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Say I've looked at three random samples, one each from three populations of students. I found that a few students in each sample didn't turn in a fieldtrip permission slip, leaving me with the following point estimates:

| Population name | Population size | Sample size | Missing Slips (Sample) | Estimated Missing Slips (Population) |
| --------------- | --------------- | ----------- | ---------------- |---------------- |
| 6th graders     | 1,200           | 200         | 2|12 |
| 7th graders     | 800             | 100         | 5|40 |
| 8th graders     | 1,000           | 200         | 4|20 |

Can I say that an estimated 72 of 3,000 (2.4%) 6th, 7th, and 8th graders didn't turn in field trip permission slips? If I can, how do I go about calculating a confidence interval for my estimate?

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    $\begingroup$ Of the $3000$ students, $2928$ returned the permission slips and $72$ did not, right? I can't see a way to say anything other than $2.4\%$ of the students failed to return the permission slips. // What information would you hope to gain from the confidence interval? Confidence intervals only make sense when you want to use collected data to infer something about some larger set of data or process from which the observations were drawn. $\endgroup$
    – Dave
    Commented Sep 10, 2021 at 16:24

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As Dave correctly points out in the comments, if your goal is merely to describe the sample then you can compute describtive statistics (e.g., the proportion of students who didn't turn in permission slips) and state those descriptive quantities without any uncertainty. Confidence intervals and other inference procedures only arise when we want to use the data to draw conclusions about quantities in larger populations.

So in the present case, you wouldn't say that "an estimated 72 of 3,000 (2.4%) didn't turn in field trip permission slips" --- you would just say that "72 of 3,000 (2.4%) didn't turn in field trip permission slips". If, on the other hand, you have in mind the task of estimating a quantity in some larger population, you will first need to specify what inference you are trying to make ---i.e., what larger population is of interest here?

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You can think of this as a Bernoulli trial process in which returning the slip is "success" and not returning the slip is "failure". Your estimate of the failure rate is:

$ \hat P_{failure} = n_{failure}/sample\,size = 2.4\%$

You can take this observed value to calculate a confidence interval for the probability of failure in the population using the following formula (95% confidence interval, z = 1.96):

$ \hat P_{failure} \pm z \sqrt{\hat P_{failure}(1-(\hat P_{failure}))/sample \, size} = 2.4\% \pm 1.3\% $

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  • $\begingroup$ I think you might be using the term "representative" in a non-standard way. "Large sample size" does not equate to "representative." Bootstrapping does not make the data any more or less representative, either. Regardless, there appears to be no call for bootstrapping in this simple situation: why not recommend the textbook Binomial confidence interval? $\endgroup$
    – whuber ♦
    Commented Sep 10, 2021 at 21:24
  • $\begingroup$ Yes, that is correct! I will revise the answer accordingly. $\endgroup$
    – Amir H
    Commented Sep 10, 2021 at 21:36
  • $\begingroup$ See this site search for a better solution. $\endgroup$
    – whuber ♦
    Commented Sep 12, 2021 at 19:35

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