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In short: I'm wondering if I'm understanding the concept of two-sided "Confidence Interval" via an example?

Details:

Suppose we observe 5 successes in 20 trials and so our observed $p = 5/20$. To obtain a 95% CI for the PARAMETER of this observed $p$, we argue this way:

What the unknown, fixed PARAMETER value of this observed $p$ could be such that out of infinitely many repetitions of a binomial experiment with $20$ trials, $95\%$ of the time we capture that PARAMETER value? Now we take two steps:

First, we find the $p$ of a binomial distribution given 20 trials which strictly allows 2.5% probability for 5 or more successes in 20 trials to happen in large repetitions (the blue binomial distribution). This $p$ can be shown (e.g., by optimization or via Clopper & Pearson) to be "$0.08657147$". This will be the smallest limit value if we were to construct a 95%-coverage uncertainty net to capture the PARAMETER value of the observed $p$. In other words, this "$0.08657147$" is the smallest possible $p$ for a binomial distribution given 20 trials that allows at least 2.5% probability for 5 success out of 20 trials to occur in large repetitions.

Second, we find the $p$ of a binomial distribution given 20 trials which strictly allows 2.5% probability for 5 or less successes in 20 trials to happen in large repetitions (the red dashed binomial distribution). This $p$ can be shown (by optimization or via Clopper & Pearson) to be "$0.49104587$". This will be the largest limit value if we were to construct a 95%-coverage uncertainty net to capture the PARAMETER value of the observed $p$. In other words, this "$0.49104587$" is the largest possible $p$ for a binomial distribution given 20 trials that allows at most 2.5% probability for 5 success out of 20 trials to occur in large repetitions.

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  • $\begingroup$ The interval will differ if you have a one-sided versus a two-sided interval. $\endgroup$ – Michael R. Chernick Jul 28 '17 at 2:45
  • $\begingroup$ I agree with your second paragraph. The interval you show is described correctly for a two-sided equal tailed interval. The figures you show are confusing. I don't see why you produce two beta curves and also don't understand what the two binomial distributions are for. $\endgroup$ – Michael R. Chernick Jul 28 '17 at 3:29
  • $\begingroup$ @MichaelChernick, that's the Clopper & Pearson method, that is what the $Beta$ distributions are for. $\endgroup$ – rnorouzian Jul 28 '17 at 3:39
  • $\begingroup$ I think there is only one distribution for which you determine upper and lower tails. Yes the Clopper-Pearson method is one way to calculate exact binomial probabilities. Also two tailed 95% confidence intervals do not need to be equal tailed. The definition allows say 1% in the lower tail and 4% in the upper (for example) rather than 2.5% in each tail. $\endgroup$ – Michael R. Chernick Jul 28 '17 at 12:32
  • $\begingroup$ @MichaelChernick, I assume by "there is only one distribution for which you determine upper and lower tails", you 're referring to "Normal Approximation" which due to the symmetry of a normal distribution doesn't require two distribution. But I did not ask about that approximation. Furthermore, generally speaking, one-distribution method has limited generality as many important statistics follow asymmetrical distributions for which an approximation is not known. One thing I do want to know, though, how a $Binomial$ distribution can directly be converted back to a corresponding $Beta$ ... $\endgroup$ – rnorouzian Jul 28 '17 at 12:55
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This image may help your understanding:

Two-sided vs one-sided intervals

Essentially for a two-sided confidence interval, it ranges from an upper to a lower limit, with 2.5% chance on both sides.

A one-sided confidence interval exists when you only look at values above OR below a certain null value (the value that satisfies the null hypothesis). The 95% CI, and the 5% outside the interval, are both above or both below the null value.

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