I am going through the book Sampling Techniques by Cochran. I'm requesting for assistance with the following question:

A simple random sample of 30 households was drawn from a city area containing 14,848 households. The numbers of persons per household in the sample were as follows.


Estimate the total number of people in the area and compute the probability that this estimate is within ±10% of the true value.

I found the estimate of the population total to be 51,473 with a standard error of 3,315.4869. I'm having trouble interpreting the second part of the question ("compute the probability that this estimate is within ±10% of the true value") as a mathematical expression.

  • $\begingroup$ What have you done so far? Show your work $\endgroup$
    – Aksakal
    Oct 14, 2015 at 13:56

2 Answers 2


Get the mean of this sample: $\bar x=3.4667$

Knowing the number of households $n=14848$, the population estimate is $\hat\mu=n\bar x=51473$

The second part is interesting. He gave you a sample of 30. This is a peculiar number. It's the common threshold after which it's customary to assume that a sum of random variables would be normal. So, he might be giving you a hint to assume normal distribution for the sample mean. Then all you need is to compute $$F\left(\frac{0.1 \hat\mu}{\sigma_{\mu}}\right)-F\left(\frac{-0.1 \hat\mu}{\sigma_{\mu}}\right)\approx 0.88$$

Here, you're stepping left and right from the mean by 10% of its value $\hat\mu$, then normalizing the step size by the standard deviation of a sample mean, which is easy to estimate: $\sigma_{\hat\mu}=n\sigma_x/\sqrt{30}$

Then you use standard normal CDF to get the probability between the left and right step.


This question seems tricky because I count three separate issues.

First, suppose you knew the true number of persons in the city $t$, and the true distribution on number of persons per household, $H$. Then, knowing that there are 14,848 households, you could compute the probability of ending up with $n$ persons in total $P_H(n)$, and integrate $P_H(n)$ over the range $.9t$ to $1.1t$.

Unfortunately, we don't actually know $t$. Fortunately, we just calculated a distribution over $n$ that also works as a distribution over $t$, so we can do a double integration, over both $dt$ and $dn$.

Unfortunately, we don't actually know $H$, which we would use to calculate the distribution over $n$. Fortunately, we have a distribution over $H$ which we can calculate from the sample, and so we can do a triple integration, over $dH$, $dt$, and $dn$.

But... is the question really asking you to do triple integration? It's likely that at least one of these steps is ignored. Most statistical computations I see ignore the uncertainty in $H$ (or the analogous distribution estimated from a sample). Or they ignore that the statement "$x$ is within 10% of $x^*$" is different from the statement "$x^*$ is within 10% of $x$," which makes the first integral considerably easier. (Then it's just computing the integral from the MLE minus 10% to the MLE plus 10%, which is two CDF lookups.)


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