And folks, we're gonna jump over and talk to our man, Teddy Kegstad as we do every Wednesday and you can read Teddy's report, folks. The Tiger Forex report, he comes out with it every week on Monday mornings. He comes out with updates when warranted through the week. You head on over to the newsletter tab at TFNN. You can subscribe, folks. It's $97 a month. That comes with the money back guarantee, folks. You subscribe for 30 days, you're unsatisfied, you don't use it. It just doesn't work with your trading style for whatever reason.  When you sign up, you also gain access to a recent webinar, Teddy did. 60-minute webinar, forex strategies and fundamentals. What is behind the Tiger Forex report? A great webinar, Teddy did. When you sign up, you get the report for a month. You get the newsletter. You get the 60-minute webinar. You can't go wrong. And boy, we got some action in currencies. We got some action in crude. Teddy Kegstad, good morning. Good morning, Tommy. Boy, we got a lot of action, Teddy. I can't help but say what's going on in that crude market with the volatility we've been seeing, where do you want to start as we chop around this morning? Well, we can start with crude. Obviously, they're banging the lows. The bears are very happy and the bulls are scratching their heads right now. So I think one of the biggest thing is supply is good and demand is actually not that strong. I think that's one of the things that are suppressing crude prices right now. I'm still a bull. I think that this is just grinding a bottom out and I don't think it's going to maintain its lower bias over there. I would say definitely the next few weeks. I appreciate the take, man. I got the chart up here right now, even on a weekly basis. We're talking about, we just made a low, man. Pretty remarkable, 72, 75. And as a consumer, Teddy, in terms of demand, I'll tell you, when I see these prices, right, I say, man, I'm going to go fill up my gas tank, man, because $72 for the price of crude seems like a bargain in terms of a risk award portfolio of where it could be. Pretty remarkable, the pullback we've had, though, if you're looking at that market as a buyer, I know you cover it in the Tiger Forex report. Maybe you could talk to us about some of the levels as we come down. I mean, boy, a $10 pullback, man, pretty decisive. We reached $72, is this an area that you'd be maybe potentially looking at for some buys here? If you're looking for that, where are you looking for on the upside? Talk to me maybe a little bit about some of the levels, because I think it's $74 risk award. I'm not a big trader, but as I said, as somebody that buys gas, man, $74, and I know it's not exact translation of what I'm paying at the pump, but it does matter. Boy, I see a risk-award balance that I like at $74 right now. Yeah, I think you got to be a buy-break scenario right now because you have to be very careful with your risk. I think that for sure that any upside potential outweighs any downside potential right now. Like, I don't see crude, even if it was to break another $5, $6, that would be a spike low. I can't see that grinding and holding down. You know, we're heading into winter and we've had a pretty mild start so far. I mean, like, I'm in Chicagoland, man, and it's like 40 degrees, which is beautiful the last couple of days. I mean, that sounds pretty cold to you in Florida, but for the first week in December, we're usually at like, you know, I mean, the upper 20s sometimes bobbling down towards zero already, you know, for the cold temperatures. So I'm liking it. I think that right now it's a tough trade without a doubt, and it's tough to be a bull, but I'm looking for it definitely to bounce back up and get above that $80 mark. You know, I think it's very, very unlikely that if it does get down and stay in these lower 70s that it's gonna hold, I think it's really gonna start to lift again. So especially you can see with the currencies too, like right now, oil is not having such an impact, but every time oil starts to get a rally, you can see it affect the yen and it does start to impact the currency markets, you know? And that's where I think you really have to be tight.  If you're really looking at in the longer term, at best we're setting up a range trade situation for crude right now. And that's where you gotta be careful trying to be a bear because if it's not a true bear market, this is where bears start to get slaughtered over the next couple of weeks, especially as we come in to roll over for December and also the turning over of the year into winter time as well. And could you talk to the listeners, Teddy, because you've covered it many times and boy, this is one of the best educations I got from you early on. We started having you on many years ago, but in terms of crude and the producers, right? And the demand and the supply. And when you look at especially the dollar versus the yen and how the US is a producer, could you just go over for the listeners and the viewers real quick for those that haven't heard it in terms of how that relationship works? Because I think just understanding the fundamentals of that market in terms of crude, the price of crude, how that impacts the dollar and especially how it impacts a currency like the yen, which has such a different part of their economy in terms of the production of crude versus like us, America. Okay, yeah, well, especially with Japan, that's a very good case example because as an ally, we also kind of guarantee Japan, meaning that they buy oil on the open market from all over the place. I mean, they even buy oil from Russia for price sakes. But the reality is we also protect them in the sense that if there's shortages or things like that, they know that they can rely on the US as a supplier. So at least they have been able to in the past. So the price of crude is very indicative of where the yen can be when it's trending. And we saw a lot of that over the last year and a half where especially because of the interest rate differential, and then you had the crude oil differential also that made the yen become very, very weak because you gotta remember that you're not buying oil to begin with, you gotta convert the cash. And then also you had the interest rate differential that slammed the yen, so your buying power also decreased. So every time crude especially would go up, that would hurt the US dollar yen because, or excuse me, the yen, meaning the yen's a bear and the US dollar yen would go up because that starts to add the value on the dollar side. So that's where crude oil definitely has an impact on currency markets, which then also flutters into other things because you gotta remember Japan's a manufacturing country. If they're an oil is something that they rely on for manufacturing, so it's not just automobiles. That's only one portion of what they need oil for. They need oil because they're a mass producer of hard goods, so that's why that really impacts that the Japanese yen currency especially. It's a great breakdown, man, and it makes so much sense when you break it down like that. The price is going up, we're selling it, they're buying it, of course it's gonna translate into it and then you just go down the rabbit hole of the effects and impacts that that can have down the line on the economies and the currencies and yields, et cetera, as we're all seeing. What do you wanna jump to next, Teddy, in terms of currencies? I know we got all the currencies kinda just pulling back dramatically from the moves we've had. You got the Euro 105, man. You got the dollar index basically at 105. You wanna talk about the Euro, the dollar? Where do you wanna jump to? Well, the Euro and the pound are the ones that are the strongest in the dollar index. Now, the dollar index hit a nice spike low on Monday, so Monday and Tuesday, the dollar was strong, obviously today it's coming back. Now, one of the reasons I think it's coming back also is that it's kind of a profit-taking break right now. You have to realize that the interest rate market was much lower earlier today and overnight and then they started a rally this morning as we opened up our markets. So that's kinda hitting the dollar, and I think it's more of a profit-taking move. I would key off of the low in the dollar index from Monday. If that low gets taken out, well, that's obviously bearish because you're making new lows. That's intuitively obvious, but I think right now that's a good pivot point that if we hold above that, if we don't break through that today and instead of new low, I would be very cautious trying to sell against the dollar. I think what you're seeing is a profit-taking move. We're heading into the Fed meeting. It's unlikely that we're not gonna see a rally in the dollar over the next few sessions. Teddy, I appreciate the education, folks. You heard it right there. Check out the Tiger Forex report, folks. Absolutely great. Teddy, I appreciate the time, man. We'll talk to you next week. Sounds good. Take care. Okay, man, have a great one. Folks, stay tuned. We'll be right back to finish up the show.


Hello and welcome to NewsClick. It is now 20 years since the demolition of the Babri Masjid in Ayodhya by Hindutva extremists. To discuss the continuing social, cultural and political impact of that traumatic event and the events that preceded and followed it, we have with us today Sukumar Murlidaran, veteran journalist. Welcome to NewsClick, Sukumar. The Babri Masjid demolition has left a permanent scar on the multicultural fabric of our nation and on our secular polity. Has India become decisively or at least more sharply communalized than before? Yeah, well, when you say a permanent scar, I would not use that word. It was a very deep scar, no doubt about it, it is a very traumatic event. These things can be healed, provided that the nation states institutional mechanism kicks in and does what is necessary and what is warranted, what is mandated by the Constitution to correct these abuses. Unfortunately, that has not happened, so the scar has remained and the scar, if anything, has got deeper. And corrective now will be a much more tumultuous kind of event than would have been the case earlier. We should understand that Babri Masjid, when it occurred, was a decisive blow against pluralism, against the multiplicity of perceptions of our history, of multiplicity of perceptions about our identity, which in a way reflected the true situation on the ground because this is a very diverse country and if we want to hang together as a country, we have to respect that diversity. But unfortunately, this was the precise opposite, saying that we want a single unitary point of view about how we look at our history. We want a single unitary pantheon of nationalist heroes, including stretching us far back as mythology. And we will have corresponding to that clearly identified set of villains who we will identify as our enemies. So, this obviously is not something that very large sections of this country can share in. And there was a coercive effort to impose this vision and the demonstration at Ayodhya was a kind of climactic act in that drama. So, in the context of trying to heal these wounds, what lessons do you think India has learned or not learned since the demolition? Well, in the immediate aftermath, you and I were old enough to remember what a deeply traumatic moment it was. There was a sense of national urgency that this has gone too far and we have to deal with it in one way or the other. And political parties united, except for the authors of the demolition who remained isolated. The others united to kind of present broad based kind of consensus that we should correct this wrong and ensure that it does not happen. Now, there were several tracks on which that could have been pursued. Firstly, the demolition itself, they should have been accountability established for that, who was responsible for it. And then, of course, the demolition was not an isolated event. It came after a long cycle of mobilization with overt acts of violence against religious minorities. And it was followed also by a long sequence of riots in which Bombay, as Mumbai was known then, was one of the worst affected cities. And that was again a Bombay being a city then that so many of India, so much of India identified with. It was very symbolic that the very heart of the nation is being torn apart. So, that moment should have been utilized and that the processes of accountability to hold those responsible guilty under the process of the law without it appearing as political vendetta. No, because there was a clear cut legal provisions under which they could be charged. And also for the riots that proceeded and followed. Unfortunately, none of this happened. Then the political deals were struck and opportunism prevailed. And the principles of our constitution were entirely forgotten. Which is why 20 years after the event, we still can't say that we are immune to a further Exactly, that is precisely my question that since the demolition and its aftermath, the riots in Bombay, there have been many other such incidents coming leading up to today where the administrative actions required have been seen to be lacking. And what is otherwise normally presumed to be the last resort, which is the Indian judicial system has also been unable to show itself capable of dealing impartially with such disputes and resolving them. And in fact, even the High Court judgment, which is the current operational judgment on the Ayodhya dispute still has resorted to arguments relying on faith and belief, rather than on an impartial judicial accounting of the system. Exactly, and what is most alarming about the higher judgment of the Lucknow bench of the Lahabad High Court is that it was a three way, there were three judgments, three separate judgments by a three judge bench. You know, it's sometimes very awkward to say these things, but there was a clear ghettoization of the mind of the judicial approach there. Because the one judge who was from the minority community had a very different perception of the whole thing. He said that this whole controversy is cooked up. He identified clearly that 1949 was an act of trespass, that the process by which the mosque was commandeered by local communal forces and the idols were inserted there was an act of trespass. And that original illegality created that entire political turmoil that followed. And in 1986 also he identified when the doors were opened by order of magistrate. The matter is the title suit pending in a higher judicial bench and there was clear political directive involved in the magistrate ordering that the gates be opened for Hindu worshippers. So these are identified, but then he said that you know, but we can't push this case too much. And he I think quoted something from the theory of evolution about the fittest surviving and so on and so forth and said, you know, let's back off and just allow this majoritarian point of view to prevail. But that is not a judicial accounting in a sense. So that judgment is entirely void of judicial logic. That's right, on either side, on any side in fact. The minority judge had more of the judicial prudence than his judgment, the other prude didn't have quite much of it. So this is what is alarming about it because under the pressure of circumstances with the coercive kind of political environment that prevails, even the judiciary, which is an institution that we all have learned to respect, and which has in several of its determinations shown itself to be worthy of that respect. In this case, it just couldn't do what is right. So fortunately, the Supreme Court has set aside the judgment, used a very strong word, said that it's a perverse judgment. But then how far the Supreme Court is actually going to correct the situation. We don't know. In fact, don't you feel a sense of deja vu in watching what is happening in Hyderabad as we speak with the dispute over a recent or fairly recent Hindu temple come up in the precincts around the Charminar. And you've got the same old arguments being raked up as to how ancient this Hindu temple was and whether it has come up now or then. And still there is no firm administrative action. And one is still waiting to see whether any judicial action will be taken to resolve this dispute and in what way it will happen. Yes, precisely. I mean, that's the most alarming development because the Charminar is as far as I can understand protected monument and there should be very strict rules about any kind of construction activity in its near vicinity. And to see that this has come up without anybody noticing it. And the Hindu has done a great public service by highlighting this issue at this stage. We have to also look back at 1949 and Ayodhya being one among many such acts of territorial conquest, so to speak. Because it was a very fraught environment and partition had just happened. There was large scale violence. Both sides were inflicting horrific violence on the other. And there was a deep sense of ill feeling towards the other. So Ayodhya, there had been a dispute brewing for a long time, but that was purely local. Now by that intrusion of the idols there, that local dispute became, in a sense, identified with the identity of the nation coming to being. So in some senses, suppressing that other identity became a way of asserting the new identity of the Indian nation. Now this has continued and if you consider the correspondence at that time between, say, Prime Minister Jawaharlal Nehru and the Home Minister Sardar Malabhai Patel, Mr. Nehru does point out that even in the vicinity of Delhi, there have been several such cases of mosques being kind of converted through ritual introduction of Hindu icons and Hindu symbols and that this is the most unsatisfactory situation. It has to be reversed. He makes that very clear statement. So this is, in a sense, continuing still even after 60 years. Well, the lesson of this is that this assertion of national identity by opposition and by antagonism towards another is still continuing and with potentially very, very dangerous consequences. Let me ask you a slightly wider ranging question. In matters relating to so-called religious or community sensitivities in India, there is clearly an increasing resort to pressure tactics, sometimes violence in order to push a particular communitarian vision or perspective on issues to push a de facto even if not a legal argument. Whether it is monuments, biographies, novels, paintings, what have you, there is an insistence on pushing one communitarian vision even at the cost of public order through violence etc. Given this increasing trend and given weaknesses of the administrative as well as judicial system in tackling this, how does the modern secular Indian state and Indian society evolve to deal with these challenges? Well, this is obviously a very, very difficult question to address and we are touching at the very heart of how we identify smaller identities and how these are allowed to allow their own domains in which they have primacy, but in some senses are willingly subject to the larger identity of the nation, which is a compound of multiple such identities arrived at through a process of consensus rather than coercion. Now, there could be coercion which induces one small community to give up its language, its observances, its ritual practices and submerge it in the larger, for instance, what is called the Sanskritization process. In some senses it was voluntary, in some senses it was driven by aspiration, in some senses it was coerced. Partly coerced as well. So at some point in India's history, I think a lot of these smaller identities got the feeling that they were getting an unfair deal. You know, if you look at the cultural production of say 30 or 40 years back, you had cultural productions like the Muslim Social, which was a very important genre in our film about the Muslim community and their own internal debates and how they're dealing with challenges of modernization and reckoning with how to preserve the best of that tradition without being bound by all that is regressive in it, you won't find that kind of cultural productions anymore. Now they're more and more focused on what they view as fundamentals and that applies for variety of communities. Now the Nadar community in Tamil Nad is on the war path because of some paragraph in history textbook which says that they were migrants in the late 16th century to what is now Southern Kerala. Now the point is that the current identities, linguistic identities are being read back into history whereas these are all modern creations and artifacts very much of modernization and there's no need to make your current claims, your current entitlements to whatever is the national product and whatever is the other rights guaranteed by the states make it subject to some kind of primeval or primordial identity. We are all equal citizens. Now that is the reality that somehow we are not able to get everybody to accept because the state is unable to live up to that guarantee of equality before the law. And in the middle of all this lack of equality before the law demands an equal recognition of evidence, of history, of fact-based solutions and that seems to be regressing in the face of communitarian pressures, emotional pressures and ideological pushes of one kind. Even if you accept that the reading of history that's reading of facts that the reading of evidence can be different depending on your point of view you have to accept that we should have discourse based on equality at this time and that the doors are open for everybody to come and place their viewpoint without resorting to this kind of political coercion or threat of violence. Now unfortunately that environment is not yet established. Now that essentially is a failure of the Indian state in the sense of its inability to live up to constitutional premises, constitutional guarantees. So it's very easier said than done but I think a lot of it is also a function of rising inequality, the failure of the media for instance to fairly represent diverse viewpoints and now you have increasing dependence on the what's called the market philosophy which has principles of exclusion. So possibly we're moving the reverse direction rather than allowing more and more voices to come into the national debate what we found over the last 20 years despite all the appearances of growing diversity of the media and so on is that the discourse is becoming more and more exclusive. Thank you Sukumar. I think we've raised a lot more issues than we perhaps intended to when we started this discussion. I hope you'll have a chance to come back to some of these issues again fairly soon. Thank you once again.


So does anyone remember Joe Biden? I think that's his name Joe Biden. He was the Democratic Party's nominee in 2020. He was the centrist clown that Obama pushed through Got everyone to drop out and back him when he won one state.  So let me read to you what he said on Twitter He wrote health care is a right for all not a privilege for the few now You might think Mike doesn't that sound like a victory doesn't that sound like a win? Doesn't that sound as if he's adopting the policy that you want? Well, no, it's not a victory. It's not me not taking yes for an answer What he's doing is is he's stealing the rhetoric of Bernie Sanders and in particular Medicare for all proponents But he's not stealing the policy. You see, I'm okay if you want to steal the policy If you want to adopt Medicare for all great I wouldn't necessarily believe you but I think that's a step in the right direction Just adopting it. I think would shift the Overton window to the left But he's not adopting Medicare for all This is the individual who said if Congress were able to in some, you know, historic feat Pass Medicare for all and send it to his desk. He would veto Medicare for all This is the individual who prior to the Democratic primary The health industry was betting on to save their asses This is the individual who literally used his dead son To make it seem as if Medicare for all would be an insult to his legacy So Joe Biden does not support Medicare for all But yet he's co-opted the language that activists of Medicare for all use And that's a problem. That's disingenuous. You see, believing that Medicare for all Is the right policy and believing that healthcare is a right These are ideas that are inextricably linked. You can't disaggregate them They go together. They go hand in hand. Now, Joe Biden isn't the first Democrat Who has used the healthcare is a right language But here's why they're wrong If you don't believe that 100% of the population should have healthcare Not access to healthcare, not access to affordable healthcare, but just healthcare period If you don't believe that 100% of the population is guaranteed that Then functionally speaking, you don't believe it's a right Do we believe that we should all have access to affordable free speech rights? Do we all believe that we should have access to You know affordable second amendment rights like these things don't make sense, right? Because when we're talking about things that Americans perceive to be rights We don't put a price tag on them That means that if you put a price tag on this right, you're automatically excluding people Because if you believe that you have to purchase something, even if it's a public option Well, that is going to exclude people. That means you have to have money That means that affordability is going to be something that the government dictates So that's going to leave some people out.  I want to make healthcare a right But the response isn't to actually make it a right functionally You just want to make it a privilege Make it cheaper and affordable which is subjective And look the affordable care act was a step in the right direction Of course, I think that everyone who is a proponent of Medicare for all will admit this But the problem with the affordable care act is that what is affordable is highly highly subjective And even if it is technically affordable that doesn't necessarily mean that it's going to be robust healthcare So when we talk about healthcare being a right as Medicare for all proponents What we're saying is every single american should have access to health care not health insurance They should have health care and they are entitled to all types of health care mental health care I care dental care. We're not going to limit what they are able to get We're not going to place, you know restrictions on whether or not, you know, they can get into this doctor based on their network No, it's a right a right is a right That means you can't deny it to people on the basis of income. So for him to say this, um, I find it highly insulting You don't get to say this unless you actually support Medicare for all now if you're going to endorse Medicare for all I wouldn't necessarily believe you as I alluded to earlier, but then You are allowed to say that you believe health care is a right, but until then Shut up about this like you don't get to say health care is a right No, that's nonsense. It's insulting and just just stop How are we gonna pay for it?


Hello, and welcome back to statistics chapter 8.1 estimating mu when the population standard deviation is known. This is Monica Wahee, your library college lecturer, bringing you more statistics. So let me just cut to the chase. What this lecture is about is confidence intervals. Okay, so we're going to be talking about confidence intervals. Let's just be honest here. So at the end of this lecture, you should be able to state the formula for the margin of error. You should be able to explain how to choose Z sub C. That's how you pronounce that Z sub C, when making a confidence interval, you should be able to state one of the assumptions behind x that needs to be met in order to make confidence intervals. You should be able to interpret what a confidence interval means. And finally, you should be able to demonstrate how to calculate sample size. So here's what we're going to talk about. First, I'm going to review a few key points from chapter seven. Now, I know you're kind of sick of chapter seven, but I'm just going to pull out a few key points that I want you to think about during chapter eight, because I think it'll help you understand chapter eight. Next, I'm going to explain what confidence intervals actually are. We're going to go through quite a bit of theory. So you understand what we're doing when we're making a confidence interval. Next, I'm going to demonstrate how to calculate confidence intervals. And then I'm going to demonstrate how to calculate n or sample size if you want to know how big your sample should be for a certain confidence interval. So the first one of how to make confidence intervals, that's after you've already had your sample and measured it. The second one, demonstration of how to calculate sample size, that's before you get your sample if you want to know how many you need. Alright, so here's chapter seven, please humor me. I'm just going to pick out a few ideas that I want you to think about. So first of all, remember, from back, the slides I showed you, there were all these different types of inferences. And one of them called estimation, I said, we're going to practice this in chapter eight. Well, I wasn't lying. Here we are. So what's estimation? Estimation is where we estimate the value of a population parameter using a sample sounds like a logical thing to do in statistics. In this situation with the confidence intervals, what we're doing, we're trying to estimate Mu from using a sample. Because Mu is obviously a population parameter. And if you just have a sample, you don't know what Mu is, but we try to estimate it from our sample. And so that's what chapter eight's really about. Okay, next, remember 7.1, right, remember the empirical rule? And remember, I asked you all these questions about the probability of selecting an X above or below a cut point or the percentage of data between a certain group. There was one question though I want you to think about. And that was this question about the middle 68%. So the question was, what are the cut points for the middle 68% of the data? And that literally meant that it had to be 34% on one side of Mu and 34% on the other side of Mu. And when I said, Well, what are the cut points? What are the X's that mark that on this particular distribution? And it was 51 was the lower limit and 80 was the upper limit. So I just want to point out here that this means that the probability of selecting an X between 51 and 80 with this particular distribution was 68%. So technically, that's kind of like you would say, if I select an X from this distribution, I am 68% confident it'll be between 51 and 80. That's another way of saying this. Okay. Okay, now we're going to go skip ahead to chapter 2.7.2 and 7.3. Now remember what we did there? We had already learned about the empirical rule. Now we figured out how to calculate X's for Z's that were in between the cut points of the empirical rule. So most questions focus on the probability of selecting X below or above a cut point that wasn't on the cut points of the empirical rule. So that was most of what you were doing in 7.2 and 7.3. However, there was one question I gave you on the slides about a middle 20%. So I'm bringing that one up. So this was the question. It's what scores mark the middle 20% of the data, right? So it'd be like 10% on one side and 10% on the other. And so what we did was we looked up we looked up what the probabilities were and what the Z's were, and we got the middle 20% of the data. And so the middle 20% of the data was 61.9 through 69.1. So I just wanted you to notice that that also could be said this way. It's like the probability of me selecting an X between 61.9 and 69.1 was 20%. Or I could say this way, if I'm 20% confident that if I select an X from this distribution, it will be between 61.9 and 69.1. So if you've ever wondered how you can feel a certain percent confident, that's how you do it, is you just say it that way, you take a middle thing like this, and you say the probability of doing this, you know, you look at the probability and you say, I'm that confident, that percent confident that the X I select will be between these two cut points. Then I want to remind you what we did after that is we went into chapter 7.4 and 7.5. And this time, we weren't just picking one X, we weren't looking at the probability of picking one X. We were looking at the probability of picking a sample or a set of Xs that gave us an X bar that was between certain cut points, right? So this one was, what is the probability of me selecting a sample of 36 students with an X bar between 60 and 65? That was the way I was doing this question. So and again, we answered the question, right? The probability of selecting a sample of 36, so that the X bar is between 60 and 65 is 41%, which is another way of saying, I am 41% confident that if I take a sample of 36 people from this class, their X bar will be between 60 and 65. So that's how you feel 41% confident. Because that's what I always had trouble with in statistics. I'm like, I don't know how to feel that way. Well, this is how you feel that way. But I wanted to point out this is not like the perfect example. Because what we're going to be doing with confidence intervals is going to be centered around mu, which is a zero. This is lopsided, this is centered somewhere else, because this was a different lecture, right? But what we're talking about is what if you wanted to find the X bar lower and upper limits for the middle 90% or the middle 95% or the middle 99%. That's what you would be saying is that, okay, well, between this lower limit and this upper limit, I'm 90% confident that the mu is going to be in there. That's what you're trying to do with the confidence intervals. Remember the central limit theorem? The confidence intervals work with the central limit theorem. And so I'm sorry, more theory. So remember how we had that neighborhood of people, and we were taking samples of five of them and measuring their BMI, and we were trying to figure out what will we get if we got the X bars for every possible sample of five from that group, right? And of course, we got millions literally of X bars. Okay, and we then we made a histogram of all those possible X bars, because each, each sample of five represented a sample we could have gotten if we'd just done one study. So in real life, we usually only get one sample. So if we get to look at all the different samples we could have gotten, we wonder, well, which one are we going to get? Which, where is our X bar going to be in our sample that we pick because we're only going to pick one. And mostly we're upset because we're not sure how far it's going to be from me. So if you look at this histogram, there's a bunch of samples where the BMI, the mean BMI in the sample, the X bar was less than 20. And there's also a bunch where it was more than 40. And that's probably not very representative, because you see where most of the X bars are between 25 and 30, right? So we get worried if we think about what if we get a sample that's on one of those tails? What do we do about that? So as you can see, just from three different samples, the wildly different X bars. So I could have gotten one of these. So how do I know, right? Like I feel uncomfortable because I know that that X bar might not be close to the mu. Well, the first thing you need to know is don't get too excited about the X bar. Remember sampling error, how you're going to get an X bar and it's probably not going to be me. Well, we know that. And we're we can handle it in statistics. In fact, that's how we handle it. It's with confidence intervals. What we just say is, Okay, X bars there. But we want to create a range or an interval around X bar with a lower limit and an upper limit. And then just say, Okay, fine, we're not we know X bars and right. But within this range, most of the X bars I could have gotten, like if I got another samples, most of their X bars would have fallen between this lower limit in this upper limit in this interval. But you can't actually say all. And I'll tell you why you can't say all is because if you came to me, and you asked me what your grade was going to be, if I said, Well, I can 100% assure you it'll be something between a and F. That's totally useless, right? And so if you said, Well, I want to be 100% certain that between this lower limit and this upper limit, I've got my BMI, I say, Okay, great, we'll make the lower limit zero, which nobody can have in the upper limit 100, which nobody can have there. Now I'm 100% confidence in there. So that's why you don't want all you want a probably you want to know if like 90% or 95% are in there. Because otherwise, the 100%, you could just take two huge numbers, you know, one hugely low and one hugely high and say they're all in there. And that just doesn't get you anywhere. So what I did was I superimposed the Z distribution, the normal distribution on top of that histogram of the X bars. And I just drew a lines for where the middle 95% would be because I just know that it's between negative 1.96 and positive 1.96. Okay, and so that's technically if you wanted your 95% confidence interval, you'd have to find the X at negative 1.96 and the X at 1.96. And then that would be your lower limit and your upper limit of the middle 95% of the X bars. And so you'd say, Okay, I'm 95% confident that the mu is somewhere in there because most of the X bars are in there. So I'm going to keep going with this example. So imagine you just took 100 samples of any size from this population, right? So maybe I'll take 100 samples of like five, we can stick with five, you get 100 x bars, okay, which is not what we normally do. But I just want to give you this example, you get 100 x bars, right? If you figured out the upper and lower limits for the middle 95% of the sampling distribution, you'll find that 95 out of 100 of your x bars were actually between those limits. And how I know this really works is that I use Excel and I can make up people. So I can make up a population of people with BMI's all different sizes. Then I can take 100 samples of five from each of those fake people I made up in Excel. And then I can take the X bars and I can actually see this. This really works really well. That's why people are so convinced of it. But of course, you got to realize that there are five that don't fall in there. So in real life, when you're doing a study, you know, you always ask yourself, am I getting one of the weird ones that fell outside or not? But you try to be 95% confident if you're using a 95% confidence interval, that you're getting one of the ones in the middle. So if you I was picking on 95% because I memorized this 1.96. So if you want the middle 95%, the Z is always the same. It's always 1.96 on either side of me, right? negative 1.96 positive 1.96. It's just always the same, right?  So people like me memorize that one, like 95% is totally the most popular one in healthcare. Everybody uses 95. In the olden days, we used to use 99% more than we do now. But 95% has always been popular. These other ones aren't really that popular, but I just showed them to you for demonstration purposes. I want to pull out some points about vocabulary. So you're going to hear me say these three things, you're going to hear me say critical value Z, and also Z sub C, which is written like the Z with little C under it. Now I remember when my professor first said Z sub C, I had no idea what he was saying. I thought he was saying Z C. So I was like, what is Z C Z C? Well, it's Z sub C. And you'll see that sometimes that whenever you have a like a letter, a little letter, and it's a little below, it's sub, right? So in it and then also critical value, Z is also Z sub C. So what what if you look at the table, you'll see that level of confidence is we use C to stand for that. So if I'm like, I want the level of confidence of 95%, I could say C equals 95%. Right. But if I do that, I'm forcing Z sub C to be 1.96. They go together. So if I pick a C of 80%, then my Z sub C or my critical value C is 1.28. And you know what it means by critical value, like it's critical, you know, that's the cut point where you're going outside the confidence level. So it's critical. That's why they call it critical value Z. So the Z you pick, or even the C you pick determines the width of the interval around x bar. So you see remember where 1.96 was? Well, that's for 95%. If you pick 99%, you'd be talking 2.58. Remember, like three is kind of big, that's kind of out there in the in Zealand, right, like that's close to the edge. So that's what happens is you get this wider confidence interval. And remember, you're going to get a grade in somewhere between a through f. Well, the wider it is, the less useful it is because the less precise it is, you don't really know exactly where it is. But in any case, whatever confidence level you pick, at the end, you're the generic thing you're getting ready to say is I am I'm using an example 95%. I'm 95% confident. The mu is between the lower limit and the upper limit. But you never actually say that you have to interpret it with the thing you did. Like, if I was doing blood pressure, I could say I'm 95% confident. The population mean blood pressure, SPP is between whatever my confidence interval was was like 110 mmhd and 120 mmhd. So that's how you would actually say it. And don't worry about it. I will give you a demonstration. So just to remind you, I kind of am going over this theory in different ways. So it syncs it. A confidence interval for mu is an interview that we've computed from sample data, where the level of confidence or the C this percent refers to the probability of having the resulting interval contain actually the value of mu. So when I say I'm 95% confident, I'm 95% confident this interval has me in it. Another way of saying it is the level of confidence is the proportion of confidence intervals calculated from a random sample size of n samples that actually contain you. So remember when I said, let's say I took 100 samples, if I had a 95% confidence interval, I would expect that 95 out of 100 or 95% of those intervals would have me in it. So again, don't get too excited about x bar. When you get it, don't go Oh, my gosh, it's high or low. Just remember, it's a point estimate, meaning a one time estimate of a population parameter. There can be others like if you've gotten a different sample, you've gotten a different x bar. Okay, so just don't get too excited about it. Because remember sampling error, right? Your x bar is not going to probably be your mu. So when using x bar as a point estimate for mu, you have to deal with the concept of the margin of error. I some books abbreviated me, but the one we're using use uses just to calculate. And that's the absolute difference meaning like it doesn't matter plus or minus the absolute difference between the x bar you got in the mu. So I'm just gonna give you an example of what I mean by e. So if we had gotten the x bar of 60 from these students, and we magically knew mu was 65.5, our margin of error or e would have been 5.5, because that was different. If our x bar had been 90, and then our mu was really 65.5, then our margin of error is 24.5, because that's a difference between 90 and 65.5. And again, it's absolute difference. You always make it plus. I just wanted to remind you what the word margin means. See when you type like a report, and you have margins, like you're supposed to have a margin of, you know, one inch or whatever, it means a little space. And that's what you're doing is you're making a little space of error, meaning somewhere in this space is where your mu is. So that's why it's called a margin of error. All right, now you can't just go around making confidence intervals without thinking about it. So first, I just want to go over the assumptions about x that you have to consider before making confidence intervals, x being the thing that you're measuring that you're making your distribution out of, right? So first, x is out there in the world and you're drawing a simple random sample of size n from a population of x. Notice this is a simple random sample. This is not a biased sample. Remember how like if you get a convenient sample, it's probably probably pretty biased. You can't do the confidence interval thing, if you're getting a bias sample. So you're trying to get a simple random sample of a particular size from a population of x values that are hopefully normally distributed. Because as you can see, under three, if x itself has a normal distribution, then we know that x bar will have a normal distribution no matter how big our sample is. So we like it if we're drawing from a population of x values that has normal distribution. Oh, and also two says we have to know the value of the population standard deviation. So if we're doing that, and especially if we meet three, that the x itself has a normal distribution, we're really excited about making our confidence interval because we know it'll work really well. But if we're in the situation where you see four, we aren't sure about excess distribution. Remember, we only have a sample. We don't have a like a lot of data from the population to even figure that out. So if we're not sure, we should get a sample of at least 30. But as you can see by five, if it's definitely not normal, like it's skewed, like remember how we looked at some parameters from hospitals, right, that are totally skewed. So if I was sampling those hospitals, I'd be shooting for a higher sample size, like 50 or 100, because we're taking a simple random random sample of size and and ends got to be pretty big. If you've got skewed data and you want to make a confidence interval. So basically, what you're thinking about is your x and if you got a simple random sample, if you have a population standard deviation line around you can use. And then if x itself has a normal distribution, you don't need to worry about your sample size. If you're not sure you want at least 30.  Now for a confidence level C, the critical value Z sub C is the Z score such that the area under the curve is between negative Z sub C and positive Z sub C. And if you're like, you're just repeating yourself, I am kind of repeating myself. But I'm doing it in different ways because I know this is really difficult concept. And so sometimes people need to hear it differently. But this is one way of saying, okay? Now I'm going to go on to talk about how to calculate a confidence interval. And trust me, it has something to do with all this theory. So here are the simple steps, right? So the step one is you make sure you have your x bar, right, because you took a sample, you better have an x bar. And of course, you're going to have an end because you took a sample and you know how many people you measured. But then you also need to make sure you have your standard deviation for your population lying around because you need those ingredients for this. Next, you have to pick C. And then that connects to whatever Z sub C you pick. So you see the table on the right, you have to select your C in the Z sub C that comes with it. So you're going to be given your x bar, your population standard deviation in your end. And then you got to get picky about C and Z sub C. But once you've made those decisions, you calculate e or the margin of error using this formula. So see that there, beautiful formula, right? Then once you get that e, that margin of error, you so you have to get the lower and upper limits for your interval. So to get the lower limit, you subtract e from your x bar, right? So that's going to make x bar forced to be in the middle, because five is you add e to your x bar to get the upper limit. So I'm going to just walk you through these steps with an example. But that's all you have to do. It's almost like the theory was harder than actually doing it. I did want to do a shout out to this because the first time I learned this stuff, I could not get over this. I was like, Okay, I get we're trying to estimate me from a sample. But if we have a population standard deviation, why don't we have the population mean, like we obviously had to measure everybody to get the population standard deviation. Why don't we just look at the mean at the same time? Like, why are we even doing this? So that was like a really difficult conceptual problem. But then I worked for a very short time at the Department of Public Health, the state of Massachusetts. And what I realized was when you're doing like population level health, the mu can and often should change. But the variation, which is measured by the standard deviation tends to say the same. So imagine that we studied at the Department of Health or Public Health, a community where the blood pressure, the mu blood pressure was just way too high. We'd study it, we find this really high mu and we'd also find the population standard deviation of blood pressure. Then let's say we did work to try and lower the mu blood pressure. And we wanted to see what it was like a few years later after we did all that work. Well, we can't measure the whole population again. We'll just take a sample of them. But we can use the old population standard deviation because that generally doesn't really change that much. It's just the mu that's going to change. So that's like this huge explanation because I had this huge conceptual problem with this. And hopefully that will solve your problem. If you have this problem. And if you didn't have this problem, you probably just sleep better at night than I do. And I just to illustrate this, I this with this slide was available online. So I took it. It's just an example of one of the findings from this article on total knee arthroplasty, which sounds perfectly painful. But in any case, as you can see, this is BMI, you know, since we're on the subject, their BMI their mean BMI changed as time went on and went up. However, I can't demonstrate this to you. But in the paper, you'll see that they're they had a standard deviation. And that standard deviation really didn't change much. So if you're making it, let's pretend you measured everybody, the population in 2003 on the slide, we're just pretending. Then you wanted to see what things were going, what was going on in 2007, you could use the old population standard deviation from 2003, and just apply to 2007. But as you can see, even if you took a sample of those people in 2007, you probably get like kind of a higher x bar because there's a higher mu. Alright. Another thing I want to get into before I actually demonstrate this is that for you standard error fans, you might have noticed that in the margin of error calculation, the standard error is hiding. So remember, I touch out a prep cook your standard error because you can use a lot and then you don't have to deal with thinking about square root of n being underneath something. So the formula actually for the margin of error could technically be written like this. So if you're doing a quiz or something, you want to keep track of your SE, your standard error, and reuse it if questions are asked about different levels of confidence, because only your Z sub C would be changing that your SE would be the same. So this is just a shout out to standard error fans. Okay, now I'm going to walk you through actually calculating a confidence interval when the population standard deviation is known to give you an example. So here's our example. So remember the class of students that I was in that was telling you about, where we had such a hard test that nobody got 100%. So in that class of students, there was we studied the population standard deviation and we found it was 14.5, right? But now it's a new term, right? Same college new term. And we have a new class of students. So we want to know if their mu is different, like maybe the teacher got a little better. And so maybe their mu is a little higher. Who knows. Okay, so step one is make sure you have your x bar, your population standard deviation and your n available. So we're going to take a sample of 100 just because it's easier that way. And we I'm going to say we got an x bar of 63, right? That's not a very good score on 100 point test. But that's it's a hard test, right? And we have our old population standard deviation to use. So we've done step one. Okay, so we have our x bar, our population standard deviation and our n. Now let's do step two, where we have to pick C, which is the confidence level. And that forces you to use the Z sub C at that confidence level. So just as a demonstration, I'm going to pick 99% for C. So the Z sub C is 2.58. But I could have picked some other one, I just picked 99% because I felt like it. All right. Step three is to calculate the margin of error or e. So remember our formulas up there. But then I you know, and standard error fans, you know, there's a standard error in there. So I just first put up the numbers we need their x bar is 63 remember, population standard deviations 14.5 our n is 100, our C, our confidence level is 99%. So our Z sub C is 2.58. So we just plug it into the equation. And we get the e is 3.7. Okay, so that's step three, we got e, which is 3.7. Now step four, you want to get the lower limit of the confidence interval, you subtract e from x bar. So our x bar was 63. And we got our new e look I put in the red of 3.7.  So that's our lower limit. And see, I put that on the slide now. Now step five, you add the e, which is 3.7 to the x bar, which is 63. And you get 66.7. So you get your upper limit. And then what do you say, you say, I am 99% confident the mu of the new class of students is between 59.3, which is a lower limit, and 66.7, which is the upper limit. And if we just think logically about that, that's about like 59 to 66. So if I was talking to you about your class, and you're like, Oh, my gosh, it was so bad, the mean was 63. And you also tell me that the 99% confidence interval is 59 to almost 67. It really tells me everybody did pretty bad in that class. Like, if 99% of the samples you could have taken would have gotten an x bar in between those two numbers, people are not doing well in that class. All right. So I just gave you one demonstration of how to calculate a confidence intervals because it's actually a lot simpler than the other stuff we were doing. Like you just plug in the numbers and you do that and you do the interpretation, it's pretty easy. But sometimes you don't want to know that the you're not trying to estimate mu. In other cases, you're actually trying to figure out how many people to have in your study in order to estimate mu. So I'm going to show you how to do that if that's your problem. So I started by just putting up the steps to calculate sample size, when the population standard deviation is known. So I just want you to look at the formula. It's actually the same formula just rearranged. So it's solved for n, right? Remember that square root of n, which is under underneath the standard error. Well, how we how that gets dealt with is you see on the right side of the question, you have to square things, right? So basically, if you just look at that formula, you realize in order to calculate n, you have you need a z sub c, you need a population standard deviation, and you need an e. So here's how we're going to go about it. Step one is make sure you have your population standard deviation around and available because you're going to need it for this. Next, you have to make a decision, you have to pick C. So this is before you're doing your study. So you're saying, Okay, how big of a confidence, how much confidence do I want when I'm done with my study, right? If you want 99% confidence, you're going to have you're going to end up with more needing more sample, right? Because then you'd end up putting that 2.58 in for z sub c up, up in that equation, because it's on the top, it'll inflate the number. If you are don't need to be so confident, like if you look at 95%, you get a discount, that's only 1.96. Then you don't need as much as so it depends on much money and effort you want to make. So so you have to pick C, so you can get your z sub c, so you can put that in the formula. Then next, you have to also pick how big of an E you want. So remember how big of a margin you want. If you pick a big margin, then you won't get a very precise estimate, right? Like you you won't get upper and lower limits that are kind of close together. So you can kind of know where it is, right? So you've got to also make a political decision of how big of an E you want. And then finally plug all that into the formula and you calculate the sample size of the end. So I'm coming up with a new question. How many students do I need? If I want a 95% interval, and I want my E to be five, that's the way I start the question. Okay, how many students do I need if I want this and that? Well, what does that literally mean? Let's just make a list. First of all, we're going to use our population standard deviation. So I threw that in there. Right. And then because I wanted a 95% confidence interval, our C is 95%. And then we're stuck with our Z sub C, which is 1.96. Like I was showing you on the last table, we were looking at. And I decided I wanted my E to be five. So I put that in there. So that's steps one, two and three. So step four is calculate. So let's calculate. Alright, so what do we do? We plugged it into the equation. So we have Z sub C, which is 1.96 times the population standard deviation. We have the product of that divided by our E, which is five. And then we take that whole thing to the second, and we get 32.5. So that's our end, right? But you need to like always round up due to not being able to have like 0.3 of a person. So sometimes people ask me, Well, do you round down? Because it's under 0.5. I'm like, No, because you need at least 32.3 people, and you can have 0.3 of a person. So you always have to round up.  It's just the way people work. So that therefore your answer is actually 33. So if you wanted a 95% confidence interval, and your population standard deviation was 14.5. And you wanted your margin of error to be five, you would need to get a sample size of 33 people. So we finally made it to the end of this lecture. In conclusion, I reviewed I started by reviewing some slides from chapter seven. I know you're kind of done with that. But I reviewed them because they related to the subject of confidence intervals. Next, I explained Z sub C, and the level of confidence and the whole concept of the point estimate of x bar, then I finally went into some application, right? Enough of the theory. Let's actually construct a confidence interval when the population standard deviation is known. And then so I did one of those for you. And then we did an example of how to calculate sample size in case you are getting ready to do a study. And you need to figure out how many people you needed it based on the population standard deviation you have, and your choices of confidence level, and of the size of e. And so if you thought you were 95% confident or 99% confident, here is someone who is 100% confident. That is a very confident peacock on that slide.


OULD alright bet ిమినిండ్హ ఆదారిక్ మ్క్టథ మిరిక్టవిట్మ్ట్ న్క్మ్టటిట్వ���ి పియదియిజా Αినా Kirstyరిక్.tight�ినేనండి పరినినిండ ύటాండంద Olafపఱ Weak people are dying, but hoped, transformed, parties and sun may go back to thefi, ozzarella farmers be looking for someone who can provide hope. మారీబలె మారీలె ఫరారె కారెహె ని యార్మినె. నారేయో కిల్టో కిస్. పెపర పయారె. Zambia today is going through economic challenges, serious economic challenges and the PF think that they can buy off everyone. That is a big mistake. That is a big mistake but PF you are making a big mistake in thinking that you are going to flash money because people are hungry, people don't have money and you are going to flash money, give personal money here, here, here and everyone will vote for you. You will be in for a rude shock. You will be in for a rude shock. Clement Temple. The money that you have amassed through your presidential initiative which you want to be giving around in Kavada. That is not campaign. That is not campaign. Zambians have grown beyond appreciating monetary gifts They have grown beyond that. They will get your things but they will still vote according to the way they want. They will still vote according to the way they want. So don't fool yourselves. I have seen a number especially so called aspirants. Aspirants. They are busy dishing out money. One good example of Clement Temple is the one I know of. He said, no, I am not the one and whatever. Come on, we have seen you guys dishing out money.