Re: normal approx. to binomial
James Ankeny wrote: > [snip] >Please tell me if this is way off, but when they say > that a binomial rv may be normal for large n, it seems like this would only > be true if they were talking about a sampling distribution where repeated > samples are selected and the number of successes calculated. A single observation may be normally distributed. Similarly, a single binomial random variable, (with large enough n, and success probability not too small) has a distribution that is approximately normal. Of course "approximately" must be predicated on the fact that the distribution is still going to be discrete. It may be easier to picture if you think in terms of the distribution of the sample proportion Glen = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: normal approx. to binomial
In article <[EMAIL PROTECTED]>, Jay Warner <[EMAIL PROTECTED]> wrote: >one tech issue, one thinking issue, I believe. >1) Tech: if np _and_ n(1-p) are > 5, the distribution of binomial >observations is considered 'close enough' to Normal. So 'large n' is >OK, but fails when p, the p(event), gets very small. Close enough by whom, and for what? The approximation is better for p=.5, but the dominant term in the error is O((1-2p)/sqrt(pqn)); there is on O(1/n) term which becomes more important for p near .5. So unless p is quite close to .5, it takes 100 times as many observations to get one more decimal place of accuracy. Even if p = .5, it will take 10 times as many. This is especially important in the tails. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 [EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558 = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: normal approx. to binomial
You may be interested in an applet I have on my website demonstrating the normal approximation to the binomial. <http://www.ruf.rice.edu/~lane/stat_sim/normal_approx/index.html> --David > From: [EMAIL PROTECTED] (James Ankeny) > Organization: None > Newsgroups: sci.stat.edu > Date: 9 Apr 2001 14:41:31 -0700 > Subject: normal approx. to binomial > > Hello, > I have a question regarding the so-called normal approx. to the binomial > distribution. According to most textbooks I have looked at (these are > undergraduate stats books), there is some talk of how a binomial random > variable is approximately normal for large n, and may be approximated by the > normal distribution. My question is, are they saying that the sampling > distribution of a binomial rv is approximately normal for large n? > Typically, a binomial rv is not thought of as a statistic, at least in these > books, but this is the only way that the approximation makes sense to me. > Perhaps, the sampling distribution of a binomial rv may be normal, kind of > like the sampling distribution of x-bar may be normal? This way, one could > calculate a statistic from a sample, like the number of successes, and form > a confidence interval. Please tell me if this is way off, but when they say > that a binomial rv may be normal for large n, it seems like this would only > be true if they were talking about a sampling distribution where repeated > samples are selected and the number of successes calculated. > > > > > > > ___ > Send a cool gift with your E-Card > http://www.bluemountain.com/giftcenter/ > > > > > = > Instructions for joining and leaving this list and remarks about > the problem of INAPPROPRIATE MESSAGES are available at > http://jse.stat.ncsu.edu/ > = = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: normal approx. to binomial
[EMAIL PROTECTED] (Jason Owen) writes: >[EMAIL PROTECTED] (James Ankeny) writes: >[snip] >>Typically, a binomial rv is not thought of as a statistic, at least in these >>books, but this is the only way that the approximation makes sense to me. >Actually, the binomial rv is the sufficient statistic for the data, >which are represented as the sequence of 0/1's from the Bernoulli >trials. Cripes... I meant that the binomial rv is the sufficient statistic for p. = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: normal approx. to binomial
[EMAIL PROTECTED] (James Ankeny) writes: [snip] >Typically, a binomial rv is not thought of as a statistic, at least in these >books, but this is the only way that the approximation makes sense to me. Actually, the binomial rv is the sufficient statistic for the data, which are represented as the sequence of 0/1's from the Bernoulli trials. >= >Instructions for joining and leaving this list and remarks about >the problem of INAPPROPRIATE MESSAGES are available at > http://jse.stat.ncsu.edu/ >= = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: normal approx. to binomial
On Tue, 10 Apr 2001, Gary Carson wrote: > It's the proportion of success (x/n) which has approxiatmenly a normal > distribution for large n, not the number of success (x). Both are approximately normal. (If the r.v. W = (x/n) is (approximately) normally distributed, then the r.v. V = x = n*W must also be; only with a mean and standard deviation each n times as large as for W.) -- DFB. Donald F. Burrill [EMAIL PROTECTED] 348 Hyde Hall, Plymouth State College, [EMAIL PROTECTED] MSC #29, Plymouth, NH 03264 603-535-2597 184 Nashua Road, Bedford, NH 03110 603-472-3742 = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: normal approx. to binomial
It's the proportion of success (x/n) which has approxiatmenly a normal distribution for large n, not the number of success (x). Gary Carson http://www.garycarson.com = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: normal approx. to binomial
one tech issue, one thinking issue, I believe. 1) Tech: if np _and_ n(1-p) are > 5, the distribution of binomial observations is considered 'close enough' to Normal. So 'large n' is OK, but fails when p, the p(event), gets very small. Most examples you see in the books use p = .1 or .25 or so. Modern industrial situations usually have p(flaw) around 0.01 and less. Good production will run under 0.001. To reach the 'Normal approximation' level with p = 0.001, you have to have n = 5000. Not particularly reasonable, in most cases. If you generate the distribution for the situation with np = 5 and n = 20 or more, you will see that it is still rather 'pushed' (tech term) up against the left side - your eye will balk at calling it normal. But that's the 'rule of thumb.' I have worked with cases, pushing it down to np = 4, and even 3. However, I wouldn't want to put 3 decimal precision on the calculations at that point. My personal suggestion is that if you believe you have a binomial distribution, and you need the confidence intervals or other applications of the distribution, then why not simply compute them out with the binary equations. Unless n is quite large, you will have to adjust the limits to suit the potential observations, anyway. For example, if n = 10, there is no sense in computing a 3 sigma limit of np = 3.678 - you will never measure more precisely than 3, and then 4. But that's the application level speaking here. 2)I think your books are saying that, when n is very large (or I would say, when np>5), the binomial measurement will fit a Normal dist. It will be discrete, of course, so it will look like a histogram not a continuous density curve. But you knew that. I think your book is calling the binomial rv a single measurement, and it is the collection of repeated measurements that forms the distribution, no? I explain a binomial measurement as, n pieces touched/inspected, x contain the 'flaw' in question, so p = x/n. p is now a single measurement in subsequent calculations. to get a distribution of 100 proportion values, I would have to 'touch' 100*n. I guess that's OK, if you are paying the inspector. Clearly, one of the draw backs of a dichotomous measurement (either OK or not-OK) is that we have to measure a heck of a lot of them to start getting decent results. the better the product (fewer flaws) the worse it gets. See the situation for p = 0.001 above. Eventually we don't bother inspecting, or automate and do 100% inspection. So the next paragraph better explain about the improved information with a continuous measure... Sorry, I got up on my soap box by mistake. Is this enough explanation? Jay James Ankeny wrote: > Hello, > I have a question regarding the so-called normal approx. to the binomial > distribution. According to most textbooks I have looked at (these are > undergraduate stats books), there is some talk of how a binomial random > variable is approximately normal for large n, and may be approximated by the > normal distribution. My question is, are they saying that the sampling > distribution of a binomial rv is approximately normal for large n? > Typically, a binomial rv is not thought of as a statistic, at least in these > books, but this is the only way that the approximation makes sense to me. > Perhaps, the sampling distribution of a binomial rv may be normal, kind of > like the sampling distribution of x-bar may be normal? This way, one could > calculate a statistic from a sample, like the number of successes, and form > a confidence interval. Please tell me if this is way off, but when they say > that a binomial rv may be normal for large n, it seems like this would only > be true if they were talking about a sampling distribution where repeated > samples are selected and the number of successes calculated. > > > > > > > ___ > Send a cool gift with your E-Card > http://www.bluemountain.com/giftcenter/ > > > > > = > Instructions for joining and leaving this list and remarks about > the problem of INAPPROPRIATE MESSAGES are available at > http://jse.stat.ncsu.edu/ > = > > > -- Jay Warner Principal Scientist Warner Consulting, Inc. North Green Bay Road Racine, WI 53404-1216 USA Ph: (262) 634-9100 FAX:(262) 681-1133 email: [EMAIL PROTECTED] web:http://www.a2q.com The A2Q Method (tm) -- What do you want to improve today? = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: normal approx. to binomial
James Ankeny <[EMAIL PROTECTED]> wrote: : My question is, are they saying that the sampling : distribution of a binomial rv is approximately normal for large n? : It's a special case of the CLT for a binary variable with probability p, taking the sum of n observations = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
