Equals one P H. 送你过去去。Point one. 81413. More than four times. P X greater. 0.1755. Less than three times. Okay, sir, I'm done. Okay, let's let's hear what you got. Okay, so for one A I got 0.2123Yeah one b is not point 1727. Yeah one c is not point 9972. Good. Yep. And for 22a is 0.1413. Yep. Good. B is 0.1755. Great. C is 0.4113. Yes. Okay. All of those sound. Okay. Okay, okay, it's great. Random variable. In 20 0.415. Smaller than equal to seven. So for three a, do I like do it separately? Like I find that probability of x is greater than three, and then I find the probability of x is smaller or equal to seven, I would subtract them. I would do probability that x is great less than or equal to three, and subtract it from probability that x is less than or equal to seven. Da. 宠。男生回头起会。嗯。The conclusion you would have reached in test that five. So I don't understand like how do you do B3p. To do a hypothesis test Yeah like test at the 5% significance level. Like what does it mean? Yeah. So that means that if your probability of obtaining the evidence as quoted is as bad or worse Yeah than this 5%, then you would reject the null hypothesis. So like 30%, order a starter one day. Random sample seven, order a starter. So to sort of visualize it, let's say you were to, I don't know, visualize it as a normal distribution or something, right? And your observation is that 30% of customers I know one day a random sample of 40 has taken seven order of starter. So how much would that be if you were to estimate it on a on a. Let's do. Yeah so okay, so observation is seven. So you need to work out the probability of obtaining evidence as bad or worse. Yeah as bad or worse will be further away from the mean, which in this case will be less than or equal to the observation of seven. Yeah your probability of success 30%. Okay, so let's write down the distribution. So you've got x. Is binomally distributed Yeah with n equals. And equals just trying to see if we're going to use the that part a is relevant or not. No, I think the part a is. One day 11 day, the English of it is a bit weird as well. 11 day a random sample of 40 customers is taken and seven or a starter. Yeah. So I think for that part a, you can ignore the distribution at the top, which they give you. So that part a is. Not really conjoined with part B I believe. Okay so you're gonna have n is 40Yeah so that's right I'll distribution 40 and then. And then the probability of success is what you're basically testing, right? So you're testing against p where p. Zero 0.30 point three that's what you're testing for Yeah and you are no ll hypothesis. In your alternative hypothesis, okay, these are what you write down is that p is equal to 0.3. And your alternative hypothesis is saying that test at the 5% level, whether the proportion of customers ordering a starter that's p test, whether it has decreased. So the alterso, the alternative hypothesis is asking is p actually less than 0.3? So this is what's known as a one sided test. So question three b. Is known as a one sided test. It's possible. I have a two sided test, and in that case, you would divide the significance level. By two that e half in each end of the tail of the distribution. Yeah but because there's an inference about actually we've got some sort of prior assumption that you know we we won't know if it's if it's actually decreased. Yeah. Now if you were to kind of visualize this just roughly to get an idea of what the expected value would be under this distribution, you can multiply the number of trials and the expected probability of success. So like the expected value of this distribution, Yeah the mean of this distribution, this is just something that you can do to help visualize that. You don't need to do this to answer the question, but the expected values is the product of those two Yeah which is np, which is twelve. So on average, if you have 40 trials and the probability of success is 0.3 on each one, you would expect twelve successes. Okay, Yeah. So let's visualize that. Let's let's put our expectation in the middle of the distribution. Yeah this would be the expected value, therefore, the highest probability of success if we're modeling it as a like a normal distribution, which it might not be, but this is just to visualize it. Yeah in order to understand this idea, which I said about obtaining evidence as bad or worse, because it needs to be shifting away from the mean Yeah, that's what I mean by evidence as bad or worse. Seven. Is our observation. Yeah let's visualize seven on here. Seven would be somewhere like here. So when I say as bad or worse, that's this way because that is away from the mean. Yeah let's say that for example, it said that you know 17 ordered a starter. Yeah. 17 will be over here. Yeah. And then not as bad or worse would be working out this probability. Yeah. So the probability of attaining evidence as bad or worse, Yeah as bad or worse, it would be the probability of x less than three equal to seven. Yeah, that's your that's what you need to compare. And you compare in there with the rejection region of 5%. And so you want to test if that probability lies within this rejection region where this probability in the tail is 5%, because then what you're saying is, well, the probability of observing seven or less. Yeah if seeing that is smaller than 5%, then we're kind of saying to ourselves, well, in reality, the probability of seen that is so so small that based on this initial prior rejection region, which we have defined, that it's so small that we just don't believe that you know that the mean was twelve or that you know the proportion was 30% and it has to be lower. Yeah. So you're kind of testing is this value and the sort of associated area under it? Does it lie within this sort of rejection region? Yeah. And you can if it's a two sided test, in this case, you would be testing whether it's under 2.5% because you wouldn't have this prior knowledge of it you know being shifted less. It could be there or it could be in the in the upper end as well. Yeah. But in this case, we're not concerned with that because it's just a one sided test. So your kind of you know summary from that is you are testing or you are investigating. Whether the probability of x is less than or equal to seven Yeah he's less than under critical region Yeah or significlevel. That e is probability of x less than or equal to seven. Less than 0.05Yeah okay, if it is. Then we reject H. In favor of H1. Yeah. If it isn't, then we accept H noyeah. Okay. Okay. And it could have sent something the similar way around. It could have said, the question should have said on one day a random sample of 40 customers and 17 ordered a starter. And the question might have said, test at the 5% level, whether the proportion of customers ordering a starter has increased. That could have been a plausible quthen you would be testing. Is the probability of x bigger than or equal to 17? You know is that less than 5%? And then youwork out the probability and the cumulative probability and youwould have to do one minus that, because when you're working at a cumulative probability, it always works out less than or equal to. So if you wanted the probability of x being bigger than or equal to 17, you have to do one minus the probability that x is less than or equal to 16. Yeah, because that's what your calculator works out. When you do an accumulative, it always sort of integrates or adds up from the from the sort of left hand side, okay? So it's like P X smaller or equal to seven. 嗯，服务器报。Seven. Point 55 no point 0.0553 so it's greater than 0.05 so so we do not reject. I got quite Yeah it's quite close. I got 0.0553, which is bigger than 5%, which is bigger than 0.05. Therefore, not enough evidence to support H1. Yeah. So we accept H, not the null hypothesis. And then you want to say some sort of wordy thing back to the context of the question. So there's not enough evidence to support that the that the proportion has decreased. Yeah, okay, okay. So have a go at part. See how would your conclusion change if well, okay, that's easy. If you were comparing it at the 10% level, so it's all based on your rejection region. Yeah. If you were instead comparing it against 10%, well, you're 0.0553 would be less than 10% in this case on that test, it would be enough evidence to reject H nand support alternative. Okay. So it's really based on on what significance level are you testing it against? Yeah, it can change the whole outcome. Okay, I would go with profour. Okay. X binomial 30, zero, five, eight. P X greater equals twelve. Greater 0.9849 on particular day, random sample, 40 customers are ordered breakfast, 19 of them ordered coffee. So x binomia distributed value 40. Probability is not point three. So then. Test at the 1% significance level. So. Increased so H H alternative H no is what is it p. Equals 0.3. Each alternative is p larger than not point three. So 19 of them ordered coffee. To be equal to I. 147148, no 148 zero point not 148 is larger than point not one. Mouth evidence. To support H1 and. 5% no point zero 148 smaller zero point not five so enough evidence to support. H no. Making you. Sleep. 真的。Over three. Coffee has increased. Okay, so. Okay, what did you get? For question four or a is not 0.9849. Can said that again no point 99849Yeah Yeah okay. And for b so I did p the probability of x greater zero equal to 19 which is 0.0148 and it's greater than 0.01. So the proportion of the customers orning coffee has not increased. And four c is because 0.01 point is smaller than 0.05. So reject H no the proportion of the customers orning coffee has increased. Yep, that's perfect. Yeah. Okay. Company produces pens. Over see that any penstrenot not sample system sting penis taken for see that two or more pens are defected. I know you're distributed. 15, 0.08, a probability x is larger than or equal, greater or equal to two. Six. 15. 0.3403. Employee claims that the publithat penis defis more than not. When not, they take a sample of 20 pounds, and three are effective. Stating your hypothesis, merely test the employees claim at 5% significance level. X anomally distributed. What you. Non? Pique? I'll turn to the p is greater than not point one big. Five. 23 are defective. Three. It's increased point 212 operate 212 square to the 0.05. So. Accept. H no. Good. That's a pain. Is not. More than. The place tennesone of serves and quarter to 60% expplanomally distributed 20 not point six probability x equals 15. For. G X larger than 15 greater than 15. No five more. And this coach thinks that a proof getting the serving ant court has changed, and he serves 50 times in a set of 35 vcourt stating your hypothesis. This, the coaches lay at 10% significlevel. 50 times. 15. So for like question six b, is it like different? Is it like a tutail test test? Yes, yes, two tail test. So that's where the idea of this obtaining evidence as bad or worse comes into play. That's how you decide about whether you're doing less than or equal to or more than or equal to. So you want to do that probability of obtaining evidence as bad or worse and then effectively, rather than testing it at 10%, once you've decided whether it's you're testing whether it's less than or equal to or more than or equal to. You'll be comparing it against 5% because you assume that there's 5% in each tail. Yeah. Non, il? Il so if probability of x greater or equal to 35 same number Trix 50. 0.0955. So I'll test that the 5% significance level. So no point zero five. Is no 0.0955 is greater than 0.05, so. We accept H normal m. And there is. Not enough evidence. To say that and is. What? Open the. Of getting. He serve. In court has changed. Okasadone. Okay, so let's say what you got for for question five. Question five, a is not point 34 zero 35b, so probability of x greater or equal to three is not 0.212. 0.212 is greater than the 5% 0.05. So we do not reject H null. And the probability that's appdence defensive is not more than 0.08. 0.08. Yes. Yes, that sounds good. Okay. And then for question six, okay, question 66a one is probability of x equals to 15 is not 0.0746 and then probability of x is greater than 15 is 0.051. Yeah see doing one minus probability of x is less than or equal to 15, 0.0510. Yeah, good. Okay. And six b is probability. I did. Probability of x greater or equal to 35, which is 0.0955 and it's greater than 0.05. So we do not reject H. Noll. And because there's not enough evidence to say that and this probability of getting a serve and court has changed, that all looks that all looks good. It all looks good from my end. So I think question seven is a little bit different. What you have to do here? Well, I think a and b. Are as normal per c. When you want to find the critical region, you need to find the threshold values into your values for x such that you're going to get and you can probably do this by trial an error and by testing different x values or by doing the inverse binomial where it's as close to possible, but each tail is going to has have as close to a probability of 0.05. It might not be super close. It might be that one tail is like 3% and another one is two. You know, and that's fine because we're dealing with integer values here. It's just a slightly different question or a slightly different type of question. So you might have to use a little bit of trial and error. Bear in mind though, the two values for x, so itbe x is less than or equal to some value, some integer value, and then x is bigger than or equal to some other value, where those values, let's say x is less than or equal to a, or x is bigger than or equal to b, these integer values of a and b are going to be sort of, if you remember that distribution which I drew here, let's say a was seven and b was 13 or 17. From this diagram up here, they're going to be roughly equidistant around the mean roughly, maybe not completely exactly, but roughly. Yes, there's a little bit of symmetry that you can that you can use to help, okay. Relative biased disliding up six, stop weight four, the dice is going to be right. 20 size, find the probability dice will land on six exactly five times. And near 20 way ability of x equals five. 46. It keeps up oversetimes on line number six is incorrect right down the hypothesis that test polysuspicion. So its H nuis p equals 4.4H1 is p is not equal to not very well. Using a 10% significance level behind the preregion for tutest. Tail, 10% of the tail, lower tail. So not point five 0.5. Upper tail. That x smaller or equal to three. Let's try it three. Okay, so this one, if it's full. That's bigger than 0.5 sets. Three, the don't is this of the tail? 13. 13. 0.0211. About 14 scene. Nope, that's too big. So x not equal to 13, x smaller or equal to three. And the actual significance level of test based on your critical region from part c. So for question, d like to find the actual significance level. Do I just add like the probabilities of the Yeah, Yeah, Yeah. So you've done the hard part. Yeah. So you've got x is less than or equal to three and x is bigger than or equal to 13. So you just need to add those probabilities, those areas which I think are 0.016 and 0.021. So that gives you actually instead of having a total significance level of 10%, you actually only got a significance level of 3.7%. And this is just because you know the we're basically stuck with integer values to work with. But I was Yeah, it sounds like you've done the that you've got x is less than or equal to three and x is bigger than or equal to 13. Yeah Yeah okay. And then Polly rolls the dice 20 times. It ends on the the dice lands on the 6:11 times. Well, your rejection region is x is less than or equal to three or x is bigger than or equal to 13 or twelve. Eleven is not in that rejection region. So you don't need to compute the probability though. It's just based on that. It lies not in that rejection region. So there is not enough evidence to say that the probability is not 0.4 based on the spce landing on six, eleven times in 20 throws that does that make sense? Yeah, perfect. All right. Then I think thatbe a good place to leave it. Good stuff on that. You managed to to get through quite a few, given that you didn't know too much about it to begin with. So that's good. And I think always, if it helps with visualizing it, I like to draw it as a this kind of normal distribution, even though it's not a normal distribution. I think estimating the expected value and then sort of visualizing where things are and think about this idea of the probability of obtaining evidence as bad or worse, which is essentially what we're doing. Yeah and away from the mean. Okay, we'll finish it there. I'll speak to you later. Okay. All right. Thank you. Okay. Bye.