Say a company runs a survey across random
N cities independently in some country estimating the fraction of males and females on each city.
- Males = $X_1$% and Females = $(100 - X_1)$% for city 1
- Males = $X_2$% and Females = $(100 - X_2)$% for city 2
Now, say that we have a independent and unbiased estimate of the fraction of males and females in the entire country, i.e. Males = $Y_T$%, Females = $(100-Y_T)$%.
Using all of this data, how can we get a potentially better estimate of the fraction of males and females in each city?
What would be a frequentist or Bayesian way of solving this problem?