So, we have understood that there is something weird going on when it comes to the atoms. So, there is one more weird thing about the electron revolving around the nucleus that is came up after Maxwell proposed his theory, according to Maxwell whenever charge accelerates it will emit E m wave. Now, electron if it is moving around the circular path it is accelerating right. So, it will emit E m wave fine. So, this electron that is moving in the nucleus like this will emit E m wave. So, if this emits E m wave what will happen to its energy? Energy decreases will go down. So, energy should go down for the electron right and what is the formula for energy of an electron minus 1 by 2 1 by 4 pi epsilon naught e square by r fine. Now if energy decreases what happens to the radius what will happen to the radius? Decrease or increase? Decrease right, suppose this is right now minus 5, this is a negative quantity no doubt suppose it is minus 5. If minus i have to decrease what it should become minus 6 or minus 7 and if it has to become minus 6 the r should be less. So, that this is more negative fine. So, electron if it is emitting E m wave its radius should decrease and slowly and slowly it should spiral and come and hit the nucleus like this. Why earth does not do like this? Because earth has no charge it is happening because of gravity fine. But according to Maxwell if charge accelerates it will emit E m wave. So, if Maxwell has to be correct according to that the electron should revolve and spiral into the nucleus but that is not happening fine. So, another weird observation. So, hence there was a requirement of a different kind of physics that could explain what is happening because nobody is able to explain what is going on. So, Bohr came up with quantum physics to explain what is going on fine. Bohr has proposed hydrogen atoms model and he has given three postulates when it comes to hydrogen like atom. So, we will first talk about three postulates or three principle on which the hydrogen atom works right down the first one electrons can revolve in certain stable orbits without emitting. So, he started with this that he was like if there are certain stable orbits electron if it revolves around those stable orbit it will not emit E m wave even if it is accelerating fine. So, the second postulate right down for these stable orbits for these stable orbits how you identify these the angular momentum should be quantized angular momentum of electron is quantized. What is the angular momentum of a mass moving in a circular orbit m v i m v which is a linear momentum cross width. The thing is this is the electron's path fine this is the velocity. So, its momentum is also along this line. So, perpendicular distance of the momentum from the center is radius only. So, m v into r is the angular momentum fine. So, basically m v r when this is equal to n h by 2 pi. If an orbit satisfies this equation if orbit satisfies this equation then that is a stable orbit. So, electron can be in that stable orbit without emitting E m wave, but if it is not in the stable orbit then what will happen then it will emit E m wave and it will reduce its radius and go to the next possible stable orbit third postulate. Electron can jump from one stable orbit to another stable orbit by emitting or absorbing a photon the focus is a photon. It cannot absorb more than a photon more than a single photon it cannot only one photon it can absorb. So, by absorbing or emitting a single photon whose energy is difference in the energy of those stable orbits. So, if energy of one stable orbit is even other is E 2 even is more than E 2 fine. Let us say energy of one orbit is even and other is E 2 and even is more than E 2. So, if it is jumping from even to E 2 it will emit or absorb photon emitting and it is going from E 2 to E 1 absorb how much energy E 2 minus E 1 or E 1 minus E 2 right and that should be energy of a single photon it cannot be that that energy is coming from two different photons fine. So, there is a fixed wavelength it can absorb which is corresponding to the energy difference and there is a fixed wavelength it can emit that is why it emits or absorb only fixed number of what is revivalence are you getting it any doubt on this three postulate no. Now, you will see that whatever was the experimental observation will be able to use these postulate to match and you will see that both are match for example, let us take postulate number 3. According to postulate number 3 energy of the photon should be equal to let us say this is h times mu this should be equal to what E 2 minus E 1 fine. So, mu is what C by lambda this is E 2 minus E 1 for the electron right. So, 1 by lambda becomes what 1 by lambda is 1 by h into C. Now, E 2 is what E 2 is minus half this is total energy minus half 1 by 4 pi epsilon naught E square by E square by let us say it is R 2 square sorry R 2 only fine minus of minus 1 by 2 1 by 4 pi epsilon naught E square by R 1 square getting it fine. So, R 1 and R 2 are what radius for energy level 2 and energy level 1 fine, but then there are no integer coming up right now getting it. So, just keep it like this we will come back to this expression once we are able to derive R 2 and R 1 in terms of an integer. Now, tell me if this is the second postulate if n v r is equal to n h by 2 pi can you tell me what is the total energy I do not want energy in terms of r I want energy in terms of n n you have one more equation. So, you can put the energy in terms of n v square by r that is 4 equation is still valid n v square by r is equal to k q 1 q 2 by r square that is valid and this also is valid. So, you have two equations to play with find out total energy equal to are you able to or should I do it ok. See we have this equation n v square by r is equal to what I will just write k here k times e square by r square this we have right. So, one r we get cancel. So, I will get m into v square is equal to k times e square by r fine. Now, continuity is very easy to find you can just multiply half to it will get continuity this which you have already found earlier itself fine, but then I want kiting energy to be in terms of n or total energy in terms of n fine. In order to do that I will square this equation let me square it I will get m square v square r square is equal to n square h square by 4 pi. So, I will get m v square is equal to n square h square divided by 4 pi square m into r square this is what we will get. So, you will get half of this is equal to that fine, but it is in terms of r only you have this second equation. If you equate them if you equate them you will get the value for r this is equal to n square h square divided by 8 pi square m r square. So, I will just expand k also now this is 1 by 4 pi epsilon naught. So, 1 pi will get cancelled 4 to the 8 1 r will get cancelled. So, r will come out to be n square h square by n square h square epsilon naught by pi m. So, this is what I will get. So, I will get n square e square fine this is r. Now, when you substitute that r there you will get the kiting energy. So, kiting energy will be what equal to 1 by 4 pi epsilon naught that is k into e square divided by 2 times r which is n square h square epsilon naught divided by 2 pi m e square pi will go away this is 2. So, you will get m e to the power 4 divided by h square epsilon naught 2 times of that epsilon naught square in fact. So, this is equal to 1 by n square fine. So, this is kiting energy all right what is the total energy minus of this minus of this. Let me just verify whether we have made correct calculation in power 4 there is a factor of 8 coming in the denominator which we have got kiting energy is this only right k e square by 2 r will be 2 coming in. So, this is 8 this completely goes away. So, this is gone this will be 4 anyways. So, we have made some cylinder somewhere this is actually 8 a factor of 8 will come here somewhere we have made a cylinder here. So, we have total energy is what this is kiting energy. So, total energy should be negative all this fine this is kiting energy. So, total energy should be equal to minus of m e to the power 4 divided by 8 h square epsilon naught square into 1 by n square. So, instead of e to n even I can substitute that fine rather than putting it in terms of radius I can put in terms of n where n is not principal quantum number that is an integer fine. So, if you put it here you will get e to which is minus of m e to the power 4 divided by 8 h square epsilon naught square into 1 by n 2 square take that common will be 1 by n 1 square fine. So, if you put minus inside if you take minus inside this thing will become n 1 and this will become n 2 fine and then this whole thing this this and this will become a Littberg constant when you put the values you get the same thing 1 by lambda is a Littberg constant 1 by n 1 square minus 1 by n 2 square where n 1 is less than n 2 fine always this remember it is this n 1 which determines what series it is the lower number if the electron jumps from 3 to 2 which series it is? If it goes from 2 to 4 what it is? lower number determines what series it is. So, now we have a model based on which we can understand what is going on in hydrogen or hydrogen like atom for hydrogen like atom the same formula becomes like this 1 by lambda is equal to Littberg constant into z square 1 by n 1 minus 1 by n 2 square this is for hydrogen like atom having atomic number more than 1 I think it and you can arrive here easily when you write this equation m v square by r if it is higher atomic number what you write here 1 by 4 pi epsilon naught q 1 is now z times e and this is e divided by r square and m v r is still equal to n h by 2 pi. So, if you start like this you will get it here. So, you will you will be able to find out radius in fact we have found radius first in terms of n it is proportional to n square energy is proportional to 1 minus 1 by n square you getting it similarly you can get velocity. So, now let us try to find out the energy corresponding to nth level we have derived an equation right how much that was equal to minus of energy in the nth all way how much this was minus m e to the power 4 divided by h square epsilon square into 1 by n square. So, when you substitute all the values you will get energy in the nth orbit to be 13.6 divided by n square electron volt in terms of joule how much this will be into charge of electron. So, energy in the nth orbit is this and if you take that third postulate post third postulate simply you have to do this you know minus 13.6 n 1 square minus of minus 13.6 n 2 square this is delta e change in energy that should be equal to h c by lambda of course, this should be multiplied with charge of electron e getting it e 2 minus e 1 sorry e 2 minus e 1 should come n 2. So, e 2 minus e 1 is equal to h c by lambda same thing we have written fine. So, the Leimer-Bahmer passion and everything was just experimental finding you can use that whenever they ask you to find out the wavelength you can use that readymade formula 1 by lambda is written constant everything else fine. Suppose they ask you to find the energy you can use this and suppose you are confused what to do then use this this is safe as well because you are going by logic here.


With advanced economies having reached the end of the process of industrialization, we've witnessed the rise of new forms of highly complex challenges that have moved to the forefront of social concerns. Challenges such as climate change, inequality, environmental degradation, terrorism, global financial instability, multicultural integration or cybersecurity. These are challenges of a qualitatively different kind to the core issues we faced during the process of industrialization. If we turn the clock back 100 years or 200 years, the kind of challenges we faced, although complicated, were much simpler in their nature. The challenge of building and coordinating the railroads, building the modern nation state, and the bureaucracy required to provide its public services efficiently. We had the challenge of building skyscrapers for the first time, of flying around the world or landing someone on the moon. Today we can manage complicated rail networks that span the entire European continent. We can expect to have a nice meal and watch a film while on flight to the other side of the planet. We can build skyscrapers virtually anywhere in the world, like cutting out cookies. All these problems are certainly not simple, but they are of a different nature to a problem like escalating inequality. And it is these complex systemic challenges that are a central part of our 21st century reality. In fact, we may note that many of these wicked problems can be understood to have derived from the set of solutions we developed during the industrial age. These industrial age systems of organization, though at once great achievements, have in many ways reached the end of their life cycle and were left with some of the limitations in their design. The combustion of fossil fuels that served us so well for centuries being a classical example. As Einstein would say, we can't solve problems by using the same kind of thinking we used when we created them. These wicked problems we face today require a new skill set and new, more collaborative systems of organization to try and tackle them. Wicked problems are highly complex problems. They are unstructured, open-ended, they are multidimensional, systemic, and may have no known solution. In many ways, we can think of wicked problems as essentially disfunctionalities within a complex system. Where we have the complex system of the human body, we have the wicked problem of healthcare. Where we have advanced economies, we have the wicked problem of extreme inequality. And where we have the internet, we have cybercrime. In all of these cases, the problem cannot be isolated and separated from the system. These wicked problems are systemic in nature. They can be understood as an emergent phenomena of how the local components interact, of how the whole system works, and not simply one part of it, that can be isolated, tackled, and solved in a traditional linear fashion. The concept of a wicked problem first arrives from social planning and political science, Horace Rittle and Melvin Weber formally described the concept of wicked problems back in the 1970s, contrasting them with relatively tame, solvable problems. So let's firstly talk about some of these tame problems. Examples of tame problems might be solving a jigsaw puzzle, finding the shortest route from A to B on a map, or repairing a computer. These standard problems share a number of features in common. Firstly, they have a clear boundary. There is a well-defined and stable problem statement that has a definitive stopping point. That is, when the solution is reached, it is relatively simple to say what is part of the problem and what is not. Next, there are a finite number of elements interacting, and given enough analysis, we can fully understand the system, the drivers in the system, and the causes and effects. Think about fixing a car. It may be complicated, but there is a manual somewhere where all of the parts and interactions within the car can be known. With these tame problems, there are typically widely shared values to the nature of the problem. Thus, it becomes a technical problem where there is a solution that can be objectively evaluated as being right or wrong. And lastly, we're typically dealing with a static problem. It does not change over time. Think of fixing a light bulb. The actual problem stays the same throughout the process. These tame problems can be standardized. That is to say, they belong to a class of similar problems, which can be solved in a similar way. And thus, over the course of the past few centuries, with the rise of modern science, we've developed a powerful and standardized set of methods within mathematics and engineering for solving tame problems, even when they're very complicated. Wicked problems are qualitatively different from tame problems, largely due to the fact that we're dealing with open systems instead of closed systems. Climate change is one of the best examples of a wicked problem, as is the challenge of global governance or multiculturalism. All of these challenges have a number of features that make them highly complex and non-amminable to our traditional approach in which problems are defined, analyzed and solved in sequential steps. Firstly, because we're dealing with open systems, they are often unbounded in space and time. They cross disciplines, borders and departments, making them very difficult to structure. You will often hear the word messy used when people talk about wicked problems. They are also open in scope. The more you probe into the problem space, the more it expands into different dimensions. Unlike our tame problems, these wicked problems have no stopping rule. Terrorism, again, might be a good example of this. It slips through the fingers of powerful governments and searching for its root cause will take you on an open-ended exploration of many different socio-cultural, political and economic dimensions. Wicked problems are multi-dimensional. Not only are they interconnected on many scales from the micro to the macro, they are also composed of multiple different social, economic, technical and environmental factors. There is no one standardized global solution that fits all. Coupled with this is the fact that these problems involve a strong cultural dimension. They are often deeply qualitative in nature, involving human values deeply entrenched socio-political dynamics and vested interests. Again, climate change is a good example as is systemic political deadlock. Next, these problems have many deep interconnections and interdependencies. The systems we are dealing with with these wicked problems are intricately interconnected, interdependent and non-linear. The problem of tackling poverty might be an example of this. It is strongly interconnected with many different areas from economic inequality to racism to education to public services and so on. With these highly interconnected problems, when you try to pull on one part, you end up getting the whole thing. Lastly, complex problems are dynamic in nature. They evolve over time. Events are continuously changing. Problems and solutions are entwined as they co-evolve, with the definition of the problem itself also evolving. Coupled to this is the fact that the different components are often autonomous and adapting to their own incentives and local environment with limited centralized authority. The challenge of global governance is a good example of this where authority to act still resides on the national level and thus the international political environment is the product of continuously evolving local level interactions and adaptations. Overcoming this to find common global consensus and coordination is proving very difficult. In response to the intimidating nature of wicked problems, many people feel overwhelmed and respond with denial of the problem or resignation and determinism resulting in inertia. Another typical response is to try to turn a complex problem into a simple one in a number of different ways such as describing a related problem or sub-problem that you can solve and declare that that is the problem or creating objective parameters by which to measure the solution's success with whatever is not measured being then free to absorb the real problem. That is to say just shifting the problem from one place to another which is very easy to do given the scale of these problems. Attempting to tame a wicked problem in this way while appealing in the short run will clearly fail in the long run. The wicked problem will reassert itself perhaps in a different form or worse the tamed solution may exacerbate the problem and divert resources. Truly tackling complex problems requires systems thinking to see the bigger picture and integrate the many different perspectives and domains to the problem. By seeing the system as a whole we can get a real idea of the true nexus that makes up the core constraints at the heart of the problem. We need to gain a fundamental perspective that encapsulates the many diametrically opposed views and then work backwards to be able to contextualize the diverse perspectives on the problem. Wicked problems involve social complexity and thus we need to develop a platform for building a common understanding of the problem. This means being able to communicate the complexity of the problem in a way that is accessible to all, creating a paradigm within which people can integrate their own context and meaning. Tackling these problems requires a strong infrastructure of trust and collaboration to overcome the need for a long term sustained investment where there is a lack of short term incentives. Analysis of the stakeholders and the power dynamics surrounding the situation are critical to successfully designing stakeholder collaboration. Empathy and active listening are required to be able to effectively map out the diverse set of views and opinions that need to be integrated. Finally, having created a platform for the co-creation of solutions we need an iterative experimental approach to developing the project due to the co-evolutionary nature of problems and solutions and we'll talk about this further in the coming modules. In this section we've been talking about wicked problems, highly complex problems that represent systemic failures within a complex system. We looked at some of the characteristics of tame problems that can be standardized, classified and solved through routine procedures. We then went on to talk about some of the features to complex problems including their open nature, their unstructured messiness, multi-dimensionality, their systemic roots and lack of known solution all of which requires a different, more collaborative approach to tackling them that involves harnessing qualitative values in order to get deep engagement, collaboration and long term commitment from diverse stakeholders.


Good morning, good day, and good afternoon. This is Monica Wahee, your library college lecturer, here moving you through chapters 7.2 and 7.3 z scores and probabilities. I decided to mush these two chapters together because I thought they actually kind of belong together. I didn't really understand why they were separated. So at the end of this lecture, you should be able to explain how to convert an x to a z score, show how to look up a z score in a z table, explain how to find the probability of an x falling between two values on a normal distribution, describe how to use the z table to look up a z corresponding to a percentage, and describe how to use the formula to calculate x from a z score. Well, that sounds like a lot, but you'll understand it at the end of this lecture. First, I'm going to go over what a z score is and what the standard normal distribution is. Then I'm going to talk about z score probabilities and what those are. I'm going to show you how to use the z table to answer some harder questions besides the ones I talk about during the z score probabilities section. Then I'm going to show you how to use a slightly different formula to calculate x from z. Finally, I'm going to just remind you some tips and tricks about using z scores and probabilities correctly. So all this talk about z scores. So what is a z score? And what is the standard normal distribution? Well, let's take a look at this very pretty thing I made. You may recognize it from the last lecture. It was my little empirical rule diagram. So remember the empirical rule. Remember how it required a normal distribution. Well, that worked well for the cut points available, right? Like mu, mu plus or minus one standard deviation, mu plus or minus two standard deviations. If we ask questions that were right on those cut points, we had good answers. But what about in between those cut points? So I wanted you to notice in this empirical rule diagram, these numbers at the bottom, like I just circled them like negative three, negative two, negative one, and then mu doesn't have a number. So pretend there's a zero there. And then there's one, two and three. Okay. That is the standard normal distribution. And that is also called Z. So these things on the right, those are Z scores. So see the green area, zero is the Z score that's on the lower limit of that. And one is the Z score at the upper limit of the green area. So you can see that this whole curve, the standard normal distribution on the right, whole the me with the whole curve is zero in the standard deviation of the whole curve is one. And that is what Z score is. So I just want you to notice the concept of standard. I'm in the US and in the US, we use, you know, the US dollar. But one of the things I've noticed is that a lot of countries see it as a standard. So they'll map their currency to the US dollar. So maybe the Euro will map its currency to the US dollar. Maybe the Egyptian pound will also map its currency to the US dollar. And once it does that, it's a lot easier to compare them, right? And so that's the main reason for the standard normal distribution is it helps you compare X's from different distributions, different normal distributions that have different mutes and different standard deviations from each other. It helps you map them to this normal standard normal distribution here that's standard. So you can compare them. So let's talk about Z scores. Every value on a normal distribution, so every X can be converted to a Z score. Just like I was saying how you can convert any currency to dollars, there's some formula for that. So you can convert every X on a normal distribution to a Z score. But you have to know how to use the formula, right? And what goes into that formula? Well, first you need the X that you want to convert to a Z score. So you need to pick one. Then you need to know the mu of your distribution, your normal distribution, and the standard deviation of your distribution. And here are the two formulas that are used. The one I was just talking about is on the left, is the formula for calculating the Z score. And we'll go over the one on the right later in this lecture. So remember in the last lecture I was talking about a class that had 100 people in it and that all took a really hard test. It was so hard nobody got 100%. And it was a 100 point test. So nobody got 100. The top score was in the 90s. So and remember in the upper right, there was, there's the mu. The mu was 65.5, which is pretty bad score on a 100 point test. And the standard deviation was 14.5. So I'm going to give you an example of calculating a Z score on that particular distribution. So let's say you got a friend, you have a smart friend. And that smart friend got a 90 in the face of all this. Well, let's calculate the Z score for 90 on this particular distribution. Okay, so here's what we're going to do is first we're going to remind ourselves, you don't have to do this in real life when you're doing it, but I'm just doing this for demonstration purposes, is what our empirical rule stuff look like. Remember at mu plus one standard deviation was 80. And mu plus two standard deviations was 94.5. So already, you know, whatever your answer is going to be for 90 is it's going to be between one and two, right? But we just don't know exactly what it's going to be. So I'm just showing you this for demonstration purposes to relate it to the last lecture, but you don't have to do this in real life when you calculate it. Okay, so we know that the Z we're going to calculate is going to be somewhere between one and two. And as you'll see on the slide here, I labeled over on the Z curve, I labeled where Z equals zero, which is the mu that 65.5. So we're going to anticipate we're going to get a z score that somewhere between one and two. And you'll see in blue, I listed the ingredients, right? So we have the smart friend score 90. We have the mu 65.5. And we have standard deviation 14.5. And then we have our Z formula. So let's do it. Okay, so x minus mu is going to be 90, which is our x minus 65.5. You do that out first, and then you divide it by 14.5. And look, our Z score is 1.69. And that's exactly where we thought it would be, it would be somewhere between one and two. And so as you can see, you can take any x and convert it to Z. Here, we'll do another example, only this friend is not so smart. This friend actually got a score that was kind of low, it was so low, it was below the mu of 65.5, this poor friend only got a 50. So let's try it again, let's do a Z score for 50. So again, you know, this is just for demonstration purposes. But remember, in empirical rule land, 51 was at mu minus one standard deviation. So we're going to expect that between again, negative one and negative two, Z is where our 50 x is going to land if we calculate the Z score. And so here we are, we calculate the Z score, we have 50 minus 65.5 divided by 14.5 and we get negative 1.07. And the reason why it's negative is, as you can see, it's on the left of the mu, so then the Z score is going to be negative. And so as you can see, it's exactly where we thought it would be, it would be a little bit to the left of negative one. So now we're going to get into something that's a little bit harder, which is the Z score probabilities, you're feeling pretty good about the Z score. But now let's talk about the probabilities. Okay, so remember the probability from the empirical rule, this is just old empirical rule stuff. So remember, I gave you a question at the end of that lecture, I said, what is the probability I will select a student with a score between 36.5 and 51. And remember, the answer was like this orange area, which is 13.5%. But what if you have Z scores like 1.69, the smart friend, and negative 1.07, which are the not so smart friend, you know, in other words, you have X's of 90 and 50, which are not on the empirical rule. How do you figure out the percent or the probability? That's the next step with your Z scores. Okay, so now let's ask this question. Let's say what is the probability that students scored above the smart friend? Now we could also ask for below, but I'm just choosing to ask for above this time. So in other words, what is the area under the curve from Z equals 1.69 all the way up. So see, like a little ways through that blue wedge, we wish we knew the area for everything up from 1.69 Z through the purple area through the little black thing at the top, we wish we knew that area. We only know from the empirical rule what's on the cut points of like one and two, but we don't know this in in between things. So how do we figure that out? Well, this is another problem here. What is the probability that students scored below the not so smart friend, right? And in that case, see the diagram, we'd have to figure out what is the part of the orange that that friend gets plus the red and plus a little black part at the bottom. What is the percent or the proportion of the curve that represents that. So that's what we're getting into now. And that's what we do is we look these up in a Z table. So what the Z table is is basically they figured out every single Z score you could have between negative 3.49 and I'll go into why negative 3.49 between negative 3.49 and positive 3.49. And they went like every 100th. So they figured out for every single one of those Z scores, what the probability is, and they actually fit that all on a table. And so now what I'm going to show you how to do is how to use that table to look up the probabilities.  It's just that the empirical rule is nice because you don't have to pull out the table. But if you have something that's not on the empirical rule cut points, get out your Z table. So how do you use the Z table? Well, the first thing is you want to figure out what area you want, right? So we're going to start and do the not so smart friend, because that's a little bit easier actually to demonstrate. Okay, so what is the probability that students scored below the not so smart friend? So which is a secret way of saying, what is the area under the curve that makes up most of that orange part, all of the red and the little black part at the bottom? What is that proportion? And so for areas left of specified Z value, you're supposed to use the table directly. So I'm going to show you how to use that table to look up negative 1.07.  Hi there. So here we are at the Z table. And if you have the book, you can look it up in the appendix in on page eight. But there's also a lot of Z tables on the internet. Sometimes they're arranged a little differently. So I'm using this one because it's from the book. So remember the Z that we're looking up, we're looking up the Z of negative 1.07. So remember, I said they had to somehow calculate all the different probabilities for every single Z between negative 3.49 through positive 3.49, every hundredth, they had to come up with that. Well, how did they fit it all on their table? Well, this is what they did. See, this is the beginning of the table. Remember, I said negative 3.49. Well, this is negative 3.4. And then to find the Z and negative 3.49, you have to imagine that the nine is here, but it's going to be the last one here. So see this nine here, this is what it would be. So just for pretend, if we had a Z score of negative 2.58, I go 2.5 and then I have to go over to the eight right here. Okay, or if I had one that was negative 2.10, right, or negative just plain 2.1, right, then I'd go over just one to this zero line. And see, these these little tiny things in here, those are all probabilities. In fact, let's go look up our probability, which is negative 1.07. So we're going to go down here, negative, here we are at negative 1.0. And then we have to go over to the seven column, right? So what's the cell, here's a cell and three from the left, I guess I could guess that. So we have negative 1.0987. So this is 0.1423. Otherwise known as 14.23%. So that's actually what you get out of the Z table. That's the probability that's the percent you're looking for. And just in case you're wondering, these aren't all negative, the first page is negative. The second page is positive is all the positive C scores all the way up to 3.49.  Okay, hold that thought. Okay, here we are back at our slides. And so look at that green part where it says for areas to the left of a specified Z value, which we're doing with the not so smart friend, use the table entry directly. So here was our table entry, it was 0.1423. So we're just going to use that number that we found. And we're going to say the probability then is 14.23%. And that kind of makes logical sense knowing the empirical rule. Now I'm going to show you an example of what why I was saying use it directly. In this next example, we're going to look at the smart friends probability. In fact, we're going to ask what is the probability that the students scored above the smart friend and the smart friends at z equals 1.69. So I'm going to demonstrate now for areas to the right of a specified Z value, you either look them up in the table, then subtract result from one, or you use the opposite Z, which is in this case would be negative 1.69. And you'll get the same answer, whether you do it the first way or the second way, but I'm going to demonstrate both. Okay. So first, I'm going to demonstrate what happens when you look up the probability in the table for that Z, and then you subtract that probability from one. So let's go look up Z equals 1.69. All right, here we are back at our Z table, only this time we're looking up a positive Z. So we don't want this first one, we want the second one. So remember, we're looking up Z equals 1.69. So we're looking under here for 1.6. And that's right here. And now we have to go over to the nine column. So that's going to be 0.9545. So hold that thought 0.9545. Okay, we're back with our probability that we looked up in the Z table. Now remember, we were supposed to look it up in the table and subtract the result from one. So that's what we're going to do now. So we found 0.9545 in the table, we're going to take one minus 0.9545 and we get 0.0455 or 4.55%. This is a little tiny piece, which kind of makes sense, because it's right at the top of the distribution just a little piece of the blue in the purple and then a little black at the top. All right. And so what you want to imagine is that 0.9545, which is like 95.45%. That's the whole piece below Z equals 1.69. That's most of the blue, the green, the yellow, the orange, the red, and the little black at the bottom. That's all in the 0.9545. Okay, so again, we were looking up in the area to the right of the specified Z value, and I showed you the first way of doing it. There's another way of doing it. And that's where you just use the opposite Z from the get go. So we're going to now use the opposite Z, we're going to look up negative 1.69. All right, here we are back at the Z table, only this time we're looking up negative 1.69. So negative 1.6 is the first thing we need to find in this column. So here we are negative 1.6. And then we know nine is the last column, I'm learning that. So we'll go over here. And so that that looks familiar, right? 0.0455. Okay, hold that five. All right, we'll back. And so as you know, if you look it up in the table directly, like the 1.69 directly, and you take that probability, and you subtract it from one, which is what we did last, we got the same answer we got now, right? 0.0455 or 4.55%. So it is kind of more efficient to just use the opposite Z. If you're looking for areas to the right of the specified Z value. But I always say, when you're done looking it up, compare it to the picture. And I always say draw a picture too. You know, I don't mind if you have normal curves, drawn over all of your homework, or all over the wall, I guess, or maybe a whiteboard, that's probably more efficient. But it's best to draw it out, label on there where your Z and your X are, and then just look at it. Because we know that the little piece above Z equals 1.69 is not 95% of that curve, it's just not it that's over 50%. And we can tell that little tiny piece is under 50%. So if you accidentally do the first way and forget to subtract from one, you know, maybe if you check it against your normal curve drawing, you'll realize, Oh, I made a mistake. So even though there's two different ways to find the probability, if it's to the right of the Z value, just try to make sure no matter how which ways you use, that you finally do a reality check against the drawing you make, just to make sure you got the right piece, because there's only two pieces, there's a big piece and a little piece of the skirt. And we got 4.55%. We know that's a little piece. And we know from our drawing that we were looking for the little piece. So that's how you do your reality check. Okay, you thought that there weren't any harder questions. Well, here are some harder questions. So this is a little bit more on probabilities in the Z table. So here's another question we haven't handled yet. What if you were looking at a probability between two scores, such as the probability the students will score between 50 and 90. So it's somewhere in the middle. Okay. Note that in that case, when you have a between one, you actually have two axes, and we'll label them x one and x two. So the not so smart friend is going to be x one, and the smarter friends going to be x two, just to keep these axes straight. Okay. So the next step is you're going to calculate z one and z two. And I'm kind of cheating because we already did these we already knew the z one for the not so smart friend was negative 1.07. And we already knew the z two for the smarter friend was 1.69. So I just put them on the diagram. Okay, and then I'll here's this beginning of the strategy, and I'll just explain the strategy, and then I'll do the strategy. So for z one, you find the probability to the left of the Z. So you find the little piece to the left. And remember, you can take the direct probability from the Z table. So that's what direct means is you just get to copy it directly out of the Z table. Then for z two, you find the probability to the right or above. So you find the little piece there. And you use one of those two methods I showed you, which we did together. And then finally, imagine like the whole curve, you're subtracting the piece at the bottom, the z one probability, and you're subtracting the piece at the top. So you're trimming with those two pieces to get the between probability. So that's the strategy is basically you find out the the size, the probability of each of the little pieces on the sides, you should track both of those from one, and that traps whatever's left in the middle. So I'll demonstrate this. So remember for z one, the probability to the left of z one was point 1423, we did that together. And then we use both of those methods. And they got the same answer to find the probability to the right of z two, which was point 0455. Okay, so that's a little piece at the top, and then we got the little piece at the bottom. And now we'll take one minus the piece at the bottom, minus the piece at the top and the total is 0.8122, or 81.22%, which kind of makes sense. That's a big piece in the middle. So it wouldn't be surprising if it was about 80% of the curve. So this is how you do a between one. Here's another question I haven't really handled. What if you were looking at a probability more than 50%. So such as the probability that students will score greater than 50, right, like like the big side. Okay, well, actually, you just do what you normally would do you say for areas to the right of the specified z value, either look up in the table and subtract the result from one, or use the opposite z, which in this case would be 1.07. So if we did method one, we end up going one minus 0.1423, which we already looked at. And we get 0.8577. We use method two, we'd take the z of 1.07, not negative 1.07, but 1.07. And we could go look it up in the z table, and we get 0.8577 again 85.77%. So it's this isn't actually a harder question. I just wanted to show you how it works when you're getting like a bigger piece, a bigger than 50% piece of the distribution. And here's another sort of similar example where we're looking at the probability that students will score less than 90. Okay. So that's easy, right for areas to the left of the specified z value, just use the table directly. So when we went and looked up z equals 1.69, we got 0.9545. So that's the answer. It's 95.45% of the curve is below z equals 1.69 or below x equals 90. So as I mentioned before, but I'll just mention again, you're supposed to treat all probabilities to the left of z equals negative 3.49 as P equals zero. So I showed you what negative 3.49 looks like in the z table, it's like 0.02. Well, there's not much smaller than that. So just if you actually calculate a z and you get like negative four, just say the P is zero. Okay, then the second thing is treat all areas or probabilities to the right of z equals 3.49 as P equals one or 100%. So as you couldn't imagine, you know, 3.49, that's at the top of the curve.  Okay, so we've gone through how to calculate z. And we've talked about looking up probabilities in the z table, and we've even talked about manipulating those probabilities to get certain probabilities. But we haven't talked about calculating x when C is given. So sometimes you're actually given a z, and you are have to calculate the x back from the z. In fact, sometimes it's even harder. Sometimes you're given a probability. And the probability is not a z, but you can use the probability. Remember, that's those little percents in the middle of the table. You can go find it in the middle of the table and look up the z that keys to it, and then put it into this equation. And so I'm going to just give you examples of some real life questions that you might see, like on a homework or on a test, probably not in real real life, that where you need to calculate x and you need to use that formula in the red circle. So let's say I was just bored, and I was wondering, what is the score, the test score on the distribution that is at z equals 1.5? Okay, so see where z equals 1.5? We never asked that question before. So let's say I just out of curiosity wanted to know what would the test score be of a student who was at z equals 1.5? So what I would do is I would take 1.5 times 14.5, because that's what the formula says, it's z times the standard deviation. And then I do that first because of order of operation. And then after doing that, I'd add the mu, which is 65.5. And I get 87.3. So the x, the student who got 87.3, that student got a score that's at z equals 1.5. Now, as you probably imagine, people don't go around asking so much about, well, I wonder what that person score is at z equals negative 2.3 or whatever, they don't usually phrase it like that. Usually, you see more like a question like this, which is what is the score that marks the top 7% of scores. Now that's a secret way of saying, we are looking for the z at p equals 0.7 Oh, so it's like we turn that 7% backwards into probability, and we say, we're actually looking for the z at p equals 0.7 Oh, so how do you do that? Well, I'm going to show you. Okay, so we're on the hunt for probability 0.0700. Okay, so let's start at the top of the table here, you'll see we're digging around the middle of the table, right? And you'll see like 0.00. That's nowhere near the ballpark because we're looking for 0.7 Oh, so let's scroll up here, or scroll down actually. So now we're more and we're in the 0.04 neighborhood. Here's 0.06. Okay, we're getting close. Well, here we have a 0.0708. And that's 0.0008 more than we want it to be. Well, here next door, we have 0.0694. And that's only 0.0006 less than we want it to be right because if it had 0.006 more, it would be 0.0700. So this is technically closer than this one because this is 0.008 off. And this is only off by 0.0006. So we're going to choose 0.0694 as the probability of record for this for the top 7%. Only, we're not going to just choose this, we're going to figure out what is Z at that score. So what are we going to do? We're going to map back here, negative 1.4. And then we got to go all the way up, which we can guess is 8. So it's negative 1.48. So hold that thought. Okay, we started out looking for the Z at P equals 0.0700. And but the closest we got was 0.0694. And that map to Z equals negative 1.48. Now, what I want you to notice is negative 1.48 is actually on the left side of mu. Okay, so that is the Z score at the bottom 7% of the scores. So we're going to use the positive version of that Z since we want the top 7%. So we're going to use 1.48. So the opposite Z. And now we're going to plug it into the equation. So 1.48 times 14.5, which is the standard deviation, plus 65.5 equals 87. So now 87 is the score that marks the top 7% of the scores. I'm going to do another exercise for you that does the this time the bottom 3% of the scores, because this is often kind of challenging for students. So I'll just give you a second demonstration. So as you can imagine, we're going on the hunt now for Z at P equals 0.0300. So let's go over to the Z table. Alright, now we're getting a little good at this, right? So we're digging around in the middle and we're looking for 0.0300. Okay, and starting at the top, we're in the zero, zero department. Oh, here's 0.01 something, zero two. Okay, we're getting close to 0.0300. So Oh, here, point, 0.0301. Could you ask for anything closer? Totally perfect. Okay, so that's what we're going to use for Z is the Z at 0.0301. So let's look up that Z. So that Z is negative 1.8. And then we look up eight. So it's negative 1.88. Hold that thought. All right, well, we were on the hunt for P equals 0.0300. And we didn't find that. But we did find P equals 0.0301 in the table. And that mapped back to Z equals negative 1.88. Right. And now we go back to the question, we see that we want the bottom 3%. So we keep the negative. Now, if I'd asked about the top 3%, we'd lose the negative, we use 1.88 in the equation. But since we want the bottom 3%, we're going to keep the negative. Okay, so now let's do the equation. So x equals and then in the parentheses, negative 1.88 times 14.5, which is our standard deviation, then plus our mu, which is 65.5. And the score we get is 38.2. So 38.2 is the score that marks the bottom 3% of scores. And just be happy, your score is not in there. Okay, now here's another challenging, hard question. What if the question on the test or probably not in real life, but on a test says what scores mark the middle 20% of the data? And so I put a little arrows on there just to point out, well, when they say middle, they mean it's hugging the mu. It's actually assuming that there's gonna be 10% on the right side of the mu and 10% on the left side of the mu. And so how you start to do this is you figure out the Z score for one minus point two, which is the 20% divided by two, which equals four, right? So then after that, you know, because one minus point two is point eight, and point eight divided by two is point four. So we get this point four. So we go find the Z score at point four, which you get at using the Z table now. So so I'm, you know, looked around and I found point 4013 in that digging around in the middle of the Z table. And that map back to negative Z equals negative point 25, right? And so that is then what I would put on for the lower limit on that one. And then Z equals point 25, the positive version goes on the other side. So once you figured out both of the Zs, the Z on the left and the Z on the right, you just have to put them through the equation. So for the left side, we use the negative Z. And for the right side, we use the positive Z. And that's how we get our limits. So what scores mark the middle 20% of the data, 61.9 and 69.1. It's not weird how that worked out. But anyway, 61.9 and 69.1 mark the middle 20% of the data. I didn't totally didn't do that on purpose. It just worked out that way.  So now is a good time to talk about just a little review just help you come down a little bit from this whole really intense lecture. Okay. So first, I'm going to do a little Z score quiz game show style stuff here, right? So if you ever get the question when you're on the test, and you're like, Oh, my gosh, where is X? Where is X? Well, if you can't find X, it's usually in the question. So usually the way these questions go is somebody like maybe me will put a meal in a standard deviation at the top of the question. And then there'll be like maybe five questions about that pertain to that meal in that standard deviation, but they ask about different axis. And when I would teach this class in person, you know, people will come running up to me in the middle of the test, which you probably shouldn't do. And they would say, Where's the axe? Where's the axe? You gave me, you know, these pieces of the equation, but I can't find the accent, I'd be like, walk in the question, look in the question, you know, because I don't want to give it away. And then they'd all run back to their seats and find it. So that's so if you're wondering, you're panicking, where is X? Look in the question, it's usually in the question. Okay, so let's say you find an X. And what do you do with an X? Okay, and you're stuck with an X, what do you will usually what you have to do is calculate a z score. So remember, if you've got an X, you probably have a mu in a standard deviation, you can calculate a z score on that. So if you're panicking on a test, and you have an X, a mu standard deviation, just for fun, calculate a z score and see if it gets you anywhere. Okay, well, let's say you have a z score, what do you do with the z score? Well, you always look it up, right? I mean, if you're if you're going this direction, if you're getting if you started with an X, and you get a z, you got to go to the z table with it. Okay, so that's your next step.  Go look at the z table. Well, what if the question asks for an x, right? Well, remember, we have a whole formula for that. So use the x formula. So if there's no x anywhere, and it's asking for an x, then use the other formula, use the x formula. And what if the question gives you a p, or I just said p for probability, but it could be a percentage like remember the top 7% and the bottom 3%. Well, if they give you a percent, just start digging around in the middle of the z table, just start digging around looking for that percent. Because once you start digging around, you realize it maps back to a z. And then you can get into the groove of using the x formula and you'll probably get yourself out of this panic. So here are some final tips and tricks for getting z scores and probabilities right. And I've said this one before, draw a picture. And what do I mean by that graph out the question, draw the curve, draw the line for me, which goes in the middle, and where the x goes above or below the me, just start with that, it doesn't have to be to scale. But mainly, you want to get those elements in there. There's one x, shade the part of the curve wanted either above the x, or below the x, you know, just color it in so that you get an idea of do you want the big part, the one that's greater than 50%, or the little part, the one that's less than 50%. If there are two x's, then shade in the area wanted, which is usually in between them. If it's a calculate the x question, put where the z or the P is. So if it was like the top 7%, you could shade in the top little part of the curve. If it was the bottom 3%, you could shade in the bottom little part of the curve. So make this picture and do it at the beginning. Okay, then note that x is usually in the question, if you can't find x, and you're trying to do the z formula, and you're saying, okay, I'm trying to make a z score, that's what it asks for, I'm trying to find a probability, that's what it asks for. I'm looking the question and you'll probably find the accent there. A big problem that I see is people mistake little z's for P's. Now obviously, if you've got a z, that's like negative, you know, a P can't be negative, a probability can't be negative. So you won't make that mistake, even if it's like negative 0.25, right? You won't make that mistake. And if the z is bigger than one, you won't make that mistake. So if you see a z equals 2.5, you're like, obviously, that's not a probability. But when you have a little baby z score, that's between zero and one, like 0.023. It looks a lot like a P, but it's still a z.  Don't be fooled.   Still go look it up. Okay. Then finally, remember how step one was draw a picture and I went on and on about that. Step 99 or the last step before you're done with the question is check your logic against that picture.   So please check your logic against the picture before you say that you're done with your question. Okay. So you made it through this long lecture about z and about probabilities. So I gave you an introduction to the standard normal curve into those two z score formulas. I showed you how to calculate z scores and how to look up probabilities. And I also showed you at the end how to calculate x if given a z score or a probability. Okay, and all I want to say is, unfortunately, those students, those pretend students on that distribution, they were another them got 100%. Okay, that's not the case in our class. A lot of times people get 100% on the quizzes. That's why I can't use your grades as examples. Okay. So good luck on the quiz.