Confidence interval on sum of estimates vs. estimate of whole?

Question

We want to know the mean income of the population but the statistics provided to us are divided into four groups, North, South, East, and West. We do not know the sample size for the four groups, only the estimated group populations, incomes, and standard errors.

If possible, how do we estimate the mean and standard error for the total population?

.Group    .Count    .Income  .Standard Error
North     10,000    100      10
South     20,000    80       10
East      20,000    150      12
West      10,000    120      15
TOTAL     60,000    ??       ??

Thanks.

[May be related to Confidence interval for sum of means which has an answer with a "-2"]

Answer 1

Work with the sums instead of the means.

To do this, convert facts like

the mean income for the 10,000 people in the North is 100 plus or minus 10

to

the total income of the 10,000 people in the North is 100*10,000 = 1,000,000 plus or minus 10*10000 = 100,000.

The plus or minus is the standard deviation of the estimated total. Its square is the variance. Variances of independent estimates add. In this case it's reasonable to suppose the four subsamples are independent, because they sample disjoint groups.

The resulting table of information about the sums is this:

Region  Population Total income  Standard deviation  Variance
North   10,000     1,000,000      100,000            1    E10
South   20,000     1,600,000      200,000            4    E10
East    20,000     3,000,000      240,000            5.76 E10
West    10,000     1,200,000      150,000            2.25 E10
-----   ------     ---------      -------           ---------
Total   60,000     6,800,000                        13.01 E10

Now convert back:

The estimated total income is 6,800,000 for a population of 60,000, or 113 per person.

Similarly,

The standard deviation of the estimated total is the square root of 13.01E10, approximately 2168333. The SD of the mean estimate is this value divided by the total population, resulting in 6.0.

(I retained artificially high significance during the calculation, to assure no loss of precision, but rounded to reasonable precision at the end.)

The answer therefore is the mean per capita income is estimated to be 113 with a standard error of 6.0