Stats: Approximating a Binomial Prob Distribution using a Normal Distrib (Part 1)

okay this video is a demonstration little discussion on how to use a normal distribution bell-shaped right bell-shaped distribution to approximate a binomial probability distribution now this is kind of strange because I've got two things going on here so far a normal distribution as you might have seen on some of my previous videos is really dealing with continuous data right anything that can be measured whereas a binomial probability distribution binomial data is mostly discrete data almost entirely discrete and so I'm going to try to use a normal distribution that really deals with continuous data to approximate it's not going to be exact but it's going to be pretty close a binomial probability probability distribution which is discrete data that's kind of interesting but let me do a real quick recap of what I mean by binomial probability distribution first of all I have a number of trials that's what little n is number of trials I have an X X is the number of successes all right the number of successes that I've had given the number of trials N and X cannot be any number greater than n okay so X is restricted by n I also have little P which is the probability of success and I also have its complement little Q probability of failure okay the other thing too that you might recall from a binomial probability distribution is if you know these values and X P and Q you can find the mean as well as the standard deviation all right the mean formula is really simple it's just n times P that's how you find the mean take the number of trials times the probability of success that will give you the average population mean and the standard deviation well that's a little bit deeper it's actually n times P times Q right you can multiply these three things the number of trials probability of success and the probability of failure and then square root that value all right so that's just a quick recap of binomial probability distributions now how do we how do we use a norm dist to approximate this well before I get into that I've got to tell you about two little caveats two little things that we have to make sure are true or else if these things are not true we cannot do what we're about to do here I'm going to go into a couple of examples in just a second here but the first thing is that you have to keep in mind is that I want to check right so we're going to put all this here as a little check we're going to check two things here we're going to check and make sure that n times P really that the mean right n times P is also known as the mean we're going to make sure that n times P is greater than or equal to five okay that's one of the checks that we're going to do we're also going to check that n times Q the probability of failure is also greater than or equal to five now if either one of these two products is not true or it's not greater than five then we're going to stop right there and we say we cannot do this all right we cannot use the normal distribution to approximate binomial distributions so both of these products must be true and I hope you see that this is really just the mean isn't it look at that it's just the mean okay okay so these two things have to be true for me to use norm s disk the other thing too that that has to come into play here is this concept called continuity correction okay so assuming that these are true that these products are greater than or equal to 5 I also have to use this other idea called a continuity correction okay this is an interesting concept because what this continuity correction is all about is taking a discrete data value right it's taking a discrete data value and turning it into kind of a continuous data value all right it's turning a discrete data value into a continuous data value what do I mean by that well take a look at this for example if I'm interested in the probability of an exact number let's use like eight for example right if I'm interested in what's the probability of that exact number coming up well continuity correction says okay that's too discreet for me because the area of this little skinny line is actually just zero so what I'm going to do is I'm going to move to the right of it just a little bit and to the left of it just a little bit and what I mean by just a little bit is by convention we're actually going to use 0.5 so just to the right of 8 is 8.5 and just to the left of 8 is 7.5 so what I'm actually going to do this is how continuity correction works is I'm actually going to find this shaded area in between 7.5 and 8.5 all right those are the values that I'm going to be interested in that is again if I was only looking for the probability that X was equal to exactly 8 right let's try out something else here we're going to come back to these numbers here in just a bit later on but what how does continuity correction work for say something like how about this X is greater than or equal to 8 you can see on a previous video that I've recorded that this corresponds to at least 8 all right at least 8 where we include 8 well picture wise all right picture wise let's say that 8 is sitting right there if I want to include eight and I'm going to shade to the right of it I hope you see that's a shading to the right right there then if I want to include eight I need to go just to the left of eight which is seven point five okay just shy of eight on the left side of it and I'm going to shade everything greater than that right because that is a greater than symbol that's what at least eight looks like as far as continuity correction is concerned all right as far as continuity correction is concerned I'm actually going to look up this all right let's see how about at most eight about that at most eight at most eight also includes eight again you can look this up on a previous video that I've recorded but at most eight looks something like this let's say eight is over here at most eight includes eight but it's shaded to the left already chained it to the left which means that if I want to include eight then I'm going to peg all right just go a little bit above eight which is eight point five and at most eight means I'm now going to shade all of this area over here okay this is what at most eight looks like so it continuity correction says this that's what I'm going to look up now everything that I've shown you so far right these two examples that I'm showing you right here and the one that I did on the previous sheet and I can bring that back up do you notice that all of these numbers here right especially these two right here continuity correction our x values okay these are X values same thing with this one look at this right these are X values right here and these two numbers are X values how do we change X values into AZ value because I can't just look up something like this I can't just look up something like this in a nor mold distribution table I can't look up an 8.5 x value I have to change those into Z values and that's the last thing I'm going to show you in this video here so we have a nice simple formula perhaps you're used to seeing this by now and if not this is what it looks like if you want to change X values into a Z value it's this nice simple little formula and it looks like this Z is equal to X minus the mean divided by the standard deviation right so if I plug in any x value that you give to me and I subtract the mean divided by the standard deviation it will change that X into a Z value now where are these two things coming from for our binomial probability distributions or remember what I showed you at the very beginning of this video that bring that back up here so you can see it that we can find the mean and we can find the standard deviation given N and P and Q all right so we can find these values the mean and the standard deviation by N and P or the square root of NPQ that gives us standard deviation now remember you can only do what I'm showing you here you can only use this thing if right using this little caveat here you got to make sure that both N and P are greater than or equal to 5 and n times Q greater than or equal to Phi right so this product and this product are both right now but you hear both must be true not just one of them but both of them must be true all right hope that helps

Channel: poysermath

Views: 99,004

Keywords: binomial, distribution, poysermath, continuity, correction, normal

Id: rPOSpI7qMl0

Length: 10min 29sec (629 seconds)

Published: Tue Mar 27 2012