Divide the population of U.S. workers into two categories: those who have a job in the goods-produci

anginih86

anginih86

Answered question

2022-06-11

Divide the population of U.S. workers into two categories: those who have a job in the goods-producing industries, and those who have a job in the service-producing industries. Among all workers, sixteen percent have a job in the goods-producing industries. For these workers, the mean of annual earnings equals $ 42 , 419, with a standard deviation of $ 35 , 572. Among all workers, the remaining eighty-four percent have a job in the service-producing industries. For these workers, the mean of annual earnings equals $ 36 , 976, with a standard deviation of $ 31 , 247. Find the mean and standard deviation of annual earnings among all workers in the population.

Is this the correct answer:

Pop mean ( .16 42 , 419 ) + ( .84 36 , 976 ) = 37 , 846.88
Pop StdDev ( .16 35 , 572 ) + ( .84 31 , 247 ) = 31 , 939

Answer & Explanation

timmeraared

timmeraared

Beginner2022-06-12Added 22 answers

Let X be the income of a randomly chosen worker in the goods-producing sector, and let Y be the income of a randomly chosen worker in the service-producing sector. Then the income W of a randomly chosen worker is given by
W = ( 0.16 ) X + ( 0.84 ) Y .
It follows that
E ( W ) = ( 0.16 ) E ( X ) + ( 0.84 ) E ( Y ) .
If we (reasonably) assume independence then
Var ( W ) = ( 0.16 ) 2 Var ( X ) + ( 0.84 ) 2 Var ( Y ) .
For the standard deviation calculation, use the fact that the standard deviation is the square root of the variance.
cazinskup3

cazinskup3

Beginner2022-06-13Added 6 answers

Your calculation for the mean is fine. But you can't just take a weighted average of standard deviations and expect to get the standard deviation for the population.
You need the square root of the pooled variance.

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