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Say out of a sample of 200 penguins, I find that 192 of them like chocolate. From this sample data, I create the $95\%$ confidence interval for the proportion of penguins that like chocolate:

$$ \left(\frac{192}{200} - 1.96 \times \sqrt{\frac{(\frac{192}{200})(\frac{8}{200})}{200}}, \frac{192}{200} + 1.96 \times \sqrt{\frac{(\frac{192}{200})(\frac{8}{200})}{200}}\right) \\ (0.933, 0.987) $$

Why would it be invalid for me to subsequently say that I am $95\%$ confident that the percentage of penguins that like chocolate is above $93.3\%$ and below $98.7\%$?

Brian M. Scott
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David Faux
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  • I think that the mean percentage of penguins who like chocolate is between 93.3 and 98.7 percent. My stats is a bit hazy though.. – Daryl Oct 23 '13 at 21:49
  • I don’t consider it an invalid paraphrase, provided that you understand what a confidence interval actually is and are reasonably sure that your listeners do, too. The problem is that many people take it to be a statement about subjective confidence, which it isn’t. (It definitely isn’t for people like me: I don’t consider ‘I’m $95$% confident that X’ as a statement of subjective confidence to mean anything different from ‘I’m very confident that X’.) – Brian M. Scott Oct 24 '13 at 15:11

1 Answers1

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The confidence interval does not rely on any prior probability distribution for the fraction of penguins ($p$) that like chocolate. You may have thought it extremely unlikely that penguins would like chocolate; if so, you will still deem it unlikely (albeit less so) that the true percentage is between $93.3\%$ and $98.7\%$, even after doing this experiment. Your confidence that $p$ lies in that interval (that is, your posterior probability) could still be low.

The confidence interval describes something different. A confidence interval is a rule that assigns an interval to every possible result of your experiment, such that whatever the true value of p, it will fall in the assigned interval $95\%$ of the time.

The difference is fairly subtle, but it can be summarized this way. A confidence interval is not a particular interval that almost surely (given the randomness associated with $p$) contains the correct answer, now that the experiment has been done. It is a rule, formulated before the experiment was done, that was almost sure (given the randomness associated with the experiment) to generate an interval containing the correct answer.

mjqxxxx
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