Calculate the P-Value in Statistics - Formula to Find the P-Value in Hypothesis Testing

hello welcome to this lesson and mastering statistics we're going to continue working with hypothesis testing in this particular case we're going to start to talk about the concept of a p-value so keep in mind that this right for now is in the context of our large sample hypothesis testing of means so we're doing hypothesis testing with means we're doing large samples for right now and so we have sample size is greater than 30 now up until now we've been doing everything in terms of rejection regions all right and that's basically where the level of significance alpha whether it's left tail right tail or to tail you have to set it all up but essentially you create these boundaries and these are rigid boundaries that you can then look at calculate your test statistic from your data see where it falls and depending on where it falls you can tell if you have rejected the null hypothesis or if you fail to reject it all right well every once in a while when I do teaching I get around to a topic that I get really excited about because p-values in this case is something that gives a lot of students a lot of heartburn a lot of you know scratching your head and trying to figure out what it really means well I'm excited because I can explain this to you I think with some concrete examples and especially after we get through this we do a couple more examples hopefully I believe that you will have very very good understanding what p-values are intuitively number one thing before you do any kind of you know diving into this is I want you to keep in the back your mind that's the purpose of a p-value really the process and sort of the reason that we do it is really no different than what we've been doing before essentially we want to figure out Lin which reject that null hypothesis and when we fail to reject it so before we we were using the rejection regions and figuring out where it falls here we're doing something very similar at first it's going to look totally different but then when I start talking about it enough you'll realize it's exactly the same thing so keep that in back your mind we're using it for the same purpose so we just go down my list make sure I say everything rejection regions work perfectly fine and statistics there's nothing wrong with rejection region but p-values are more common to real research so if you read a real research paper in statistics they do a big study they figure out what the hypothesis is and reject a null hypothesis you're going to see p-values running around there their explanation so it's much much more calm and I'll explain why it's more common in real research and so that's why we learn it here and I've already said this once I'll repeat it again p-values are just another trigger to decide when we should reject that null hypothesis and when we should fail to reject it first I need to write down a definition of what you're going to see in a book what a p-value is so let me get that down for you but just keep in mind don't worry too much about the definition as we write it down I mean I'll kind of explain it but as we go through it you'll get a much more intuitive understanding of what a p-value is that will be much more concrete than what I'm going to write down here the following is what you'll typically see in a book it will say p-values and a book will typically define it as follows this is a good definition there's nothing wrong with it it's just I need to show you some pictures for you to really understand it it's basically the probability and by the way that's called a p-value because it's basically P for probability of obtaining of obtaining a sample I'm going to put in quotes here because I need to explain it more extreme then then the ones observed in your data assuming that the null hypothesis is true the crucial part of what we're reading here is that the concept of a p-value is just a probability and you know what probability has been talking about that for ages in the class probability right it's a decimal between 0 and 1 the probability of obtaining a sample more extreme than the ones observed in your data now what do I mean by observed in your data it's because all of these hypothesis tests involve you know you write down your null hypothesis you write down your alternate and then you go get some data because you need to try to you know just prove the null hypothesis or reject it or whatever so you last 23 or 28 or 99 people what they had for breakfast that morning or whatever that's the data so you collect that data that sample data that you have whether it's 50 samples or 60 samples that's your sample data and you have all of those different values they're typically you're looking at a mean in this case we've been talking about hypothesis testing of means so you're looking at the length of pencils on an assembly line volume of water being filled into you know the bottles of water in a factory or something you're talking about numbers and the hypothesis test that we've been doing so far have been all about the mean values of those things so we go and select some to study and sample to try to test that alternate research hypothesis and we get the values back the p-value is the probability of obtaining a sample more extreme than the ones observed in your data so you have a collection of data that you get from the assembly line or whatever the 25 samples are the 50 samples that you have that's the data that I've collected the p-value is giving a representation of what would be the likelihood of getting a data even more extreme than the one that I actually did collect and the reason I put more extreme imporant in quotations here is because it kind of depends on the problem that you're doing as to as to which way is more what more extreme actually means in other words if I'm doing a right tailed test right tailed be that way then basically I'm measuring the length of pencils and I spend my research hypothesis is that the pencils are greater than 3 meters long right so more my data more extreme is going to be in there in the right hand direction more extreme towards that tail but if I'm doing a left tailed test then more extreme means more direct more strain in the left-hand direction I think a lot of this can be simplified by writing some of this down so let's say that I have a left-tail test right very common thing that we do in statistics left-tail test so let's draw a picture of it real quick so we have what we have is a little distribution like this this is a V distribution or a normal distribution centered at zero okay so what it's basically saying is if I'm doing a left tailed test don't forget what I'm really doing let me switch over to red here on the left tailed test there's always a little region here off to the left right that we shade right now typically in previous problems this shaded region has always been the level of significance right that's that's what I always told you your level of significant goes goes in your tail and then your your test statistic just lands wherever it lands and you draw your conclusion here we're doing things a little bit different I'm just explaining what a p-value is to you if you're doing a left tailed test and by definition the null hypothesis is here and we suspect that the candy bars in this case but the left tail tests are shorter than they should be in other words we think they're getting smaller then the null hypothesis said so we're moving this way okay now when we do the sampling and we get all the values of the candy bars we calculate a test statistic that's this number right that's what we've been always been testing based on the sample mean hypothesis mean and the standard deviation of the sample and the number of samples we get a test statistic here so I'm going to put that Z down here and this is what we get from our sample data this value of Z comes from the sample data right so it's kind of a representative of this value of these kind of like a representative indication of what the sample data really is telling you it's taking into account the mean the standard deviation the number of samples so that's what this kind of means this whole time we've been comparing this number to the level of significance basically and where that falls on the chart to figure out if we reject the null hypothesis or not so forget about rejecting anything forget about testing it right now the concept of a p-value is basically this value of Z comes from the test statistic it is the test statistic and it represents your sample data so when we say the p-value is a probability of obtaining a sample more extreme than the ones observed in your data what I'm basically saying is that this value of Z is called the test statistic and this comes from my sample data so all of these values here to the left all of the Z values to the left these are more extreme all of these possible values of Z to the left are more extreme than this one and the reason I'm counting to the left is being more extremist because this is a left tailed test right so the bottom line is the p-value is well geez we just switch over to green the p-value is this area that's shaded right here right it is the probability remember probabilities are areas under a curve of obtaining a sample more extreme than the ones observed in your data my data gives me a sample standard deviation of sample I'm sorry a sample mean a sample standard deviation in the number of samples here's what we're relating to the hypothesis the null hypothesis we get a z value back this represents my set of data I'm saying that it falls right here I'm not doing any testing yet I'm not testing any null hypothesis saying the data that I get back it's kind of represented in the chart here it's far enough away from the null hypothesis to the left we're doing a left tailed test there you go data points more extreme than the ones that I've actually collected or by definition to the left and the area of all of those possible data points that I could get to the left is what we call the p-value right more extreme means to the left in this case now let me go over here and we'll do now a right tailed test and hopefully you kind of have an idea of what it's going to be before we actually do it but let me go ahead and do it just to be absolutely explicit in a right tailed test we have a distribution same as we do before which is always by the way centered at zero member dot and these are no these are normal distributions because we have large sample sizes right so I collect my data let's say I'm doing a right-tailed test and I have you know candy bars coming off an assembly line my research hypothesis says these candy bars are longer than 10 centimeters that's the research the alternate hypothesis so longer than 10 centimeters that would be a right hand symbol so I would be doing a right hand I'd be doing a right hand till a right tailed test okay so I would collect all that I would go and look at 35 candy bars off the assembly line and I would get information from that I would get a sample mean I would get a sample standard deviation and I know how many samples that I collected this is the hypothesis means the null hypothesis mean I would calculate this number and I would get a value of Z this value of Z is kind of a represented it's like one number that generally represents the entire set of data that I collected it's one number right so this value Z goes here all right and what I'm basically saying is that all of these values to the right or more extreme than my data right and a p-value is the probability of obtaining a sample more extreme than the ones observed in your in your in your data set so my data set returns a value of Z here everything to the right we're saying is more extreme because it's a right tailed test right and the probability of getting something more extreme than my data set that I had here is what we call the p-value so it's literally the area under the curve to the right of the test statistic Z that you calculate for your data in this case over here it's the area to the left of the test statistic that we calculate for our data so I know because I'm using red shading and I know because I'm shading the tails some of you guys are thinking that this is the level of significance it's not the level of significance all I've said is that I calculate the test statistic from my sample data it's representative of the data that I you know have measured I put it on this guy and then more extreme to the right for a right-tail test is called a p-value so it's the area to the right more extreme to the left is going to be in that case for a left tailed test that's why I put more extreme than quotation now there's one more case I want to show you or in fact actually before we get to that let me go and give a little bit more concrete examples of left and right tailed testing okay let's say as a actual example that the null hypothesis is that the mean is greater than equal 0.15 and the alternate hypothesis is that the mean is less than 0.15 so this is a typical problem that you could actually have the alternate hypothesis is to the left so you know you're doing a left-tail test that tells you that all right also given to you in the problem you're given that from the data the test statistic Z is negative one point three four this doesn't fall out of thin air what this basically is is you collect all of the lengths or the volumes or whatever here's your measuring and you dump that information the sample mean sample standard deviations in the number of samples you stick it into the test statistic and out comes a value of Z that number of Z that that value of Z is representative of the data set that you have it kind of takes into account all the data points the spread of the data and everything and out comes one number that's kind of representative of that whole data set that's what we've been using it for all along we've been using that one representative number to tell us if we're in the rejection region or not okay so that is all given to us we haven't done anything yet but we know we're doing a left tailed test so if I were to draw this I would draw something like this and I know that I'm doing a left tailed test okay so the bottom line is the value of Z that comes about from the sample data negative one point three four and since I'm doing a left tailed test I come up here and I shade this guy to the left because I'm getting the probability of getting a sample more extreme than the data that I actually collected so this area here is called the p-value all right now how do we actually find the p-value we haven't actually calculated anything yet how we actually find it well this is a normal distribution you have a chart of the normal distribution in the back of your textbook every statistics textbook does don't forget the normal distribution gives you the area to the left it's different than the T distribution which gives you the area to the right I know it gets a little bit confusing you always have to remember that the table for a normal distribution is giving you the area to the left so what if what do we do if we want to find this p-value well we have the value of Z and it has the area shaded to the left of Z so all we literally have to do is go to the table right and find the probability that Z is less than negative one point three four so literally all we do is look at z- 1.34 in our Z distribution table the area that it gives us is the area to the left which is exactly what we want and I get zero point zero nine zero one so that means the p-value zero point zero nine zero one that is the p-value for this problem so if you were given a situation where somebody says here's the null hypothesis here's the alternate hypothesis here's the z-score that comes from the sample data calculate the p-value for this problem now notice we haven't done any hypothesis testing yet I haven't even gotten to that yet but I just want you to get practice with finding the p-value well the only reason you need this is to know that it's a left tailed test this is representative of our sample data so we plop it on the chart and since it's a left tailed test the probability of getting a value more extreme than our sample data would be the probability of to the left of this value of z which I can readily look up in the back of any book so that's how you find a p-value for a left-tail test right now what happens if we have a right-tail test okay what happens if we have a right tailed test well for a right tailed test okay for a right-tail test let's pretend that we have a null hypothesis which as a mean of less than or equal to zero point four three and an alternate hypothesis looks to put an A there a mean greater than zero point four three and let's say from the data Z is equal to two point seven eight so all we have is this and the question is what is the p-value for this problem notice we don't have a hypothesis we have some hypothesis on the board but we haven't been asked to test it we haven't really been given a level of significance we haven't really been told exactly everything about the problem all I want you to do for this problem is find out what is the p-value well you have to know what a p-value is it's the probability of getting a value more extreme than the sample data that you collected now we don't have the raw data but we have the Z value the test statistic that came from that guy there so let's go and draw a picture and get our bearings for what we actually have and what we actually need so here is our distribution this is a Z or a normal distribution and notice that this is a right tailed test this is a right tailed test so zero goes in the center and this value of z that I got from my data was two point seven eight right that is a representative number in terms of Z that kind of represents the whole data set that I've collected we put it on the curve right there what will be the probability of getting a value of next time we select a sample or whatever of being more extreme than that well this is a normal distribution so we go up here we shade to the right more extreme in this case means to the right because it's a right tailed test more extreme in the previous case was to the left because it's a left-tail test so the phrase more extreme really depends on the kind of problem that you're dealing with that's why I put it in quotation marks so what I'm looking for is the probability that I've select my next sample and I get a value more extreme than the data set that I previously collected so that means that this area here is going to be the p-value the area under the curve to the right of this value of Z so in order to do that you can use your normal distribution table in the back your book the probability of getting a value of Z greater than 2.7 aims when I'm after but remember that when you're using a normal distribution you can't just put the number 2.78 in there and circle the answer because what we're looking for is the area to the right right but if I actually put 2.78 n into the normal distribution it's going to give me the area to the left in other words it's going to return the area of everything over here so if you remember back to the very beginning of mastering statistics like the very first volume when we start talking about the normal distribution basically said when you're using that table if you want the area to the right the way you actually handle it is you basically say that's going to be equal to the probability of Z you flip the sign around negative two point seven eight because remember everything symmetrical this is two point seven eight here I'm interested in this area of the area the only thing I care about if I look on this side right around here Z is negative two point seven eighty and the area to the left of that negative value of Z is going to be exactly the same as the area that I care about here so when I'm trying to find areas to the right with a normal distribution I flip the sign around the the inequality round and I change the sign of Z there so I go into my chart and I look up the value of negative two point seven eight in my chart which is somewhere over here it's going to return an area to the left that area is the same as what I care about so whenever I do that I'm going to get zero point zero zero to seven so all you have to do is say that that is the p value zero point zero zero two 7s the p-value the p-value literally is the probability of obtaining a value like the next sample I were to take the probability of obtaining a value more extreme than the data set that I've collected the data set that I've collected is represented mathematically by this test statistic it contains kind of all the raw information and boils down to a to a value of Z so that's what a p-value is I keep saying it over and over again I want you to visualize that the p-value is an area to the right or it's an area to the left and when we get into situations where it's a two-tailed test it's going to be very similar and I'll get to that whenever I get to that but for right now I want you to understand the concept of a p-value notice we haven't done any actual hypothesis testing it I haven't even told you how to use p-values to make decisions right but I told you at the beginning that the big overall arching concept here is that p-values are going to be used to tell us if we reject or fail to reject the null hypothesis just like the rejection regions we did before I promise you that we are going to get there you are going to understand but you have to take it kind of baby steps with me first understand what the p-value is it is the area to the right of the value of Z that you get from your test that you get from your sample data if it's a right tailed test or to the left if it's a left tailed test all right make sure you understand it so we've done two problems we've calculated the p-value for one four right one for a left I want to stop here go on to the next section where I'm going to show you how to use what these p-values are in order to make decision in other words to tell us if we reject or fail to reject a hypothesis

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Id: KLnGOL_AUgA

Length: 22min 42sec (1362 seconds)

Published: Thu Jan 19 2017