Construct confidence interval from proportionsSuppose you have a population of count data, i.e., 1,2,3,…,k, you have a sample of the population of size n, and you have a confidence interval for the proportion of 1's , 2's,dotsn's in the population. Is it possible to construct a confidence interval for the sum of the counts in your sample.For example,Suppose you have a sample of size 1000 from a population of count data that can take any value from 1 to 5 and the95% CI for 1's is [.75,.76]95% CI for 2's is [.12,.13]95% CI for 3's [0.04,.045]95% CI for 4's is [.048,.058]95% CI for 5's is [.03,.035]Can I construct a 95 percent confidence interval for n μ, where n = 1000.

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Construct confidence interval from proportionsSuppose you have a population of count data, i.e., 1,2,3,…,k, you have a sample of the population of size n, and you have a confidence interval for the proportion of 1&#039;s , 2&#039;s,dotsn&#039;s in the population. Is it possible to construct a confidence interval for the sum of the counts in your sample.For example,Suppose you have a sample of size 1000 from a population of count data that can take any value from 1 to 5 and the95% CI for 1&#039;s is [.75,.76]95% CI for 2&#039;s is [.12,.13]95% CI for 3&#039;s [0.04,.045]95% CI for 4&#039;s is [.048,.058]95% CI for 5&#039;s is [.03,.035]Can I construct a 95 percent confidence interval for   n  μ, where   n  =  1000.

Abdiel Nelson · Accepted Answer

Step 1Roughly, yes. For example, take the CI for the proportion of 4&#039;s. It is centered at 0.053. If these are the &#039;usual&#039; kind of CIs used for a sample of size   n  =  1000  ,, then the number of 4&#039;s in the sample must have been   0.053  (  1000  )  =  53.Step 2In some cases, because the endpoints are rounded, you will not get an integer answer, but you can come very close to finding the frequency       f    i   with which each of the digits i for   i  =  1  ,  …  ,  5 appeared among 1000. From that information you can construct a CI based on the whole sample nearly as well as if you had the original data.Just what it means to get a CI for the &quot;sum&quot; is unclear. Ordinarily, in such a situation one would want a CI for the population mean. Maybe your final sentence should have been &quot;construct a CI for the population mean   μ based on the sum of the counts.&quot; (For samples of size n, the expected total would be   n  μ  .. You could get a CI for that.)

Suppose you have a population of count data, i.e., 1,2,3,…,k, you have a sample of the population of size n, and you have a confidence interval for the proportion of 1's , 2's,\dotsn's in the population. Is it possible to construct a confidence interval for the sum of the counts in your sample.

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