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I have a sample of $n=100$ counties and I'd like to estimate a total number of veterans in a country (U.S.A.) of 3414 counties. To do that I use ratio estimate. The sample is drawn eith simple random sampling method. The total population is known and the total population in each county is used as an auxiliary variable. To estimate the total population of veterans, I use this formula: $\hat{\tau}_r = \tau_x \cdot \frac{\bar{y}}{\bar{x}}$, where $\tau_x$ is total population in U.S., $\bar{y}$ is the mean value of veterans per county and $\bar{x}$ is mean value of total population per county. The estimated ratio coefficient $\hat{r} = \frac{\bar{y}}{\bar{x}}$. Then I compute the sample deviation $s^2_r = \frac{1}{n-1}\sum^n_i (x_i - ry_i)^2$ which comes to be a very large number (e+11).Then I calculate Variance of $\tau_r: Var(\tau_r) = \frac{\tau_x}{\bar{x}}(\frac{\tau_x}{\bar{x}} - n)\frac{s^2_r}{n}$, which also is a humungous number (e+16) and then $SE(\tau_r) = \sqrt{Var(\tau_r)}$ which is a number larger than the estimate of the total population of veterans. Where have I gone wrong?

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This can be tricky, as you have to be very careful exactly which values are being averaged. You seem to be trying to apply the principles on this page, from a Penn State course on sampling theory. As you don't provide data, I use 2018 data on Veterans and US population by county for all 3142 counties or equivalents, obtained as explained here.

I took a random sample of $n$ = 100 counties. I got a mean Veterans per county of 3761.9 ($= \bar y$), and a mean population per county of 52663.4 ($= \bar x$). The ratio $\hat r$ was 0.07143291. The variance estimate $s^2_r$ calculated by your formula is $2,126,099$, much lower than the e+11 you report (evidently where you are going wrong). Note that this is the variance for the per-county number of veterans based on the ratio estimate. It's less than one-tenth of the among-county variance in veteran population in this 100-county sample, $29,977,614$, showing the advantage of the ratio estimate.

The total US population in this 2018 data set is $\tau_x = 326,691,703$. With the above ratio estimate $\hat r$, you predict $23,336,539$ veterans in the total population ($=\tau_r$). Proceeding to the formula for the estimated variance in $\tau_r$, I get $805 x 10^9$, with a square root of about $897,000$. That standard error is only about 4% of the estimated total veteran population, not too bad.

This estimate is much too high, however. The total Veteran population in this data set is $20,163,806$, with the difference of $3,172,733$ (below the ratio estimate) representing 3.5 standard errors of the estimate. The actual 2018 ratio of US Veterans to total population is $0.0617$. Over 999 random samples of 100 counties each, the mean ratio was $0.0634$, and 63% of samples overestimated the true ratio. This bias seems to result from a tendency toward higher veteran percentages in counties with smaller populations. Although I favored a ratio estimate in another related answer, you might find that a regression estimate helps counteract that bias even if it means "predicting" non-zero Veteran populations in counties with 0 population.

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