# Poisson or Normal distribution?

In this problem:

The respiratory disturbance index (RDI), a measure of sleep disturbance, for a specific population has a mean of 15 (sleep events per hour) and a standard deviation of 10. They are not normally distributed. Give your best estimate of the probability that a sample mean RDI of 100 people is between 14 and 16 events per hour?

I solved the problem by using the central limit theorem, i.e. that the sample mean approaches the population mean, even if the population is not normally distributed.

> pnorm((16-15)/(10/sqrt(100)))-pnorm((14-15)/(10/sqrt(100)))
[1] 0.6826895


Can I model the problem with a Poisson distribution or would I require explicit information that the RDI follows a Poisson distribution in order to meaningfully do so?

Poisson or Normal distribution?

Possibly neither.

I solved the problem by using the central limit theorem, i.e. that the sample mean approaches the population mean, even if the population is not normally distributed.

That's not what the CLT says. What you have there -- "that the sample mean approaches the population mean" sounds like the law of large numbers, but it's not sufficient to solve this problem.

What you need to use the normal distribution is the assumption that the sample mean is near enough to normally distributed to apply normal calculations. The central limit theorem doesn't tell you that either.

There's not really enough information in the question to make much of an assessment (not even to apply Berry-Esseen which does give one specific sense of how far off the normal might be).

I presume the intent here was to apply the normal approximation. You're left simply to assume (with no good reason to do so) that the sample size is large enough in this instance to apply a normal approximation.

However, because you have a fairly short interval about the mean (only going out a standard deviation), this may work fairly well for a wide variety of cases. You might also consider whether to apply a continuity correction.

[Edit: After playing around with a variety of more-or-less plausible possibilities for the original distribution consistent with the information given (including the obvious condition that disturbances per hour must be non-negative), I think the normal approximation should generally work quite well for this specific instance. However, it's possible to construct distributions consistent with the information where that probability is much smaller.]

[There are some results for bounds on the tails of distributions of sums (e.g. Chernoff bounds), but I don't think that's of substantive help here.]

Can I model the problem with a Poisson distribution

Counts may well not be Poisson. Consider, for example, a situation where individuals have Poisson distributions, but each individual's Poisson parameter is drawn from a Gamma distribution. Then the population distribution is negative binomial.

Now if the population distribution were Poisson, one thing you should see is that the population mean and variance should be the same. However, $15 \neq 100$.

or would I require explicit information that the RDI follows a Poisson distribution in order to meaningfully do so?

You have explicit information that the population distribution of counts is not Poisson. It's not a plausible approximation.

Further note that disturbances per hour is not necessarily an integer, since presumably disturbances are counted across a period other than exactly an hour (I presume across the duration of sleep for each individual) and then scaled to 'disturbance per hour' so as to be more comparable. As such you have (differently) scaled counts for each individual and would not expect a count data model to necessarily fit well. You're effectively dealing with rates rather than counts, and so a Poisson would not be expected to fit.

• Just a note that your initial skepticism of use of the CLT seems off base to me. A sample size of 100 is enough for the sampling distribution of the mean to be nearly normal for all but the most diabolical base distributions. Jan 12, 2015 at 3:49
• @goodepic I think such skepticism may not have been intended by the question writer, though at least they only demanded an approximate "best guess" answer, which is an acknowledgement of some of the issues involved. Suppose the population of interest is a mixture divided into a majority who on average will record low scores and a minority with sleep disorders whose distribution is over much higher scores. Depending on the composition of mixture and severity of disorder (as measured by RDI) the distribution could be quite nasty. Jan 12, 2015 at 4:30
• goodepic -- your lack of skepticism seems off base to me (and the CLT itself only refers to the limit as sample sizes goes to infinity). A sample size of 100 is not adequate to use a normal approximation in many very real situations to which the CLT nevertheless applies. I deal with situations where sample sizes in the thousands are not sufficient. Jan 12, 2015 at 4:43
• You might like to consider what happens with a population where there are two (previously unidentified) sub-populations, the first with a tight concentration of rates just below 14 (~99% of the population) and a small subgroup (the remaining ~1%) with a tight distribution of rates a little above 114. If you sample from that population you'll match the conditions of the problem (population mean~15, sd~10), and the normal approximation to the mean will be very misleading. Such mixtures occur frequently enough that you should be wary of them unless you know they cannot be present. Jan 12, 2015 at 4:49