Adjusting percentages for local vaccination rate

This is an answer to a question on Stats Overflow.

I want to estimate the probability of a person aged 40-49 in Delaware to be vaccinated, but I only have nationwide statistics on vaccination levels by age, and a level of vaccination in Delaware, but no age breakdown for that state. So the question is, how can we combine those percentages, for the agegroup and Delaware, into a specific percentage for that agegroup in Delaware?

I tried doing that in my

previous

post, but, I came up with a more straightforward method.

To begin, from the official statistics, we get the percentage of people vaccinated in Delaware, which is 56.6%. Let \(D\) be the total population of Delaware. Then there are \(0.566 \cdot D\) vaccinated persons in Delaware.

The number of people in the US in the age group 40-49 is 12.2%. But they make up 14.2% percent of the people vaccinated. Let's assume these percentages hold in Delaware as well.

Then the total number of people aged 40-49 living in Delaware is \(0.122\cdot D\). And the number of people vaccinated aged between 40-49 is 14.2% of vaccinated subjects. So the final percentage is

\[ \frac{0.142 \cdot 0.566 \cdot D}{0.122 \cdot D} = \frac{.142 \cdot .566}{0.122} \approx 65.9\% \]