Welcome all back to the channel everybody. We are going to go down a different avenue today of the outdoors called bushcraft and the first video is starting fire. First of all, let me say I am not an expert on bushcraft, woodcraft or survival. This is something that I have been reading about and practicing on my own for about a year now and was inspired about two years ago when I went to New Zealand and started backpack hunting and I've made that a part of my life. I love to do that every year and go in camping and kind of simplifying things in the backwoods. I love doing that and I think it's an important skill bushcraft is to learn as an outdoorsman. There's a lot of aspects of it that will help you be a better total outdoorsman which I want to be. I want to be a great total outdoorsman. Hopefully when I die my children will be able to say that. I thought this would be a fun series to share with y'all as I start out doing this, the first one being fire. We're doing this in the backyard very controlled environment. The next step is to go into the back country, the actual back country in Texas and setting up a camp and that's going to be a new experience for me in terms of setting up a really primitive style camp. So when I've done backcountry hunting, you know, I've taken a lot of gear. I've take really nice tins and equipment and burners for food and all this stuff. Bushcraft is different. Bushcraft is about simplifying things and kind of making your way with very limited tools and supplies and as Nessmuck would say that's called smoothing it. You're not roughing it. You're smoothing it. Survival skills and bushcraft are a little bit different. So bushcraft is more about enjoying your experiences with little devices and survival, you know, I mean that's basically like worst-case scenario type stuff. So I'm not going there. We're just going to enjoy doing bushcraft here. First step and probably the most important is making fire and how to do that with a Ferris Aram rod or a fire steel, whatever you want to call it and just your natural materials. So first of all, what is even a Ferris Aram rod? I'm just going to take the one off of my knife sheath here to show you as an example. First of all, if you're going to do bushcraft, you want to have a lot of fire steels. Don't just carry one, carry multiples, but the best thing about a fire steel is it goes, you know, hundreds, maybe even thousands of strikes making sparks and you can do it in freezing cold temperatures. So lighters, as I found out in Colorado last year, don't always work in cold temperatures. You can definitely carry lighter. Kind of the rule of thumb is though, whenever you're starting a fire, you want about five seconds of that light to really get things going. The fire is not just about getting the spark and the fire steel. Even if you have a lighter, you want to be able to get that fire going within five seconds. So we're going to go to that next, but a fire steel is basically magnesium. I'm not sure what other types of metals are in there.  So you see these sparks coming off here. That's basically the idea with this and it burns very, very hot. I think it's around 3,000 degrees and it's super easy to carry.  It doesn't take up much room or weight in your pants pocket or your pack or whatever you're carrying and it's very, very reliable. So that's why you want to use those. So the most important thing that I've learned so far about making fire especially here in Texas is having the right tender. So the way you start a fire, get a good fire going, is you have tender, which is going to be your really flammable natural materials like grasses and barks, things like that. And then you're going to have kindling. Kindling is going to be small groups of very dry materials, sticks like this. And then you're going to have your fuel and your fuel is your actual logs like water behind me. So those are the three things that you really need to make fire. So this bag right here has tenders in it. Some I've used before, some I've never used and we're going to try them and see which ones work the best. But you know here in Texas we have a few trees that allow us to use the bark of the tree to start fires. Probably the most common, if you're in Central Texas, I have some here is cedar.  And it's a real dry, fine, crispy material. That's what you want. Another common one that we're going to have is dried grasses. So I've got some dried grass and this stuff is going to be extra dry now that I've picked it. I put it in this burlap sack and it's been hanging up for more than a day now. Dry rotted wood is one that I've never used before. I've only read about it. But you can use that and since it I got it when it was sort of wet. I pulled the driest pieces off but you can take this dry rotted wood and now it's I mean it is just super lightweight and dry now. You can kind of crunch this up and you can use that as a tinder. So I have quite a bit of that and you'll find this on the ground after it starts decaying and it gets spongy. Another one I saw on the ground that was even despite it being wet. This stuff was still dry and that was willow leaves. So old leaves off a willow tree. We're in winter right now so none of the trees have leaves but those dry, crunchy leaves off a willow tree. That's basically it right there and what we're going to do is make a bird's nest out of these materials and a bird's nest is used to carry an ember and then eventually light on fire. Another thing that you can do that I've learned in the backyard here in Texas, we don't have that many good trees that will feather stick nicely and light on fire. But a feather stick is another way that you can start a fire and you want to use your knife and create these curls that just increase the surface area of the wood material that you're using and it helps it light on fire a lot quicker. And the one species that I know of that you can do this with is ash. So I have quite a few ash trees in the yard. They are like ugly spindly looking trees. It's one of the few woods that you can take even when it's wet and you can still make feather sticks and get a fire going. This one has been dried out and it should light on fire pretty quick if I put some sparks on it. So let's get out the fire still and see this knowledge we've obtained, see if it works and we can make some fire with these tenders.  I know some of y'all are going to say that in the comments.  So to start off we're just going to take our knife and to do bushcraft you just need a good carbon steel knife or a good hard steel knife. I am a knife nut. That is one reason that I started getting into bushcraft. I have dozens of good outdoor knives, bushcraft knives. They don't have to be crazy expensive. I'm using a more bushcrafter which I think is like $60. You know you don't have to have a three $400 knife to go out and do this. You just need a good decent knife. This one's not full tang but almost. Anyways I'm really not sure the best way to process this but I'm going to try this right here. This looks pretty daggum good. I don't know if you guys can see that. You got to have a pretty sharp knife to be able to make these little curlies like this. Man I'm telling you what, this right here, this might be the best thing around. Dry rotted curlies you can see the holes in it. This reminds me of like little chocolate shavings. That's how smooth this is coming off of this this spongy wood.  Now with some other woods, you know if they're hard you can just take the spine of your knife and you can just run that on there and that's making a really fine little powder but this stuff is just so spongy. It's actually hard to do that so I'm just going to take these little shavings and see what we can do with them. Also what I've learned about bushcraft is it's just being resourceful so you want to take whatever you've got around you and that's what you're working with. It's not carrying in a bunch of stuff and getting bogged down with a heavy pack. It's just having the knowledge to go about and do this type stuff and it's a lot. It's a lot. I haven't until this point felt sort of confident that I can go do it on my own in the wild. Don't blow a key thing when using the fire still. When I first started using it I was trying to take the knife and throw the sparks at the the tender with the knife like push push that amount and usually what happens is the embers will hit and they'll start just bouncing around and usually if you try to do it really with the knife pushing it it's going to go over it and it'll it'll just go around it. The easiest thing to do is to put the fire still where you want that spark to go like really point it at it and pull back so you're pulling back like you're starting along more and more than you are pushing forward with your knife. You're going to have more control doing that. Let's see if we can do this with the wind. All right that's what you want right there I definitely missed but that ember staying lit like that that means you got a good piece flying off the steel hit it nothing there it goes okay that's lit that's a good that's a good ember on there that's lit so I could move that into my it's my grass bundle and I could probably get it going here I'm just going to see if I can get one more piece going that's still on fire right now I'm going to show you guys what it looks like to oh wow it's still going could actually light up there's a fire well those little embers got me so that's going I could put my my dry grasses on there and that's holding pretty nicely how I actually got it's a lot on fire I put my rod on the wood and I was able to get a lot of pressure on my knife to then make a good good curly bit come off of that fire steel try to do it again there it goes they're just lit so that is just literally spongy dry wood from the woods take the old crock stamp that out there we go wow that was really good stuff guys because it's it's staying it's got enough um it's got enough meat on the bone there to keep a good ember lit where you could take this you know I could take this and move this entire thing into uh into a fire if I wanted to so really impressed with um what I just did right there quite honestly because I've never done that before and that material seems to to hold that uh that heat hold that burn for a longer time than say a smaller dryer material which we're about to use gosh this wind is freaking crazy so the way to really do this if I if I really wanted to get a fire going was take the dry spongy rotted wood and I would place that into the dry grass once that is going all that will catch you know you can literally find little sticks feather sticks they take a long time to do and if if you're in a hurry what I would do is find small diameter twigs that are dry like this if you're in a hurry if they're snapping like that and they're you know less than pencil diameter you can use that just as good as you can this if you have a little time and you want to break down a piece of wood like this you can take your knife and you can baton the sticks or take your axe whatever you want to do and uh and you can make you know just start processing the wood down it's really easy to do you just take a heavier stick and you knock on your knife if y'all want to see it how to do that I'll just show you real quick just take your knife and you'll knock your knife into the wood you can also lay it down and split it you can split it that way there's uh there's there's you know a couple ways that you can split it and if you got an axe you can break down bigger pieces of wood but anyway that's just something you can do with your knife to uh to help get those those kindlings going what I've been doing is just stacking my kindling in a little square like this just like you're playing Lincoln logs I think it's easier than trying to build a tp real quick because it's going to happen fast your tenders on fire and then you're trying to like build that so this is just super simple you know make your little Lincoln longhouse set it on fire so with the dry rotted wood it took me like 10 strikes to get that thing going which isn't terrible I mean when I used to try to get fires going I would do like 30 or 40 strikes on the fire steel you just really start removing a lot of material and at that point you need to step back and go okay my my material of my my tender is not good this is probably 1095 high carbon steel I have I have other knives that are really really hard steels like L max that's my favorite steel and there's another one called AEBL steel that is my absolute favorite for just for striking a fire steel there's something there's something about it the hardness that it just really comes off but you can do it with any carbon steel knife high carbon steel knife it's just some are a little bit better than others so that's the idea though you want to have a great 90 degree spine even on your striker if you get a striker a lot of these will come with strikers you're going to use that 90 degree spine and get it at a sharpest point and try to get the most material off there and it's not about speed it's just about pressure and being smooth with it and just fast enough to where it will really pop off a good chunk of that you want that hot spark to stay going as long as you can to really ignite things now what I have here is bark off a cotton wood tree cotton wood as the the name implies I'm assuming maybe a long time ago people found out that this this material is really good for it's really fibrous so if I if I pull this material apart you'll see these inner barks of the tree and there's there's layers of these inner barks and they are just really dry fibrous just great material for starting fire so that's that's one of the few trees we have around here that would be a good example of that so inner barks you can get them off other types of poplar trees things like that this this is a type of poplar tree but cottonwood if y'all are from Texas cottonwoods are they're very noticeable they they run along creeks and they are huge they're one of the biggest trees that we have in Texas so you'll know when you see one and you can know if you open it up and you see this this kind of bark right here that's a cottonwood so almost looks like a corn husk we're gonna take that and see if we can get that to light just by itself I'm gonna take that outer stuff off and I'm gonna take this and I'm just gonna get it real crunchy and a lot of these pieces will actually fall on the ground so if you have like a like a dry bag or a sack or anything I have a piece of wood right here that I've cut but if you're out in the bush you know you could use the outer bark of a tree to do this over and that way you keep this in here that's actually a really good good idea to do didn't get a spark that time here's one two nothing yet three that was a weak one nothing there we lit lit but I don't think we're gonna go that one's weak too I'm hitting those big pieces I need those little fibers in there I can get hit okay that's a good one not catching like the dry rotted wood there we go that took like 15 strokes right there that should just continue to go and I could take this and I could move this to do my fire lay or I could just start placing the kindling right on top of this I'm gonna have to use the croc again stomper out the croc might as well add that to the to the bushcraft tool list cedar is going to be probably really close to this category same sort of thing and you're just collecting the the outside bark of that cedar and crunching it up this same way and I've started a couple of fires doing that just the way I just did there I want to try to get these little willows in here and start and start these because this is something that I just saw and I thought man that looks like excellent tender material but we'll give it a try I mean this stuff is just about as crunchy as it gets some of these actually a lot of these are on stems they're just going to crunch them up they almost look like chili peppers if you were to take a dried chili pepper looks kind of like that but they are extremely light and look extremely flammable so I'll tell you guys what I actually really like is using the inside of this cottonwood to help hold that stuff I'm just going to do that well y'all it's looking like the stems are too much like it's not letting the uh it's not letting the spark really get in there good enough it's hoping that would work out and kind of enclose it but I don't think it's going to work okay willow leaves thought it would work but not working the best that was something I've never tried before not saying it won't work but if it's not lighting up after like 20 strikes I'm just gonna say it's not as good as some of the other things we could find so here's what we got laid out we have our bird's nest nice and tidy little bird's nest that is probably two fists size with the dry grasses and some of that inner cottonwood bark and then we have our spongy rotten wood that we've made shavings out of we're going to put that into the bird's nest right here I would do it right now but the wind's probably gonna blow it away and then we have our kindling we've got some of it stacked on top of there ready to go and then a feather stick we got a feather stick so once we get our tender on fire we'll stick this above that that'll really get this going it'll catch on the kindling really well and then we have fire then it's just adding the fuel so logs like that you're ready to go okay now that it's calmed down just a little bit we're going to put this all together rotted out wood right there to the center I really like the way that stuff caught knife fire steel and we're just going to pull back on that fire steel oh ember went out got it feather stick under there should catch really good and there we go that is a well-formed fire so that's catching and then we can just add our fuel from here we'll probably add just a little bit more kindling and then we're all set so that ladies and gentlemen is starting fire with your tender bundles now I want to show you guys a cheat code because I've used this cheat code and I would still suggest and I will carry cotton balls with me because a cotton ball takes one of these sparks off a fire steel so well and it will hold that flame long enough where you can really get a fire going if it is wet and as an addition to that some Vaseline if you carry a little bit of Vaseline with you which is also good for you know cuts and putting on your nose and your lips to suffer from being windburn and sunburn and all that so it has multiple uses but putting Vaseline on a cotton ball it is literally like a little torch and it'll stay lit for minutes at a time and it gives you a chance to to really get a fire going in damp bad conditions one of the tenets of bushcraft is really extending all of your resources and materials as much as you can so last resort would be would be using a lighter something really valuable like a lighter or these cotton balls for example but the next thing that I really want to learn is using an actual flint and steel to make sparks but cotton ball if you guys are getting into this want to learn like the basics of fires and get things going plus just have an emergency um uh thing to resort to this is it right here oh there it goes there it goes even the tiniest of sparks will get that going still going that is about a minute's time so that's about all you get if you put some Vaseline on there you're getting I'm going to say two minutes so that is step one so I feel pretty confident in making these fires out in the bush so that is the next step I'm actually going to go out into the bush and I'm going to set up a camp in the wilderness in the wild and in fact I've camped around this area before many of you saw it and it's a little sketchy you guys want to stay tuned for that go ahead and subscribe to the channel I want to know in the comments if you think this is something cool if you think this is a cool skill to learn with me here on the channel if you want to see more if you want to see like all of my knives and all the stuff that um you know the little knowledge that I've accumulated over the last year or two I'm happy to share it either way I think it's going to be entertaining just just me going out of the woods and trying to do these things especially for the first time make sure to smash that like button for creating fire out of just sticks and natural materials and stay tuned for more outdoor greatness right here on the channel y'all god bless see you soon


Welcome to our lecture on the normal distribution. If you looked at the lecture on probability distributions in general, you'll remember that for any continuous probability distribution, and the normal distribution is an example of that, the probability distribution is actually a curve. It's called a probability density function. The probability that X, a random variable, is equal to any one particular value must be zero because it's a continuous random variable. The probability that it takes on any one value is so minuscule that the probability of it taking on that value is zero. We're talking about a continuous random variable that can take on any value, including fractional decimal values inside of an interval. We talk about intervals. What's the probability that the random variable will take on a value between zero and one, let's say? We ask probability questions. We frame them in terms of intervals. In addition, we talk about the probability as the area under the curve because if you see the shaded area in the curve on the slide, if we're looking at the probability that the random variable will take on a value like that in that interval, let's say that interval is between zero and one. If we can get the sum, the area in that interval, the area under the curve, that would be a definite integral, and then we can get the area under the entire curve, divide one by the other. We get the proportion of times that the random variable takes on a value inside that interval. Well, a proportion is equal to a probability, and so that's how we get our probability. Of course, you'll also remember that the way the formulas for the continuous probability distributions, the way the formulas are constructed, the area, the total area under the curve is equal to one. So if we want to take the area under the curve that's shaded in and divide by the total area, that's just dividing by one, and it makes our lives a little bit easier. Let's look at some of the properties of the normal distribution, the famous bell-shaped curve. The three key characteristics. One, it's symmetric about the mean. That means on the right side of the mean, it's exactly the same as the left side. So the symmetry here. Also, you can see the highest point is right in the center. The mean equals the median equals the mode. That's why it's called a bell-shaped curve. And finally, fx, as you move further and further away from the mean, fx gets smaller and smaller and smaller, same as right as the left. And the way we show that is x goes from plus infinity to minus infinity. It's asymptotic. It means it's getting closer and closer to the horizontal axes, but never quite touches it. There's another way of understanding that if the IQ is normally distributed, let's say it's 100 is the mean. You have very few people who are going to have IQs over 150 and very few who have IQs below 50. The bulk is always going to be the earth of 100. What you see here is the formula that produced that curve, that bell-shaped curve for the normal distribution for the probability density function. I like to scare my students when I teach in class. I can't do it here because you'll figure it out in a minute. I like to say, oh, yes, we're going to use that formula and we're going to use calculus. You'll have to do integrals. So it's not going to work here, but basically I do want you to look at it. You're never going to use the formula. Again, thankfully, everything we need has been tabulated for us and we're going to be able to look up normal probabilities in a table. We'll see how very shortly. But do take a look at it. That curve is the y value for any particular x value, any value of the random variable x. It computes f of x. f of x is the relative frequency. So it has to be relative frequency, again, because we know the probability that x is equal to any particular value, right? Zero. We can only look at values inside of intervals if we're looking at probabilities. But that's not really the most important thing. The most important thing is to know that there is a curve and if we have the right information, every particular normal distribution can be drawn. You can do it yourself. You can do it with a computer program. Look at the formula. What do we have? We've got a lot of constants, right? We've got 1, we've got 2, we've got pi, we've got e. Those are all constants. We've got a random variable, x, that's the random variable. What's left? The only things that are left that are not constants and not the random variable itself are mu and sigma. These are the parameters of the normal distribution. Mu is the average of the normal distribution that you're working with. Sigma is the standard deviation. Remember from the very first lecture, mu and sigma are parameters of the population. For every mu and sigma combination, we have another normal distribution. Every normal distribution is characterized by a particular value of mu and a particular value of sigma. Really, there are an infinite number of normal distributions. How are we going to use one table to help us compute probabilities from the normal distribution we're interested in? We're going to see that shortly. As you've been told, there are infinite normal distributions. You can create your own. There are programs that will do a few. The program will ask you, give me the mu, and you can say mu is 12.67 and sigma is 3.99. It'll draw you a normal distribution. It's symmetric. You know the properties. There's only one we call z. That's the standard normal distribution. The standard normal distribution has a mu of 0 and a sigma of 1. That's why you have a z table. It's essentially showing you, it's like our template, and that shows you what the normal distribution looks like with 0 and sigma is 1. And by the way, because of that property, you can transform any normal distribution. It doesn't matter what you're working with. As long as you know it's normal distribution, you can transform it so that you can use that z table using the formula z equals x minus mu over sigma. That's called, we discussed this already early in the course, standardizing the data. And this way, you don't never need to use calculus in this course. Instead, you use the template, which the template, the z distribution, the standard normal distribution. I'm going to look at the standard normal distribution, the z, and see how to read this table. Very important. Maybe one of the most important things you can learn is how to read the z table. It's quite easy. Okay, look at the table. And suppose you want to know how much area do you have from 0 to 1. In other words, z is going to be 1. How much area do you have from 0 to 1? By the way, it'll be the same answer, the 0 to minus 1 is the same as 0 to plus 1. Unless you don't have negatives. It's the same thing. It's symmetric. Let's do 0 to plus 1. Now look at the, it's not really a column. See, underneath the z, you see a 0.00, 0.10, 0.2. Those are row headings. So now you want to know from 0 to 1. It's actually 1.00.  Look at the 1.0. Now you want the second decimal place. The second decimal place comes in those columns on top where it has 0.00, 0.01, 0.02, 0.3. That's your second decimal place. So if you want to know how much area is there between 0 and 1.00, you see it's highlighted. It's circled 0.3413. 0.3413. If I ask you how much area is between 0 and minus 1, ignore the sign. It's symmetric. So it's the same answer, 0.3413. In other words, how much area do they have between 0 and 1? 34.13% of the population will be between 0 and plus 1 standard deviation away. How much between 0 and minus 1? Also 0.3413. I'm going to study how to use this Now first, before I even get to a particular problem, if I ask you how much area between 0 and infinity, the answer is half. Let's say from 0 to minus infinity, it's a half. Because the area under the whole curve is 1. So those numbers in that table, the four decimal places, you can see it as a proportion or a probability. So suppose you're asked what is the probability or area from 0 to 1.28. You get to find 0 is the middle. Remember this is the z-table. The mean is 0. That's where there's a 0 in the middle. Now how do you find 0 to 1.28? Now look at, but actually the first column has a z. Those are the row headings. Think of that as row headings. Think of the numbers that actually are columns. That's the second decimal place. So it's 0.00, 0.01, 0.02. That's where you read the second decimal place. For 0 to 1.28, first you go to the row heading underneath the z of 1.2. You don't have the h yet. We need to get that extra 8. Well, you go to the column that has 0.08. See that column has 0.08. And you look at 1.2. Over the intersect, that's 1.28. And you see the answer is 0.3997. That's a probability or a proportion. So the answer is what is the area from 0 to plus 1.28? 0.3997. And guess what? If you need to get minus 1.28, you don't need another table. Because it's symmetric. So 0 to minus 1.28 will also be 0.3997. Almost 40% of the area is between 0 and 1.28. That's another problem. How about 0 to 0.87? Okay. So again, you look at the z, the row headings underneath z. You go to 0.8. Now we need the second decimal place. And now you figure it out. You're going to get that second decimal place. The column has 0.07. That's your second decimal place. So that's where 0.87 is. And notice, it circled for you. There's your answer. How much area do you have from 0 to 0.87? 0.3078. Or 30.78% of the area of the curve is between 0 and 0.87. Now we're asking you to figure out how about 0 to minus 0.87. Think about it. You'll know the answer. Let's do an example. We're going to take a weight. The weight of adult males. Suppose we know that the weight of adult males is normally distributed with a mean mu of 150 pounds and a standard deviation sigma of 10 pounds. That's a particular normal distribution. Now the question is, what's the probability that a randomly selected male will weigh between 140 pounds and 155 pounds? Now notice, we have an interval because we know that that's the only way that we can ask probability questions for a continuous random variable. Now we do have the standard normal distribution tabulated so that we can look things up. But how do we get the probabilities for a normal distribution with a particular mean and a particular standard deviation? Not normal. Not standard normal. It's the standard normal. The mean would be 0 and the standard deviation would be 1. That's not what we have here. That's basically what the solution is all about. Let's take a look at the solution. Now you see a picture of the normal distribution. You're going to see that repeatedly practically all throughout the semester and not just because we like to make pretty pictures. You must do the same thing when you solve your problems. Anytime you're looking for a probability from the normal distribution, draw the picture. The very first thing you do, draw the picture. It's almost impossible to answer questions correctly. Certainly not every single time without drawing the picture. Don't be lazy. Don't try to speed things up by not drawing a picture. You will be sorry and you'll have to do things all over again. You'll end up wasting time. Let me tell you what I do. Here we have the picture of the distribution in question. Mu is 150. You see mu is at the 50% mark. The distribution, as always, since it's a normal distribution, is symmetric about the mean. The right side is 50%. The left side is 50%. Each side is the mirror image of each other. What are we asking for? What we're asking for is the probability, the area in the interval between 140 pounds and 155 pounds. In essence, we're looking at two different non-overlapping areas. Why? Because we want to use the z-table that gives us the area under the curve between 0 and z, the z-table that we've been working with in the integrated area and the picture on the top. For every z-value in the table, you can get the area under the curve between 0 and it. We have to work with what we can get. We can't, in one step, get the area between 140 to 155 pounds. We could, if we wanted to use calculus in the formula that was up a few slides ago, but if we don't have to and if we don't want to, then let's not. We have one piece of the distribution between 140 and 150 on the left side of the mean, one piece of the distribution between 150 and 155 pounds on the right side of the mean. Again, now we have the problem of not having this particular normal distribution tabulated. Here, as we know, we can transform any normal distribution into a standard normal. We have the formula to do that. What we want to do is translate this picture from an x-distribution to a z-distribution. That's why you see the additional scale that was drawn under the x. On the left-hand side, you see the formula. We have the 140 into a z. So z is equal to 140 minus 150 in the numerator, divided by 10. 10 was the standard deviation. And the result is negative 1. That's why there's a negative 1 on the z-scale underneath lining up with the 140. The area under the curve between 140 and 150 is the same as the area under the curve in the z-distribution between 0 and negative 1. And when we look that up in the z-table, we find an area of 0.3413. In fact, I think we saw that for a different problem a minute or so ago. On the right side, we want to standardize that 155, turn it into a z. z is equal to 155 minus 150 divided by 10. And we end up with 0.5 for the z-value. And you see the 0.5 on the z-scale underneath the x underneath the x of 155. And so again, the area under the curve in the x-distribution between 150 and 155 is exactly the same as the area under the curve between a z-value of 0 and a z-value of 0.5. And when you look that up in the table you see that that value is 0.1915. It's important to note that these are non-overlapping areas. Because of that, they're mutually exclusive and we can add them. We can use the adding rules, the rules of addition of probabilities. 0.3413 plus 0.1915 gives you 0.5328 for the answer to the question. And so the probability that an individual male chosen at random will weigh between 140 and 155 pounds that probability is 0.5328 or in other words 53.28% of the population will be in that area. And here's another problem. Say we know that IQ is normally distributed with a mean of 100 and a standard deviation of 10. What percentage of the population will have A, IQ's ranging from 90 to 110, B, IQ's ranging from 80 to 120? What's the first thing you have to do? Always, always, always draw a picture. Take a look at the picture that's there. This is the same kind of picture you're going to be drawing all throughout this topic and other topics too. We've got a picture of the normal distribution, the original one, the X. We've got a Z scale drawn under it. For part A and remember if you have a multiple part problem you draw another picture for each one and take the word for it. If you think you'll be saving time by not doing that you're actually going to be wasting time because you'll eventually have to go back and start the whole thing all over again. So the area where we have to standardize the 90 and the 110 for part A much like the problem we did before where we've got non-overlapping pieces. We're going to be looking at the piece between 90 and 100 on the X distribution and we're going to be looking at the piece between 100 and 110 on the X distribution. But first we have to turn those into Z values so that we can look things up in the Z table. At 90 the Z value is 90 minus 100 over 10 or in other words negative 1 and we actually saw that value before and the area was 0.3413. At the 110 we have 110 minus 100 over 10 and that works out to a Z value of plus 1 which makes sense because we're talking about values that are symmetric about the mean the areas that we're interested in are non-overlapping areas about the mean but they're the same size. So it's 0 to negative 1 on one side 0.3413 and 0.3413 on the other side the answer to the question what percentage of the population will have IQs ranging from 90 to 110? You have to add those two probabilities up and when you take 0.3413 plus 0.3413 you get 68.6 826 but of course the question asked about percentages and to turn from a probability which is a proportion and convert to a percentage all you do is multiply by 100% and the answer to part A is 68.26% Part B the problem is very much the same but the two pieces the two non-overlapping areas on either side of the mean are a little larger that's really all it is at an IQ of 80 the Z value ends up being negative 2 at an IQ of 120 the Z value ends up being plus 2 the area under the curve looking at the table the Z table between 0 and 2 is 0.4772 so since the normal distribution is symmetric around the mean that's 0.4772 on the right side 0.4772 on the left side add those pieces together you end up with a really big number 0.9544 so in this particular example with this particular mean and standard deviation 95.44% of the population will have IQs between 80 and 120 in this example we're looking at the salary of auto mechanics suppose we know it's normally distributed with the population mean of $40,000 and the population standard deviation of $10,000 so question A asks what proportion of auto mechanics will earn $24,800 or less B asks what proportion of auto mechanics will earn $53,500 or more C is what proportion of auto mechanics will earn between $45,000 and $57,000 D is asking for the 80th percentile and finally E asks for the 27th percentile since we know it's normally distributed we can use the Z table question A was what proportion of auto mechanics will earn $24,800 or less so you see the diagram always draw the picture we've been told this several times it's called the left tail see the way it is the table only tells us 0 to Z so we took the $40,000 that's going to be a 0 using the conversion it's always going to be 0 what is $24,800 in Z value so the formula you're going to use you see it there so it's $24,000 minus $40,000 or minus $1,500 this is just saying that $24,800 is minus $1,500 standard deviations away from $40,000 now you can use the Z table and look up minus $1,500 $1,500 plus or minus so we go from 0 to 1.52 by now you know how to read the Z table and you'll find the area is 0.4357 so the area from 0 to minus 1.52 is 0.4357 or $40,000 now we're in X values $40,000 to $24,800 is the same as 0 to minus standard deviations and we know that area is 0.4357 but we know that the entire left left of 0 is half and from 0 to plus infinity is half alright so you take half minus 0.4357 and then you get the answer in that left tail let's call it the left tail the answer is 0.0643 or 0.043% of auto mechanics are going to earn less than $24,800 be asked what proportion of auto mechanics are going to earn $53,500 or more that's a right tail problem always remember this half and half so that 0 cuts the normal distribution half on the right, half on the left always keep that in mind 0.5000 on the left just remind yourself now we want to get from $53,500 remember it's zero to Z so $40,000 to $53,500 well that's the same as 0 to plus 1.35 in Z value converting the X to Z values so $53,500 is a Z value of plus 1.35 how do I get that you take $53,500 minus the mean of $40,000 divide by $10,000 and then you get that value plus 1.35 now the table is always zero to something so from zero to plus 1.35 using the Z table we find that the area is 0.4115 now we need the right tail remember the entire from 0 to infinity is 0.500 subtract 0.4115 from 0.500 and then you get the answer in the right tail which is 0.0885 so the answer to the question is 8.85% of auto mechanics will earn $53,500 or more this is the hardest problem to solve and if you don't draw a picture you're not going to get it right the question asks what proportion of auto mechanics are going to earn between $45,000 and $57,000 now remember the way the table works is zero to Z zero is like base okay so we want to turn everything into Z values so the 40 as you know becomes zero the Z table has a mean of zero so you just turn the 40,000 into zero the 45,000 becomes a 0.5 it's half the standard deviation away in standard deviation units okay how do you know 45,000 minus 40 that's x minus mu over sigma and you get 0.5 $57,000 turns into a Z score of 1.7 how do you know 57,000 minus 40,000 over 10,000 that's 1.70 now you can't get that area between 0.5 and 1.7 directly so what you gotta do is you do zero to 1.7 looking at the table and you get the bigger piece which is 0.4554 now you subtract see it's in red there the lines are in red now you take the smaller piece okay to get that red area the smaller piece is zero to 0.5 so we take the zero to 0.5 which is 0.1915 okay so now you have basically two pieces to get that in red the one you want you take the big piece 0.4554 subtract the smaller piece 0.1915 and what's left what's left is going to be the answer to the question 0.5 and 57,000 so the answer turns out to be 0.2639 again there's no way to do this without having the picture and you kind of highlight what you're looking for and you notice you need the big piece minus the small piece and you get the area between 0.5 and 1.7 this is the most difficult question and it's not that difficult if you draw on the diagram distribution problems are in terms of percentiles parts D and E of this example ask you to compute percentiles of this distribution it sounds complicated you may get a little nervous but there's really nothing to stress out about you already know how to do this what's the 50th percentile in a normal distribution that's the mean in the Z distribution that means that we have a Z score of zero if you take a standardized exam and you're at the 50% mark for the 50th percentile mark that means you have scored at the median and the mean and the mode if the test scores are normally distributed because as we know in a normal distribution the mean is equal to the median is equal to the mode let's look at how we can get other percentiles using the Z table D is asking the 80th percentile remember what a percentile is the 80th percentile is that value of the distribution for which 80% of the values are smaller 20% are larger so we split the distribution as you can see from the picture on the right side 80% are smaller 20% larger 80% break down well remember at the mean which in this case is $40,000 everything to the left of the mean is half of the distribution 0.500 plus 0.3 gives you 0.8 that's the 80th percentile so just take a minute draw your picture make sure you understand how everything fits together now we know if we know the percentile we know the area under the curve we're going to be using our Z table kind of backwards in reverse of the way we used it before the Z table we've been using and that we're going to continue using as examples in these lectures is the zero to Z table where for any Z value you have an area under the curve between the mean and it but in this case we want to know we need the Z value we want to know what's the Z value for the 80th percentile we know the area under the curve and you can see from the picture the area under the curve which would be the blue shaded area displayed on the table is 0.3 0.300 so we're using the table backwards and what we're actually doing is sort of mucking around in the middle and finding the area the probability closest to 0.300 0.300 and when we do that we find that the Z value associated with it is 0.84 0.8 in the row heading 0.04 in the column heading so we have a Z value of 0.84 that's not the complete answer yet what we wanted to know is the 80th percentile of the X's of the salary of mechanics but that's just simple algebra in step 4 if we know Z which is 0.84 and we know mu and we know sigma we just plug all those values in and solve for X using simple algebra and we end up with an X value at the 80th percentile of $48,400 does that first of all take a look at the picture and just make sure it's on the correct side of the distribution a very quick check we're supposed to be on the right side and 48,000 is higher than 40,000 so yes at least we see we didn't make a gross error that's the answer the 80th percentile of this distribution is $48,400 okay now we want to calculate the 27th percentile first we have to find a Z score that's associated with the 27th percentile every Z score is associated with some kind of percentile okay so which Z score is associated with the 27th percentile so look at the diagram you always draw this and you can figure out that roughly 23% of the area is between 0 and minus 0.61 right you can see that we have the Z table for you so the area between 0 and 0.61 or minus 0.61 is 0.2291 which are rounding to 23% that means the Z score a negative 0.61 is associated with the 27th percentile bear in mind that any percentile below 50% has a negative Z score a 0 as you know is the 50th percentile a Z score of 0 at the 50th percentile if you below the 50th percentile it's got to be negative so in this case we know it's going to be a negative number it turns out that it's minus 0.61 and I'll just do the algebra minus 0.61 equals X minus 40,000 over 10,000 that's the X minus B of a sigma we've been using and then when you solve it you'll see the answer is 33,900 that's X and as a check notice you below the 40,000 40,000 would put you at the 50th percentile 33,900 is the 27th percentile this is the way to solve it if you have access to a cumulative Z table and I think it's available, we think we provide if you want it, if you have access to it you can get the answer directly you can look up any kind of percentile and from the actually any kind of Z score from any Z score you can get a percentile if you know that you're below 50 you look for the negative cumulative and you'll see, you'll look at the table and you'll try to connect the Z score with the 27th percentile and you get the answer directly okay if it's a value that's a positive Z score then we know that it's above the 50th percentile so that's another way to do it we're doing it this way because many times you don't have access to that to the cumulative remember you'll see there's negative Z scores and positive Z scores if you're using the cumulative table practice, practice, practice you will learn this material much better if you do many, many problems, the more problems you do the better, the easier the faster it's going to be for you to complete the problems and get them correct


It's going on, everyone. Sam here, late in sports cards, ripping 2020, 2022, Panini Chronicles, baseball, hobby, eight box, half case break, number 20, pick your team style and read off our list of names and teams. Andrew L with the Orioles, Arthur Y Metz, Bradley R, Braves Cubs, Brian K with the Rays, Dean E Cardinals, Gary A Reds, Jeff H Giants, Jonathan Q, Red Sox Rangers, Joven H Dodgers, Ken S Padres, Kenneth H with the Nationals, that is Kevin D with the White Sox, Kiotaka M Angels, Liz C Twins, Mark H Athletics, Michael W Pirates, Michael H Mariners, Rockies, Astros, Marlins, Royals, Richard P Bluejays, Roger W Brewers, Sean S Pillies, Shane W Diamondbacks, Troy D with the Yanks and the Guardians. Thanks for selling us out, Troy. Appreciate you, my man. Let's do it. Good luck, everybody. Got three and two coming up next.  Then we got Museum to end baseball off, full case tiered random team style. That's always got some good hit potential. Dylan Coleman. There's a Josh Lowe, Red, 73 out of 199, Redemption and Jordan Alvarez. Good luck on the redemption. Origins, rookie jumbo material autographs. Guard number one, Wander Franco. Boom. What a way to start this off. Congrats, Brian K with the Rays. There you go, buddy. I believe this will be numbered to 99, but that is the Origins jumbo. Very nice. Yeah, Ken, we have one of our customers has 24 spots in it. Yeah, we sold out very, very early on. Sick Wander right there. You can mag that. Yeah, sorry, my bad. I meant. Greg Dieckman. There's Reaver San Martin going to the Reds auto right there. Congrats on that. And Juan Soto. Gold Rush, Bryce Harper on the case hit. Gold Rush, that's what it is, this one. Congrats on that, Phillies. Nice one there. Red KB for the Rockies to 199. There's a Kevin Smith jersey, not numbered, limited for the Athletics. Congrats, A's. Cody Ballinger and Edward Cabrera Rookie. Spencer Trofelsen Rookie. The Doll Bruhan Rookie on the back, not numbered obsidian. Yeah, Bryce, it's nasty. That Bryce was cool. That was you, right? I want to say, Sean. Yeah, there you go, Sean. Congrats, buddy. That's sweet. Yeah, I love those gold rushes, man. O'Neal Cruz Rookie. We've got Edwin Nunez for the Cardinals one out of 10 on the building blocks ticket. Congrats on there. Alex Bregman. Yadir Molina to 100 color match for the Cardinals. Spencer Strider. There's a Shohei. 19 out of 25. A little bit of a patch there. Nice one there. Angels. Congrats, L.A. Very cool. Spencer Trofelsen, Silver Rookie and Cabrera. Oh, wow. There is this garbage wind stuff. Yeah. There is nice for the Guardians. Gabriel Arias. Congrats on that Troy. Last spot, Mojo. Nice one there, man. That is sweet. Awesome, Gabriel Arias, Otto. Chris Bryant. Nothing crazy, I wouldn't say, Kyle. Cosmic Chrome was pretty good. Hit like four supernovas in one case. Baseback. Peter Lanzo to 100. There's a Wander season ticket. Another redemption. And Shane Boz, not numbered. Let's see what we got here on the redemption. Prospect draft ticket blue. Card number one. Gyropalmeras. Nice one there for the Giants. Congrats, San Fran. Gyropalmeras. I want to double check that really quick. Hold on before you write it down. Okay. Also, can you get those pens? Yes. Horting the pens over here. We got any luck with origins tonight? There you go, Kyle. I'm like 99% share, Palmeras is a giant. Yeah, he's a giant. All right. Just want to double check on that. Because, you know, the one time I assume that I'm right, I'm going to be wrong. So always good to double check. Bobby Whit rookie. Wander silver rookie. We've got Alex Manoa on the color graphs for the Blue Jays. That is numbered, not numbered. Congrats, Toronto. I love these color graphs. So cool looking. Jaren Doran rookie. I'm so pumped you ended up getting it too. Kyle, that's awesome, dude. Francisco Lindor, Blue Ice, 25 out of 25 Mets, Chris Sale and Bladdy G. Yeah. So Kyle and one of the 10 box Tuesdays hit a Joe Tory, who was like the manager of the Yankees when they won their five World Series in like the late nineties, early 2000s. And it has a ton of inscriptions on it. That's like how many wins he had all the World Series champs like Yankees manager, all that kind of stuff. He's giving it to his mom for Christmas. Yeah. So it's going to be awesome then. Yeah. Let us know what she thinks of it for sure. To 199. Mookie Betts, Aaron Judge, Yachty, Carlos Correa. Thick. Thick one coming up here. This looks like a regular Spectre RPI. Matt Brash, Brandon Marsh. There's a Wander rookie. Silver Frias. Got Camillo Duvall to 199 for the Giants. Rookie Jersey Audit. Question. Oh yeah. Yeah, Gabe. We go back through it. It goes into sorting and shipping and they look through it. Yeah. In case anything was missed. Yeah. There's always stuff that's getting picked up. There's Dustin Harris. Nice on the draft ticket. Cracked ice four out of 23. Going to the Rangers. Congrats Rangers. Tray Mancini. Yeah. Camillo Duvall. Kyle Mohler to 99. Aaron Judge. O'Neill Cruz. There's a Jaren Durand rookie. America's pastime jersey card. Jordan and Jazz Chisholm. Spencer Strider Rookie. Three Debtmares. Kyle Mohler. There's a Luis Frias. Nice one there for the D-backs to 99. Three colored RPA. Congrats Shane. Yeah. No problem Gabe. Jonathan India. Like you've done that before you found cards that needed to be top loaded. Yeah. We'll take care of it. Absolutely. There's Bobby Whit. Casey Meiss. And Wanda Franco obsidian rookie. Patrick Mazzica. Josiah Gregg. Bass back there. What's up Chad? How you doing man? Hunter Green. Kevin Smith. Red Josiah Gregg. 151 out of 199. Nice one here again. Going back to Brian. Vidal Bruhan. On the obsidian rookie auto. Raze. Congrats on the Vidal. And not numbered of Anthony Rizzo. Caleb just hit something. Caleb really hit something. Kevin Smith to 99. Caleb is screaming right now. Congrats Oakland. Trey Mancini. We'll have to go and investigate that. Shane Boss. Jayrod Rookie. And Wanda. Another obsidian Wanda rookie. Mike Trout. Red. To 100. The Angels. There's America's pastime. Chuck Nasty. Charlie Blackman. To 99. Going for the Rockies. There's Noah Cindergaard. Not numbered. And Josh Chisholm. I'll find out after the break, Jarts. See what it's all about. Shane Boss. There's Josh Lowe. Nice. 23 out of 30. Going back again to the race. Congrats on that one there again. Brian. Jonathan India. Porcosin on top. There we go. Nice. Congrats Chad. It's going to be fun, man. Looks like a Origins RPA coming up here. Come on baby. Big name. Big name. Rees Nair. Matt Brash. Nice. For the Mariners. That's sick. Two out of ten. Congrats Michael. Four colored RPA on that one. I love Mariners patches. So clean. Congrats on that one. Michael. 13 more days man. That is very close. That's awesome. Aaron Judge. There's Cade Cavalli. Nice. To 99. Obsidian Otto for the Nationals. Nice Nationals hit there. Luis Heel. Is that is numbered as well. To 99. Almost missed that. Yankee. Yeah. Buccaneers have great patches too. Jake Berger to 50. Most teams if you get the right part, it's a nice patch. Kyle Tucker. Not numbered jersey. Zach Wheeler. And Reed Dettmers. Two out of ten. That's right. I was going to say I think we pulled the another map, that exact same card in the last half case. The Matt Brash. Gold. RPA I think we pulled that in the last half case. Yeah we pulled some seal. I think it was him. Brian De La Cruz. Five of five on the green for the Marlins. Congrats on that Michael. Nice De La Cruz. Jazz Chisholm also for the Marlins. Michael had a strong box there for himself. A couple of his teams. Vidal Bruhan. There's Wander Franco. Silver Bobby Witt. Gaussman and O'Neil Cruz. Good pack there. Wow. Three big rookies in that. All chrome, chrome product too. Suzuki. Juan Soto. Vidal Bruhan. There's a Alejo Lopez Pulsar. Jake Berger for the White Sox on the Zenith Auto. Not numbered. A nice one there. And Hobby Bikes. O'Neil Cruz. Jackson Coar. Nice one there. That is 35 out of 49 Royals. Congrats Casey. Michael H again. And Mike Trout. Thomas Sapucki. Aaron Ashby. Base pack right there. Thanks for becoming a member there Ryan. Welcome in. Welcome in. There's Red Freddie Freeman to 100. Reed Betmer. Chas McCormick. That's Tim Anderson. America's pastime jersey for the White Sox. Nice Timie. Alcantara and Corbin Burns. So happy Sandy Alcantara won the Cy Young. He's awesome man. He should have been a Yankee. Zack Short. Kyle Lewis. Raleigh Welker. There's Josiah Gray to 49. Going to the Nationals. Lot of Obsidian autos in this half case. Not complaining. But that is awesome. Nice Josiah Gray. And Nolan Aranada. Down to 12 total group break spots tonight guys. Eight to go on Obsidian 21 and 22 mixer.  De La Cruz. Suzuki. Silver Jake McCarthy. Juan Soto. And Cabrini's. Thick one coming up here. Looks like a studio maybe. O'Neil Cruz rookie. Yes Luis Frias. 80 out of 99. Nice one there. Studio rookie autograph for the Diamondbacks. Congrats Shane. Shane with the B-backs. Jaren Duran. Wander Rookie. Camila Doval. Silver Jake Berger. There's Reid Detmerz. Nice for the Angels. Four of five. Congrats LA. Nice Detmerz. One of the only no hitters thrown this year. Awesome card there. Luis Heal. Torkelson Zenith. Blue Nick Fortez to 99. Bobby Whit Jr. America's pastime jersey for the Royals not numbered. Dylan Carlson and Ryan Belate. There's a Jeremy Pena on the pinnacle rookie. Got Drew Ellis going to the Diamondbacks again. Rookie ticket not numbered. Congrats again Shane. Nice auto there. And Shane Buss. That's that that sets off the the pinnacle. That's just one you have to like kind of clean off. Torkelson Zack Gallin and Vladimir Guerrero. All right last box of the break guys. Good luck everybody. Roanzee Contreras. O'Neil Cruz. Luis Heal. O'Neil Cruz uh Refractor. Jonathan India and Matt Manning. There is Tyler McGill. Another Obsidian Auto this time going to the Metz. Rookie. Nice one there Metz. Alex Bregman. I think one coming up looks like we got a Spectra RPA. Bobby Whit on the Vertex. Red Shohei Otani. Nice one there. That's the 199. This time we've got a pink Craig Deekman RPA going to the Cubs. Number to 49. Three color. Congrats again Cubbies. I think that was Bradley R again. Josiah Gray. Roanzee Contreras the 25 Pirates. Elite. Rookie. There is Jordaan. Jersey Limited not numbered for the Astros. Lubebop and Salvy. Torkelson Rookie. Max Crannick. Silver Buxton. Rafael Devers. And last pack of the break guys. Good luck everybody. He bets we've got Connor Siebold. Nice. Eight out of 25 going to the Red Sox on the Origins Auto. Tal Marta. That'll do it for the break guys. Quick Rikaperoni here. Jersey's Jordaan Alvarez. Bobby Whit. Tim Anderson. Kyle Tucker. Charlie Blackman. Sharon Durant. The 25 Shohei Otani. And Kevin Smith. Yeah not too bad at all. Congrats again Shane. Got a couple of nice autos in this one. Connor Siebold. Greg Deekman. Tyler McGill. Drew Ellis. Reed Detmers. Luis Frias. Josiah Gret. Jackson Coar. Jake Berger. Brian De La Cruz. Kate Cavalli. Matt Rash to 10. Josh Lowe. Kevin Smith. Vidal Bruhan. Luis Frias. Dustin Harris. Camilla Doval. Alex Minoa. Gyro Palmeiras. That is Gabriel Rice. Edwin Nunez. Riva San Martin. And then a Wander Franco. Number 2199. That'll do it for the break guys. Thanks again for the fill. Next up three and two.