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After years of "abstinence" I dabble back into statistics of population proportions and have some trouble making sense of critical values $z^\star$ for high confidence levels:

The critical value $z^\star$ is needed for the calculation of confidence intervals. I compute $z^\star$ from the confidence level cl with the Python module scipy via

cdf = 0.5+cl/2
z_star = scipy.stats.norm.ppf(cdf)

This z_star (i.e. $z^\star$) can grow to infinity, because

scipy.stats.norm.ppf(1)

outputs inf.

On the other hand, I'd say the confidence interval

$CI= \hat p \pm z^\star se(\hat p)$

can reasonably not exceed $[0\%,100\%]$.

How do these things fit together?

  • Is the scipy calculation an approximation with goes astray for confidence levels close to $100\%$?
  • Or is my idea that confidence intervals not exceed $[0\%,100\%]$ not general enough?
  • Or is something with my cdf calculation wrong?

I would love some pointers here!

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  • $\begingroup$ The confidence level has to be between 0 and 100, but the confidence interval can be any interval. $\endgroup$ – Dave Nov 11 '19 at 6:25
  • $\begingroup$ Why can the CI be any interval? I'd say CI can be maximally [0%,100%]. Then it captures always the population proportion $p$ and therefor CI=[0%,100%] gives us a confidence level cl =1, doesn't it? $\endgroup$ – halloleo Nov 11 '19 at 6:40
  • $\begingroup$ Oops, I hadn’t noticed that it’s a proportion. $\endgroup$ – Dave Nov 11 '19 at 11:46
  • $\begingroup$ @Dave My bad. Added this crucial bit of information to the question. $\endgroup$ – halloleo Nov 12 '19 at 0:29
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At the end there you're using a normal approximation to a binomial proportion to obtain a confidence interval. The normal is continuous and has an infinite range while the binomial proportion is discrete and has a finite range; if you go sufficiently far into the tail of the normal you can go outside the possible range of the binomial parameter, $p$.

As the Wikipedia article Binomial proportion confidence interval puts it:

A commonly used formula for a binomial confidence interval relies on approximating the distribution of error about a binomially-distributed observation, $\hat p$, with a normal distribution. This approximation is based on the central limit theorem and is unreliable when the sample size is small or the success probability is close to 0 or 1.

Here's an illustration of the problem:

plot of cdf of the binomial(10,0.1) proportion, X/10 with normal approximation with the same mean and variance, showing a distinct amount of the left tail of the normal below 0

If that occurs, you're outside the region of values for $p$ and $n$ where the approximation is good for those percentiles. Indeed you would usually want to stay a fair way from the 0/1 endpoints (a fairly large distance in terms standard errors of the proportion, though the absolute distance from the end might be small in a large sample)

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  • $\begingroup$ Thanks for this. Your pointer helped me to read up on it on Wikipedia, etc. It makes sense now. $\endgroup$ – halloleo Nov 12 '19 at 0:37
  • $\begingroup$ Sorry to lose your suggested edit: I was editing it myself at the same time and posting my edit meant yours was lost. I will put your edit in, though. (Edit: now done) $\endgroup$ – Glen_b -Reinstate Monica Nov 12 '19 at 1:05
  • $\begingroup$ No problem. Your graph helps a lot! Thanks! $\endgroup$ – halloleo Nov 12 '19 at 1:13

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