compute probabilitythat estimate of total is with in +or-10% of true value

Question

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.

   5,6,3,3,2,3,3,3,4,4,3,2,7,4,3,5,4,4,3,3,4,3,3,1,2,4,3,4,2,4 

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.

Answer 1

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.

Answer 2

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.)