estimate population sum given population size and a sample mean and variance












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I have a question which seems easy but I'm not sure how to solve it.



Assume we have a population of $N$ integer numbers, $x_1, x_2, ..., x_N$. So $N$ is population size. I'm interested in the sum of these numbers (i.e. $S = sum_i^N x_i$), however, $N$ is large so I'm going to take $K$ samples randomly (uniform) and compute the sample sum as $m$ and the sample variance as $sigma^2$.



What is the most accurate estimate (with bounds/confidence-interval/expected-deviation, etc.) we can have for $S$ given $m$, $sigma^2$, $K$, and $N$?



The important thing is that I want to use all of this information (i.e., $K$, and $N$ and $m$ and $sigma^2$) to make the estimate more accurate with smaller bounds.
I don't have any assumptions about the population distribution.



Update:
Since no one has answered this, I'm going to explain the background: What I actually want to do is to count the total number of words with a particular characteristic in a book with $N$ (e.g. 3000) pages. It's really time-consuming to find them so I plan to count such words only in $K$ (e.g. 30) pages. Given these $K$ numbers, I want to find the total number of such words in the book. I assume the answer will be the sum of these numbers scaled by $N/K$ but what are the error bounds?










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    1












    $begingroup$


    I have a question which seems easy but I'm not sure how to solve it.



    Assume we have a population of $N$ integer numbers, $x_1, x_2, ..., x_N$. So $N$ is population size. I'm interested in the sum of these numbers (i.e. $S = sum_i^N x_i$), however, $N$ is large so I'm going to take $K$ samples randomly (uniform) and compute the sample sum as $m$ and the sample variance as $sigma^2$.



    What is the most accurate estimate (with bounds/confidence-interval/expected-deviation, etc.) we can have for $S$ given $m$, $sigma^2$, $K$, and $N$?



    The important thing is that I want to use all of this information (i.e., $K$, and $N$ and $m$ and $sigma^2$) to make the estimate more accurate with smaller bounds.
    I don't have any assumptions about the population distribution.



    Update:
    Since no one has answered this, I'm going to explain the background: What I actually want to do is to count the total number of words with a particular characteristic in a book with $N$ (e.g. 3000) pages. It's really time-consuming to find them so I plan to count such words only in $K$ (e.g. 30) pages. Given these $K$ numbers, I want to find the total number of such words in the book. I assume the answer will be the sum of these numbers scaled by $N/K$ but what are the error bounds?










    share|cite|improve this question











    $endgroup$















      1












      1








      1


      1



      $begingroup$


      I have a question which seems easy but I'm not sure how to solve it.



      Assume we have a population of $N$ integer numbers, $x_1, x_2, ..., x_N$. So $N$ is population size. I'm interested in the sum of these numbers (i.e. $S = sum_i^N x_i$), however, $N$ is large so I'm going to take $K$ samples randomly (uniform) and compute the sample sum as $m$ and the sample variance as $sigma^2$.



      What is the most accurate estimate (with bounds/confidence-interval/expected-deviation, etc.) we can have for $S$ given $m$, $sigma^2$, $K$, and $N$?



      The important thing is that I want to use all of this information (i.e., $K$, and $N$ and $m$ and $sigma^2$) to make the estimate more accurate with smaller bounds.
      I don't have any assumptions about the population distribution.



      Update:
      Since no one has answered this, I'm going to explain the background: What I actually want to do is to count the total number of words with a particular characteristic in a book with $N$ (e.g. 3000) pages. It's really time-consuming to find them so I plan to count such words only in $K$ (e.g. 30) pages. Given these $K$ numbers, I want to find the total number of such words in the book. I assume the answer will be the sum of these numbers scaled by $N/K$ but what are the error bounds?










      share|cite|improve this question











      $endgroup$




      I have a question which seems easy but I'm not sure how to solve it.



      Assume we have a population of $N$ integer numbers, $x_1, x_2, ..., x_N$. So $N$ is population size. I'm interested in the sum of these numbers (i.e. $S = sum_i^N x_i$), however, $N$ is large so I'm going to take $K$ samples randomly (uniform) and compute the sample sum as $m$ and the sample variance as $sigma^2$.



      What is the most accurate estimate (with bounds/confidence-interval/expected-deviation, etc.) we can have for $S$ given $m$, $sigma^2$, $K$, and $N$?



      The important thing is that I want to use all of this information (i.e., $K$, and $N$ and $m$ and $sigma^2$) to make the estimate more accurate with smaller bounds.
      I don't have any assumptions about the population distribution.



      Update:
      Since no one has answered this, I'm going to explain the background: What I actually want to do is to count the total number of words with a particular characteristic in a book with $N$ (e.g. 3000) pages. It's really time-consuming to find them so I plan to count such words only in $K$ (e.g. 30) pages. Given these $K$ numbers, I want to find the total number of such words in the book. I assume the answer will be the sum of these numbers scaled by $N/K$ but what are the error bounds?







      statistics estimation standard-deviation






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      edited Dec 3 '18 at 12:11







      Hossein

















      asked Dec 2 '18 at 16:37









      HosseinHossein

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