The contractor has captured 802 stallions, 880 mares and 217 foals as of February 9, according to figures at the gather page, with youngsters representing 11.4% of the total.
If the death rate is 5% per year, the birth rate would need to be around 25% per year to achieve a 20% growth rate, the amount commonly used by land managers to predict herd sizes.
The observed percentage of foals is not even close to 25%, so you might be tempted to say the herd is not growing at a rate of 20% per year, but technically it’s not correct.
You have to account for random variations in the sample results.
If you tossed a coin 100 times and found 42 heads, the result would be attributed to chance. It’s not far enough away from 50, the expected number of heads, to be an indication of anything.
You could get anywhere from 35 to 65 heads in the experiment. A result in that range provides no evidence that the process is not centered at 50% heads / 50% tails.
The expected range of variation for other cases can be computed from basic statistical formulas, where p-bar is the expected proportion and n is the sample size.
In the Pancake Complex, the hypothesized proportion (birth rate) is .25 and the sample size is 1,899. Every horse in the sample is a foal or it is not.
Is the observed proportion, .114, far enough away from .25 to cast doubt on the assumed birth rate?
The range of variation attributable to chance, using the formula above, is .220 to .280, with p-bar = .25 and n = 1,899.
The observed proportion is outside this range, meaning that it is sufficiently far enough away from .25 to suggest that the assumed birth rate is not correct, which means the herd is probably not growing at a rate of 20% per year.
If the herd was growing at a rate of 15% per year, you expect to find around 20% foals, assuming a 5% death rate.
The expected range of variation would be .172 to .228, with p-bar = .20 and n = 1,899.
The observed proportion of .114 is outside this range also, suggesting that a birth rate is of 20% per year is not valid, along with a herd growth rate of 15% per year.
What about a growth rate of 10% per year? The observed proportion is not consistent with that either!