Being up to and including three because keep cumulative to the three. Oh, okay. But if we want to do like something but with like f is bigger than five, then is that pdf? No so it will always be cumulative pretty much for these but the value of x that you put it. So for example, if we look at example seven, so we've got spin arts designs so that lands are red point three J S twelve spins find the progress is that J attains this. So the first thing you do is you get Oh okay, well this is binomial because either it lands on red or it doesn't. And we've got a fixed number of trials. So what you dry, if you dry x is binomally distributed where n number of trials is twelve and the probability of success is 0.3. So this is the first step to define the distribution. Then we want to work out these two specific probabilities. So part a, no more than two reds. What is no more to me, excuse me. At least then that twelve and. What's no more than two? A more than two, the neccan only be smaller than two. Should it be too? It could be two Yeah so no more than two could be zero, one or two Yeah because two is not more than two. So part a is the probo belto. The x is so no more than two. That's going to be less than or equal to two. And then we've already got a less than or equal to so putting this into our calculus, it's quite easy. So as you say, when we're going to it was by arm, cdf. And then it will ask you for n, which is our number of trials, p is our probability of success and x is that value that we're going up to less than zero, equal to like cumulatively, up to you Press equals and hopefully you get whatever answer we're aiming to get. So if you put that in what you get, what values did you get? 0.253Yeah brilliant point two. Yeah so in stats, we tend to go for significant figures, but Yeah so 250 28I've got here. Yeah part b is then it's it's the same distribution. So x is still binomally distributed with twelve trials and the chance of success as 0.3 this time is at least five. So what does at least five mean to be five or bigger than five? Yeah, brilliant. X is greater than or equal to five. Trouble is now put this into our calculator. We can't put in five because our calculator works out the cumulative probability of being up to and including that number. So we need to manipulate this inequality so that we can get equivalent statement to this so it has a less than or equal to in it. The one line is. P equals sorry, sorry, one minus p bracket x smaller or the equal to five, equal to one value or so. We want to include five here. So great ace lyrics, five. This is five, six, seven, eight, nine, zero, eleven, twelve. So we already do one minus, which values Leys. One man. Smaller or you could to four sorry Yeah because we want all the ones Yeah you can you could you could do this to help you visualize it and go. You know, you know, you don't have to write them all out, but well, I'll do it this time and go, okay, well, Yeah, we want. Those ones, so it's one minus two other ones, just one minus then and then you get your calculator, you go, okay, n is twelve, piers point three x is this time four. And then you have to do one minus the value you get there. Okay, I won't make you do that one because it's the like that there not my awrong this bit here is the hard bit if you can if you can convert to this but it's calculate is the same as always but that is apparently 0.2763. And then last part part say Jane decito use this spinner for a classic competition shows the probability of winning a prize to be less than 0.05. Each member of the class will have twelve spins and the number of reds will be recorded. Find how many reds are needed to win a prize. So this one was sort of working backwards. So instead of being told, I want the probability of at least eleven spins, they're going, okay, I want the probability of winning to be under 5%. So we're looking for what value. Does that so algebraically? If you're to win, you need to get a value or higher than that value. Are you happy with that? Yeah. So we've got the probability. X has to be greater than or equal to some value for winning. So let's call it up. No W for winning. Okay. So the probability to the x is greater than equal to this. W has to be, as they've said here, less than 0.05. Okay. So we now have to manipulate this inequality again. There's a greater than or equal to in it. We don't like greater than our equal to s. We want less than our equal to s because then we can use our calculator really easily minus good P X. One minus equal to W minus one brilliant. Yeah W minus one. Okay. And this thing has to be less than 0.05. Okay. What could I do with. That inequality now without that bit. Can do. One minus so it's 0.95 equals p. Yeah, we got that. Yeah. Yeah. So we need this. To be not put the inequality the wrong way. Isn't it going to be that way? Right? Not Yeah. Yeah, there we go. So we're looking for what value of W makes it. So we get a probability of over 0.95. It can only be twelve. I want to know let's let's check on the ci'm. Not sure. So now just input some values. So literally, again, we've still got we're still under this same distribution. X is still binomally distributed. The still twelve trials, still probability 6.3. But just guess the value of x. And we're looking for the first value. So the smallest number that takes us over that 0.95. Okay. So what value are you trying first? Oh, twelve. Yeah, between twelve. Well, the chance of being less than twelve, that's going to be one. I can tell you that one already. So it might be twelve, but we're looking for the smallest value that is over that 0.95. See, let's try eleven. What is anguus? 0.9 is it? I'm expecting 0.99 or something. Wait, but how do you know? I don't know that. Yeah because I know it's going to be basically one just on the distribution, but how can you sorry, well, how can you calculate it? I haven't calit. I just estimate it. Okay. So I know it's going to be pretty much one because the chance of getting twelve reds in a row is pretty pretty low. Okay? So this is over 0.95, that's good, but there's probably something smaller than it that's also over 0.95. So try a different value, put a different value of x. Yeah, try not. What did you get if you put in nine. But how do I do my calculate or so again, it's the same as up here. So you're in binomial cdf. N is still twelve, p is still 0.3 with an x. Just you're just guessing your value. Okay, so you said nine to put an exit nine? No, we won't. What did you get when x is nine? Well, I'm not sure, but when I say one x nine, sorry now, because I when I say nine, I meant like W was nine. So like whatever. Sorry. So when I'm talking about, look, it's really big. I talk about that number. Okay. In the orange box. So that that's the number we're going to change. So what did you get for nine? Oh, I didn't get because I pay eight that's fine, right? It's 0.998. Okay. So it's still it's still really high 0.998. So again, there's probably something below low up so try try another number below low up. Seven is 0.99 okay so it could be seven well by six. 0.961. Okay. Is that the smallest? Yeah, because five is 0.88. Yeah, brilliant. So I would always encourage you to show, so don't write them all down. Obviously you've tried twelve, eleven, nine, seven, six, five, don't write them all down, but we want the ones that are at the boundary, okay? Because this this shows the examiner and it shows you that this six must be the the first one that's over 0.95 and because five is below it. Yeah and then lastly, it's just then using that to find a. So this is why I would encourage you to do this algebraic step that we did in the red and then onto this yellow line here because now we can easily tell what W so what value of W with that? Guus? Yeah, brilliant. Do yourself. I can show you down. I'll show you what they did for their working out. So aren't they are not W be okay. The smallest number of reds needed to win a prize. So Yeah, then they've had exactly the same thing as we had, but they then haven't shown that algebra. But I would encourage you to always do this these three lines, because I feel like these really help you to cement how you go from six to the answer of seven. They've used the tables, but we've got a calculator that does it for us. And Yeah 0.8, 8.96, 14 like you said. And then they're just showing that if less than every extra to six is this down pis, that greater seven is smaller than 5%. You don't need to do that. We got the same answer. Yeah so that's binomial to be honest, or hypothesis testing. It won't be as complicated as that question c. It would be questions like amb for hypothesis testing for the actual binomial distribution within hypothesis testing. However, you know we're we're here now. So it was it was worth having a look at that sort of question as well. And then I believe six c is the last one. Yes, then it's the mixed exercise after. So that is the end of that chapter then. So you'll see there's lots of similarly worded questions. So though exactly four, I've got almost most three. We've not seen one that does that yet actually. How would you work that out? It has if it's if it has to be three ker and six, then just the next is three. So we've got okay, that's part of it. So we won the values three, four, five, six. Oh sorry Oh was inclusive sorry I just I just ignore the inclusive it the nuts. So. I it mean so those are four values of x. How would I use the cdf function on my calculator to work this out? But we can sort of see an inequality in there. We're like we like the less than our equal to is dwe. Yeah, that's good. So we could work out less than equal to six, but that would give us six, five, four, three, two, one, zero and we just want six, five, four, three. Over minus minus one minus p bracket x smaller or equal to two. Minus what sorry minus one minus minus sorry you right if you want me that. I'm not sure that's minus. Oh Yeah is. You're almost right because it's a Yeah you don't need the one minus. Why? Because blastering equator two is already in the form that we're interested in. Okay, just so I guess I to switch it. Just to switch what? Just add like the one minus. So the one minus is for when you've got greater than or equal to. So if you've got a greater than or equal to or a greater than, that's when you have the one minus. But we haven't got any. Like we've got two less than I'm equal to. So we're already quite happy. So we're just taking these away. So the reason this works because we so we've got zero, one, two, three, four, five, six. That's what this bit tells us. We don't want zero, one, two, so we need to somehow get rid of them. So we need some way to describe zero, one, two. And that's where this book comes in. Okay, again you not this doesn't come up in the hypothesis testing chapter. This is this is like a morso for the actual binomial distribution chapter. So for our purposes of hypothesis testing, this bit isn't relevant. However, again, as we spent an hour or so to get to this point, I think it's worth just having a quick look at them. Okay. But that that then is is as far as they go for these questions. I one, not quite nice actually eight, but so this is a similar thing to that one with the the spinner. But we did the example seven at part c. So we want the value of k such that we get about probability below 0.02. Part b, similar thing would being over 0.01, I'm sorry, been under point zero one. And then part c and then combining that a bit like in this part c here, combining you two answers to get a new probability of being between these two things. Yeah, we don't have to do that now if you don't want if you want to move on to the hypothesis test so we can know if youlike. We can do another Oh, should we do eight? And Yeah, we do eight. Let me let me do that then so we can actually run the actual board and let's have a look at eight. So for eight, I would encourage you to do the algebra stuff that we did appear first to manipulate to manipulate these into the form that we like. But Yeah, go on and take it away. Good. Yeah. Yeah, that's right. Yeah and then for this, we want as it says in the question, we want the largest value of k okay, so that there might be multiple values of k that have a probability below 0.02. We want the smallest one. No, we don't one the largest one. Sorry, I can't read. My mind so weird. Oh my. What? Have you done? Hmm. You're let tting me try again, you on cdf. Yeah, what have you got for np and x? Please sorry one. Oh, no, mind. It's okay. It makes sense. Oh, I think I actually clicked fcdf before, when that's fine. You are in cdf. Yeah, we put that like the lowest thing I can put. Is one right or can I because don't you can't put zero but it would the technically it would tiyou the probability of x being less than or equal to zero but in this case that would be the same as x equal and zero. I'm going to put 0.5 no. So because it's discrete values, they only ever so for binomia, it's always just integer values. It makes a bit more sense in questions that give you a bit of context. So for example, if you look back at question six, we were looking at plants that have blue flowers. If you've got 15 plants, you can't have half a plant that has blue flowers. Either it does or it doesn't. Okay. So it will only be I imagine you calcullator will tell you there's an error if you try and put a decimal in. Then the Lois is one which is 0.08. As in so you've gone so you're right you definitely on you can't put negaeither now imagine it will have a mental breakdown you with that as well. So you definitely aren't binaal cdf Yeah lovely. And then what have you got for Anne? What's n there are number of trials which do you put in for n in your calculator we'll price 40 good yet 40 and then p 0.10 point one more done and then we've got that's because x is bonnedistributed with 14.1 that's where they come from. And then you've tried what value of x you said one. Yeah what did I give you? Like something like 0.08 something. 0.08. It's back a smaller value than that. Let me have a look. I'm going to install school on this table. And it's 40 first point one. Yeah point zero eight zero five I've got Yeah okay, that way that's too big. Yeah, so that means it's probably going to be zero, but let's just, it might it might have been none of them. It might be that there's no values that give a probability of that. So what you get if you have probzero zero, one, four, seven lowhat, I goes, well, I eight, eight, okay? Which means this k minus one must be zero. So k minone equals zero. It's okay. Let's be one. Yeah, that's fine, but sometimes these critical ranges are really small. And we could have actually got to a point where Kate, so the question is not worded correctly for it to work in this case, which sometimes you will get, that value is also over the 0.0 to realize I've missed out zero, in which case there just wouldn't be a value of k such that exists. But we didn't end up at that. Okay, whatthat be you could do a similar thing for b bea bit harder because that inequality ties do the way around. Yeah. Good Yeah Yeah because greater than I is the same as one minus less than equal to R or don't. Oh sorry, No Okay sir, but yes. I think you were right the first time. I think your inequality was right the first time. Oh, Yeah, okay, sorry, that's all right. Brilliant. Okay. So we want the smallest value of R such that we get a probability over 0.99. So what value should we try in the calcua? Okay put into. It what you get one so we know it's not one we saw one earlier that was the 0.08 so there's a lot there's a lot of tracks with gold words of 40 so to just just pick it you know I like to pick somewhere in the middle to start with so I would probably pick 20 initially just to see what happens. We better give me one for like 32. Okay? So that means so it wasn't actually one, but this is because if you look at our data, it's really, really skewed. What I mean by that is the probability of success is really, really low. It's really near one of the ends. Point one, when that's the case, what happens is the chance of getting 30 up to and including 32 is effectively one because the chance of you I don't know what the context is, but the chance of you getting something that has a chance, a 10% chance of happening to happen 32 times out of 40 is basically zero is why that's happening. So quite a lot of these will be one mathematically because our calculators don't go to the like to the lengths that they do, essentially. So you tried theta. It's going to be something lower than thetwo. It's actually to core a bit lower. I five and that's 0.79. Okay so it's over five. Quite a bit overhigh actually. We've certainly another 20% over that. Go 0.9. Okay. So we're getting closer observer six as well. Seven is 0.958. Okay, we're still on to over that 0.99. Oh Yeah, ten I think. What was the probability of access? Ten, 0.998. Okay. What was it when it was nine? Was that but was that definitely below 0.99? That that was ten. That was ten. That was R when R was so I agree that ten works, but we want the smallest value of R that does this. So so just check the one below it just to check it below point 99, okay? Because if it isn't, then ₩10't be the answer. If it is, it proves that ten is. But if it's not below 99, then it won't be the smallest. Answer, what did you get for nine, 0.9, nine, 49. Okay, so it's not ten, it could be nine. So lost the chance of being eight. A try, but I don't think so. Not sure. Eight was 98 0.98 brilliant. So it's nine. Okay. So you always want to show the first one that goes over and the last one beyond under to show that like that boundary crossing. But let me check the answer. Let's see if it was that. So what do we say? We said k is one. Yeah. And R is nine. Brilliant. And then part c, if lost me question now, what's it going? Part c, one probes that x is between k and R, inclusive. Let me scdown for. Us. So how we're going to go about that one. So. We could. So we know kr and I don't make we work that out. Yeah, so we could put our numbers in. Yeah. Okay. Okay. We worked out about you said it was I think you said it was one. Here they go, catable. Do we just PaaS them together? So we do we do what it says here. That's going to look a bit similar actually to six c. So it has to be minus. Well, let's start off with the original inequality and then we'll look at manipulate that. It does. Yeah. So k was one. So that means it's if we take away the probability being a less than x to one, you see how in the original question they want us to include one. The one to include one. So in the original question, it's the probability that x is between 19 like that. Yeah. So if you take away the chance of being less than or equal to one, that includes one, but we want to include one in our answer. Oh okay, yes, a we're after ter and zero. Yeah zero. So we're after one to nine inclusive. This bit Oh, this includes zero. So we just need to get rid of that zero. That's 0.140 point zero 14 right, I've got Yeah. So 9949 minus zero 148. You've missed that zero. 0.0148. Yeah there we go. So that's 0.0 point. 98. I've got nine, 801 here. 98 zero one. Yeah, but Yeah Yeah brilliant. Okay. Just one thing I would add at the start of your binomual distribution questions, you want to we didn't do it here. We just want to define the distribution. So in these questions, they've written it for us, so it would feel a bit weird to do, but particularly in questions like nine or ten, you want to define it at first. So you've got it really explicit that you know Oh Yeah, this is a bunenable distribution where n is whatever and p is whatever, partly because it shows the examiner you know what's happening. But also in year two stats, you get the normal ll distribution added in to the mix as well. And we want na be able to show that we understand whether something is binomally distributed or normally distributed because it won't always be a. Okay, but Yeah, admittedly that's more important in questions such as six, whether they don't define it for you than it is in eight because in eight years it's written in the question. Okay, okay. Should we should we start looking at hypothesis testing? Yeah, sure. Yeah. So obviously we've whizzed through the content there. I would encourage you at some point to have a look at questions. You by no means would I say, well, know we've looked at all the content, but I definitely I definitely want you to do more practice to become confident with it. But we have locked up the content, okay. And done we've done enough to be able to do hypothesis testing. So our about to session in is the last chapter and stats. And it's one of the I suppose bonnumber distribution was no, but this is completely new because it's it's a sort of question that you won't have seen the style of before. This is what most people would consider the hardest past stats. I would say this and the large data set, the two things people really dislike in stats because they're harder. However, I think once you get the theory of how a hypothesis test works down, I think the actual math itself isn't that hard. Okay, okay. So for the next 15 minutes for today's thing, we'll focus more on the on this bit, the concept as opposed to how to get the Marks in an exam. So as I said, I think I said on Monday on our last Monday, Tuesday on whatever our last lesson was, I said I don't like the order the textbook does it in. So we're gonna to bounce around a bit. So for example, this bit, this bit I think should come later, but the textbois pretty good. It's just I personally like it in a different order. So a bit of language first. So we'll read through it and then we'll talk through what that actually means. So hypothesis is a statement made about the value of a population parameter. This sounds needlessly wordy over here. It tells us what a population parameter is. So that's the probability of something happening. Probability p, in this case, it's going to be a binoial al distribution in year two stats. There's two more types of hypothesis testing. One includes the normal distribution and one includes something else which we're are getting right now, but that all of these ones are going to be binomial distribution. Okay, so you can set an hypothesis about a population by carrying out an experiment or taking a sample from the population. The result of the experiment or that's or the statistic that is calculated is called our test statistics. That's like your answer, the actual math you Carry out. And then you have two hypotheses. So let's go. I'm going I word this differently. So the way hypothesis test works is you will have some information that is deemed correct. So for example, if we jump straight, the example, so example one talks about John. John wants to see whether a coin is unbiased, whether it is biased towards coming down heads. Okay? So what you do is we've got some information that is deemed correct. In general, we assume coins are unbiased, okay? So in general we get, Oh well, the probability of a coin landed on heads. Oh well, that's 0.5. This thing that you assume is correct is known as a null hypothesis, and it's given by H naught. So the very first, or maybe not the very first thing, but one of the first things you do in a hypothesis test question is you write H, not you're a colon, and you go, okay. Well, the thing that's is deemed correct is the probability equals 0.5. You actually know hypothesis. This is the thing we assume to be correct. Then the hypothesis that we're looking at testing is this alternative hypothesis. So this will be given in the question. So what is what's John saying? Saying. Wait, was it? John saying this coin might be biased. Yeah, good towards heads. So this is our H1. H1. John is saying the probability is greater than 0.5. Okay, the north hypothesis is always talking about the same things. It is always p. And for the for the normal ll hypothesis, it's always an equals. So italways be p equals something. For the alternative hypothesis, it's always the same value. But it would either be a greater than a less than or a not equal to because there always a choice. You've got to decide whether this person is saying progress ahead ds is greater than 0.5 less than 0.5 or not equal to 0.5. So that not equal to we scroll back up to the these bits is what is the is this bit so it's talk about two tail test. So initially we go all of this for a one tail test. So that means when you've got a greater than or less than for your alternative hypothesis, however, it could be a third option, but that's more of an add on that comes off at the end. Okay. So I think there that I know we've got to have order of the questions. So these are our non hypothesis and alternative hypothesis. We then need to look at what the test statistic is. So we'll go back to these definitions at the top. It's the result of the experiment that is calculated from a sample with our test statistic. So in the context of John. What would you say? I test I can't speak test statistic. It's going to be. Ability. Of the in nearly looking on his zmanso, the test statistic, it's really hard to say test statistic. It's always so in our so if we've got our distribution so later on does do a number of things. So he does eight, eight coin tosses. And 0.5. X is our test statistic. So in this question, so this is the number of successes. Okay? So the x is always the number of successes in context of this question. What's classias a success? That to this hypothesis test. So like in the context of this question, we've got like what classes has a success in this question when the coin finds on heads more than four times? So not necessarily more than four. It's just the number of heads. So success is getting heads. So x is our test statistic. Ck, and this is. The number of heads. So the number of heads he gets. Okay, we go down, consider, see how they've worded it. Test, Oh mi hate it when they write test statistic, I really can't say it x the number of heads. When you say, Yeah, it's all right. But when you say it multiple times, stastic, Yeah, it's because of this into a stit now, can't do it. But Yeah, null hypothesis is always in this form. So it's always H zero ught colon p equals. In the alternative hypothesis, it's always H1, that's what we call it's always p, it's always the same number. And then it will either be a less than, not equal to or or greater than. And that's always given by the wording in the question. Some of them are really, really, really subtle and it's all in the worin the question, but they're never greater than or equal to is all less than or equal to. It's always a strict inequality. It reminds me of computing. I didn't know why. Reminds you who are sorry. It reminds me of computing. Yeah just feel very Yeah Yeah with the coons and everything. Yeah now I can see that. And then Yeah, as we said a minute ago, if you've got an interquality, it's a less than or greater than it's a one tail test. It's a not equal to we get two tail test. Two tail tests work pretty much exactly the same as one. There's one change which we'll look at at the end, however, but we still haven't really looked at what hypothesis test is. So the way hypothesis test works, we go back up here, the way hypothesis test works is there is some assumed to be. Parameter that we that someone is saying, you know what? I don't think that's so in this question. The default is we assume coins are not biased. Chance get ahead is a half. John is saying, I don't think that's. So what John then does is he has a choice. Either he can do an experiment where he performs the test himself, or if we're looking at something that's a bit larger scale, then you could look at a sample that somebody else has done an experiment on. So what John would now do to test, it's not in this question, but what we'll talk for it. So what he wants to do now is he's gonna to test his coin, so he's gonna to get the coin, he's gonna flip it eight times and he's gonna record how many heads he gets. That's his experiment. Okay. And it makes sense. I get you got Oh, you said four he gets four heads over four heads. There's biased. However, even if you had a like, like let's look at a dice. For example, if you order a dice six times, a fair one youexpect every value to go once. But it wouldn't would it it like chance it wouldn't happen. Yeah. That that's because probability doesn't care about what the answer is. If we rolled it 6 million times, then Yeah sure, the values would get really, really close to their actual values. So on six rolls, chances are we're not going to get a one, one, one, two, one, three all the way through. Same is for this coin. If you get a coin and flip it eight times, you probably don't get four heads, four tails very often, even if it is a fair coin. So what we do is we look at. We look at trying to I don't word this. So we can never say with certainty whether it's biased or not, but we can say there's evidence to suggest it is or not. In a full question, you'll have what's called a significance level. So this this isn't a full question because it doesn't ask us, do the full thing, but that's fine. So youhave, what's called a significance level. A significance level is always a percentage is usually 5%, 10%, 2% or 1%. But it could technically be any value. What you do with that is you compare in it with what's the chance that happened. And what happened actually happens under write down somewhere. Didn't we? We write down our distribution under this. So this is our distribution. Let's imagine, let's go with five. Let's say, let's imagine, John gets, let's do that so I can write. So John flips the coin eight times. Eight flips. Let's say he gets five heads. Okay. I completely get what you're saying where you go. Well, he's got more than half. It's biased, but five heads isn't actually that unlikely to happen. So what we do is we assume that the null hypothesis is the null hypothesis says that the probability of Genner heads is 0.5. We assume that this is then what we do is we work out if this is what's the chance that I actually get five heads to. We look at what's the chance of getting five or a more extreme value than five. Okay, so we go, okay, what is the chance that we get five heads or more? I've gone for all more because six heads is more extreme than five heads. Okay, so you always go, if it's yes. So we look at what so essentially what we're doing is we then work this out. This gives us a number. Okay? We go, what is the chance of me getting five heads or more extreme if the coin is unbiased? So let's put a number to that. Let me see if I can get a number for that quickly. Then shthere be none. Shouldn't there be zero? Should this be zero? Yeah. Well, today, like, well, it won't be zero because fereously, because like you could flip a coin eight times and get, you could get eight heads. Unlikely, but it could happen. Yeah. So what is the chance of that? So that is going to be one minus parahundred 30x. So that's no to four, which is one minus. Oh, now one equals one -0.6, three, 67, which is what's that going to be? Three, three, six? So the chance. Okay, five or more heads is 36%. That's quite high. That's quite likely, isn't it? Really? If you think about that, that happens quite a lot. Yeah what we then do is we compare that with the significance level. Okay? So the significance level will always be a percentage given in the question and that is the minimum. Oh no, it's the maximum probability allowed for us to say. The alternative hypothesis is so let's say we were doing 10% significance level. We compare 10% with a 0.36. So we go or 0.36, well, that's greater than 0.1. What this is saying is we're going well. If the coin isn't biased, five heads or more happens 36% at the time, which is quite likely to happen because it's. Sorry, because it's over the significance level. We say, Oh, well, this value isn't therefore significantly happens a lot of the time normally. So we say there's not sufficient evidence. Not sufficient evidence. To support H1 on not there's not sufficient evidence to reject the null hypothesis. Really. We should say to. Reh zero. Okay, so so the way hypothesis test works, so just to put it all together, is someone has some theory about something that has a agreed upon value. So for example, John thinks his coin is biased, when normally we assume coins aren't biased. That gives you your normal ll hypothesis and your alternative hypothesis. Then they perform an experiment on a sample. So in in the example one, John has tossed the coin eight times to see what happens. Okay, in the question, in a full question, theytell you how many, in this case, how many heads John gets? We pretended he got five. Then what you do is you have to work out the chance of that happening, or a more extreme version. So in our case, we went five or more, because as we said, six is more extreme than five in this context, sometimes thatbe a less than. So for example, if you got two heads, then less than or equal to two would be more extreme. We'll talk more tomorrow about how we decide which race extreme. But you look at what's the chance that that happens under the null hypothesis? So what's the chance of that happening with an unbiased coin? If you get a really high chance of that happening with an unbiased coin, we go, coinprobably not biased, because that that happens quite a lot for an unbiased coin if we get a really small value. So for example, if I made that odno eight, let's say you got eight heads, chance that it gets eight heads out of eight on a fair coin is 0.0039. So 0.39%. So that's that's really, really low. If we get something that's really, really unlikely to happen, assuming it's a fair coin, then we go, you know what? I think I agree with John. The coin's not fair. The critical value, the turning point of whether you decide it's a high percent, like high enough chance of happening or not, is given by this significance level. Okay, so thatbe given in the question. Okay. So if I show you a full question quickly, Hey, guys. So if you just look at question six quickly, again, it's a cotoss 20 tosses, they got six heads. It's two tobut. That's not relevant. They always just give you a significance level. Okay? So there's a 5% level of significance. Okay? So thatjust be given in the question. Okay? So you're comparing, what's the chance of this happening in real life? If we assume it's not biased, we compare that to the significance level. If it's a higher percentage, we go, John's talking rubbish. It's probably not biased. If it's under that significance level, then we go, you know what? John's got a point. This chord might be biased because we can never say with complete certainty because even eight out of eight heads it could happen, but it happens 0.39% at the time. But if you get eight out of eight heads, chances are as a bias coin, which is where this wording comes into play. It's not I haven't said the coin isn't biased, but just there's not sufficient evidence to say it is. Okay, okay, we'll leave it there for today and then tomorrow we'll pick this up. Ward, in looking at how we write it down, how we word it to get the Marks, but essentially that's how our hypothesis test works. Yeah. Okay. So the significance level will be given to the question you compared to the trans happening yet in real life, which is getting a biased result when the coin is unbiased. And then we always, again, I'll show you the wording of it this bit and here. So we're assuming. H nor is. Okay. So we assume that nonhypothesis is we test how likely it is to happen under the null hypothesis. We compare that with a significance level and then we have some sort of conclusion. So we either go, Oh Yeah, it's over the significance level. So the null hypothesis is probably always it's under the significance level. And again, okay, the null hypothesis is probably not depending on that value that we get, okay? But 36% that the time that's quite high. You know that that that that happens a lot. That's over a third. Okay, Yeah, I'll got you again tomorrow. Tomorrow same time. So we'll jump straight up back into it then then okay, thank you. Well done. And I'll say it's tomorrow. Bye.