# Combining two estimates

The number of participants in a particular sport in the USA in 2014 was estimated by a research group at 3,000,000. This estimate came with a standard error. The number of hospitalizations associated with that sport in 2014 was estimated by another research group at 15,000 and this estimate came with a coefficient of variation. I wish to combine these estimates and to estimate the likely number of hospitalizations per 100,000 participants, with a 95% CI. How do I go about this?

You are asking about pooled means and variances. As far as I understand, the data you have is two means $m_1$, $m_2$ and two sample sizes $n_1$, $n_2$. Variances $s_1^2$ and $s_2^2$ can be easily obtained from standard errors ($\mathrm{SE} = s_i / \sqrt{n_i}$) and coefficient of variation ($\mathrm{CV} = s_i / m_i$) since you have all the needed information. Having all this information you can obtain pooled mean from $k$ sources

$$m_p = \frac{\sum_{i=1}^k n_i m_i}{\sum_{i=1}^k n_i}$$

and pooled variance

$$s_p^2 = \frac{ \sum_{i=1}^k (n_i - 1) s_i^2 }{ \sum_{i=1}^k (n_i - 1) }$$

Having this information you can easily calculate confidence intervals.

The question that you have to ask yourself is if the information from both sources equally reliable? Simple pooled estimates use information about sample sizes, but if their reliability differs then you may need to account for it by using weighted mean with weights proportional to their reliability (see here, here, here, or here for learning more about combining estimates in different areas).