the book. I guess the author (a renowned Berkeley physicist) used the following approximation:

$$\bar{\sigma}N=\sqrt{\frac{\hat{p}(1-\hat{p})N^2}{N}}$$

When $$\hat{p}$$ is small with respect to 1 (ironically, the very case in which the Wald confidence interval gives its worst!), we can neglect the $$\hat{p}^2$$ term in the numerator and get

$$\bar{\sigma}N\approx\sqrt{\frac{\hat{p}N^2}{N}}=\sqrt{\frac{\hat{n}N^2}{N^2}}=\sqrt{\hat{n}}$$

Is this correct? Is there a more rigorous way to prove this approximation?

• Maybe by using a Poisson distribution. If $X \sim \mathsf{Pois}(\lambda = 4400),$ then $E(X) = Var(X) = 4400$ and $SD(X) = 66.3.$ A 95% Wald CI for $\lambda,$ based on observing $X = 4400$ would be $4400 \pm 1.96\sqrt{4400}.$ For numbers this large the (asymptotic) Wald interval isn't bad. However, physicists often prefer, as you say, 'one SD' intervals. – BruceET Aug 24 at 8:12
• @BruceET nice. If there aren't any questions about proving the Poisson approximation to the Binomial distribution, you could expand your comment into an answer. I'd accept it gladly. – DeltaIV Aug 24 at 11:29
• Thanks. The remaining issue to be resolved, if we are use a Poisson model, is to specify the time period to which the 'rate' 4400 refers. Hope it's not an annual rate. (Because this is science 'for Presidents', maybe it's 4 or 8 years, times spans they easily understand.) Probably more like 50 yrs. – BruceET Aug 24 at 17:30
• @BruceET no time interval is mentioned. Since the discuss is in the context of the effects of the Fukushima meltdown, I assume the author is thinking in terms of lifespans of the Fukushima area population. However, I don't know about the age distribution in that area, this I don't know which an average remaining life expectation would be. – DeltaIV Aug 25 at 7:43