How language transfers cultural sensitivity. As I've had to study several foreign languages, I've realized many things about what's necessary and helpful in studying foreign languages. And I think many of these things could come under the heading of sensitivity to the culture that the language one is studying comes from, as well as an awareness of one's own culture. Cultural attitude. One thing I've noticed about much of the course material available for some of the foreign languages I've studied, as well as educational texts on other subjects written in English, is that one can see the culture and cultural attitudes, as well as things like the walk of life of the authors of these texts. A very simple example would be a modification in recent years to some of the standardized testing in America. Students were given analogies as part of the test, but one of the questions used the name of a participant in a sport that was only played by privileged students. In response to critiques concerning this, these questions were changed to make them equally accessible to privileged and less privileged students. How cultural peculiarities are transferred by a language. But any course will have an aspect of culture embedded throughout. Managing this is perhaps a critical part of teaching a language. Languages especially reflect the culture of their origin. Language books meant for intermediate learners may have stories from the cultures that speak that language natively. These stories may have moral lessons or descriptions of historical events, that figure critically into the backbone of the language and questions culture. Other course books may lead the student on a journey, taking them through everyday life within that culture, from simple daily activities to more advanced negotiations of life's twists and turns within that society. Throughout the phrasing, even that which is used to teach these lessons, conveys the attitude of a native speaker of that language and of that culture, and the circumstances they find themselves in while transcribing that course material. The importance of learning about a different culture. To make a language accessible to students of a foreign country, I believe one must understand the cultural differences that must be communicated. As mentioned earlier in the teaching English as a foreign language course, a typical say, old orthodoxy approach to teaching language is grammar translation. One thing that comes with, this is the idea that each word of one language has a direct translation to another. I've found this not to be the case. Further usages vary greatly, even of words that are ostensibly defined with the same meaning in dictionaries. While there are many basic concepts that people have in common, once passed a very beginning level of language education, words, and culture may be vastly or subtly different. Further, even with basic concepts, attitude towards those concepts may be different. Do you want to teach English abroad? Take a TEFL course today. Teaching a course to students of another culture, perhaps requires understanding their culture enough to establish a rapport with students, and educate them in their useful acquisition of English, but also not short changing a communication of cultures where English is a native language. For a student to effectively communicate in English, they will have to understand not just vocabulary, grammar, pronunciation, etc., but also the culture informing the usage of English where they might be putting their education into practice. You can also be certified to teach English abroad and online.


Welcome. Good afternoon. Thanks so much for joining us today. This is our second of four webinars, our horse management series from NDSU Extension. So we're really glad you're here today. We are today talking about fencing, a little bit of grazing stuff, and then of course we can't do any of that without talking about winter. And so you get to hear from me again today. So I'm Mary Keena, the livestock environmental management specialist at the Research Extension Center here in Carrington. And so I will be talking later about manure management and such. With me today is my co-host Paige Brummond. And so Paige is our extension agent in Ward County. And Paige is managing the chat. So if you have any questions, if you have concerns of anything is going on, you can either private messenger or just put it up for everyone to see. And Paige will be running the chat today. With that, I also said don't feel bad if I mute you, if you unmute yourself to talk, if I mute you, don't feel bad. We'll get to your question after a while. We just want to make sure that everybody can hear. And so we'll go ahead and turn your mics off for you. Up on the screen right now is the demographic survey. And then Paige also put that link in the chat. And so that helps us to do our federal and state reports if you fill it out. Awesome. If you don't, that's okay. We're still glad you're here today. So like I said, this is a second of four that we're going to have in the series. And so we're talking about fencing. So let's see if I can move this on. Okay. Okay, and so first up today, we're going to have James Rogers talking with us. And so James has not been on this webinar series before, unlike the rest of us speakers that you've heard. And so we're excited to have James with us today. He's our forage crops production specialist at the North Central Research Extension Center. And so that's located in my not. And so James is going to talk to us about fencing or fencing specifically. And so I'm going to go ahead and turn this over to James. I'm going to share my controls with him. And then he'll take over from there. And like I said, if you have any questions, certainly feel free to put them in the chat and Paige will make sure she takes care of you there. Okay. Thank you, Mary. Appreciate that. Pleasure to be here with you today. And we're going to talk about, as Mary mentioned, horse fencing.  And if you think about protecting your house from a two year old of all the things that they can get into and you do all that. And then you think about everything else that they couldn't possibly get into. Well, a horse probably can. So that's kind of the background for fencing. And there's lots of ways to do fencing. This is not going to be all inclusive of every method or methodology into fencing. But hopefully it'll give you a good basic framework for fencing for horses that will protect them. Keep them from getting injured if they were to run into a fence and also keep them in the place that you want them to stay. Just to start off with the truth test of a fence is worth is not when horses are peacefully grazing. But when an excited horse contacts the fence in an attempt to escape or because he never saw it during a playful romp. And this is from Penn State Extension. This is a really nice quote because it points out a couple of things about horse fencing to keep in mind. One, they need to be visual so the horse can see it. The other thing is if they do come in contact with that with that fence, it's going to restrain them or give somewhat so that if they do hit it hard, it's going to minimize or prevent any injuries. Here's some pictures of horses that have got themselves into a bit of a predicament or in the case down here on the left about to get into a wreck. So this horse here is scaling this gate. You can see he's not clearing it. He's going to get tangled up in that gate. One good thing about the gate area here is that it looks kind of old. It's beginning to come off the hinges. So when he hits it, it's probably going to collapse. So that's not what we want, but it probably is going to minimize injury. This horse here is crawling through a high tensile fence. Obviously, this is not an electric high tensile fence and he's crawling through it. I really like high tensile fencing. It's fairly economical to put up. Once you get into it, it's fairly easy to install. It's a smooth fence. So you can see that this one doesn't have any power going through it. So he's crawling through it fairly easily. It will give. If an animal were to hit it hard, it does give. What's nice about high tensile fencing is when it does give and you get slack in the fence, it's pretty easy to go back in, tighten it back up and off you go again. This little dude here is crawled over a gate. I guess he was trying to jump. Didn't quite make it. Apparently he was a little bit more confident in his athletic abilities than what he actually was and he's hung up here. This is just an old board fence. It's probably seen better days and just needs to be repaired. I've been staring at this at this picture down here in the right hand corner for for a little bit of time. And I have no idea how he managed to do that and remain on four feet. I cannot figure that one out if anybody wants to put in how they think he got this accomplished in the chat. I'd be more than interested in learning. I do like the commentary here from the from the bovine here on the right. So just some real basics to keep in mind on fencing. We want to keep the horses and keep undesirables out. If you live on a busy highway or any kind of road, keeping the animals in and keeping them off the road is going to really help you with any liability issues should animals get out. Another basic about horse fencing is it needs to be smooth. As I mentioned earlier, sometimes we don't anticipate things that horses can get in and get hurt on. So we want to keep that fence smooth. In relation to that, no barbed wire. Barbed wire works great for cattle fencing, but it doesn't work well with horse fencing. They can really get cut up if they were to get tangled up in in barbed wire fence. Making that fence visible is very important. Extremely important. It's important with electrical fencing. Making that fence visible for high tensile fencing and electric fencing. It's very important to introduce those animals to that electric fence so that the first time they have an experience with it. It's not in a paddock that you're wanting to contain them in. Just practical experience with high tensile fencing or polywire fencing. We've actually put ribbons, plastic ribbons or something tied onto a fence. Just so those animals will come up and they're curious about it, maybe grab it, get a little bit of a shot. So that they know it's there and that they know what it's going to do to them if they come in contact. Fencing should be strong and flexible. It should be smooth. In terms of paddock shapes, we really would like to avoid V-shaped corners if we possibly could. If you think about that, think about a horse running down a perimeter fence and into a V-shaped corner. They're just going to get stuck in that, could get injured, maybe get trapped. So if we can avoid V-shaped corners, that would be ideal. Keep it smooth. A woven wire, I really like woven wire for perimeter fencing. It makes a very nice, attractive horse fence. You should limit the size of the woven wire to about 3-inch spacing. This is going to keep that horse from pulling through and maybe getting a hoof caught in that woven wire fence. So the smaller the opening on the woven wire fence, the better it is to prevent that hoof from going through. It should be durable, built to last a long period of time. Fencing is expensive. Cost per foot has really risen in the last 15 years, 15, 20 years. But if we do it right, make it where it's durable and last over time, you're going to spread that cost over a period of years. So we really want to do it right when we initially build it. Again, smooth. If you haven't picked up on that, a big emphasis on smooth fencing to eliminate or prevent as much injury as possible. Some common issues and mistakes and solutions or preventions to those issues. Very first thing, whether you live in North Dakota or another state, be familiar with the fencing laws within that state. It's very critical that it's especially important if you're sharing a perimeter fence with a neighbor, what your responsibilities are, what his responsibilities are. If you're initially putting up a perimeter fence, you might be able to share cost with your neighbor if it has benefit to them as well as you. North Dakota Fencing Boundary Laws can be found online. You can consult the North Dakota Century Code, or if need be, you can contact Legal Aid. The most important part of a fence is going to be the corners. Building a strong corner brace is going to be key to the strength, durability, longevity of that fence. Many times, many mistakes that I've seen, we just don't get those corners built correctly, get them built strong enough. We use two-lighter materials. This is just some guidelines here. So if you're using net wire, probably a minimum 8-inch diameter post for your corner post. For 5-6 strand high-tensile wire, you can go down a little bit smaller on post. You don't have to. You can maintain the same 8-inch diameter post for all applications if you want to do that. But as you decrease the number of wires in general, you can decrease the diameter of the post. Your brace post, if you're making an H brace, in general, the longer that brace post is across the top, the more strength you're going to have in that corner brace. So at least 8-foot length, 4-inch brace, you can also use an 8-inch brace and a brace post. And I've got a picture of that later on. But really, that top brace is very important to the overall strength of that corner post, corner brace. For our power fences, sometimes we come up with an energizer that's just too small. We're not pushing enough voltage down that fence for it to be a deterrent to the animal. So on our electric fences, we're not really looking at the wire as being the barrier for the animal. It's the power. It's the voltage. It's the shock that they receive. It's the deterrent for them to keep them from going through that fence. Energizers have really, really improved over time. The most common type now and the type that I would recommend would be a low impedance energizer to go with a high tensile fence. The amount of power that those energizers are putting out is measured in a term called joules. And just to have a rule of thumb, when you look at an energizer, you want one joule per mile fence output. Many of these energizers will give you an output of stored output joules or output stored joules or output joules. You want to look at output joules and that's how you're going to compare one energizer to another is in terms of output joules. The voltage that they're putting out and pushing down that fence about two to three thousand volts is plenty for horses, beef cattle, people. I've been hit by these high tensile fences and it will rattle your teeth for sure. So two to three thousand volts for horses and beef cattle. Post spacing. This is just kind of a general rule of thumb here. Line post about every 10 to 12 feet on your perimeter fence if you're using wire high tensile high tensile fence. I'm not going to use just a high tensile fence alone for a perimeter fence. I will use it for interior cross fencing. If it is used for perimeter fence, I want it to be in combination with another type of fence and I'll show you some pictures later on for that. But on interior fencing, if you're using high tensile, remember that the shot going through that wire is the deterrent for that animal. So we're just using, we're just using posts to kind of keep that wire up off the ground. So you can kind of use your own judgment for how often you need to put a high tensile space post in 25 feet. If you're going on straight, straight level ground is probably adequate. If you're in hillier type ground, you're probably going to use more more space post on a rail fence. Your main post, you're going to be 8 to 10 feet, depending upon your board length. This is probably the most common problem with electric fencing is the grounding. You want to make sure that you have adequate grounds and follow your controller instructions for that. In general, we want at least three ground rides. They're at least 8 feet long, faced out 10 feet apart, connected with copper wire if we can. And we want one lightning arrestor per ride. That's very important as well. Many times these high tensile fences or fences in general will get hit by lightning. And a lightning arrestor can really save your energizer from getting burned up. The galvanizing on the wire is also important. Galvanizing generally is going to be classed from 1 to 3. Class 3 galvanizing will have three times more galvanized coating than a class 1. It's going to cost you more, but the more galvanizing that is on that wiring, the longer it's going to last for you. So in my opinion, I would go with class 3 wiring if you can. The other thing about class 1, whenever you wrap that wire, that galvanized coating is going to crack as you wrap it around. So that's going to be a weak point eventually in that wire is where that galvanizing cracks. Speaking from a voice of experience, I can be kind of cheap and when I started out, you know, newly married and trying to build stuff and I didn't have any money. I tried to do things as cheaply as possible and I did that with fence too. And it wound up costing me just from personal experience. I tried my darnedest to build as cheap a corner post as I possibly could. And I wound up rebuilding it too in a very short period of time because it just didn't last. So it wound up costing me more than if I just figured out how to get the funds or just build it the right way the first time, I would have been much, much better off. So lesson learned from me, build it right and build it one. Paddock design, I need to watch my time here. I've already gone down too many rabbit holes but Paddock design, your gates in your corners, that's the best location for ease of use, ease of access to get livestock in and out of. Make sure they're wide enough that it's easy for you to get equipment in and out of and it's easy enough when they're open and you're trying to move livestock through that they can see that opening. Your lanes, if you're using lanes, make sure that they're wide enough for the equipment that you're going to get down 14, 16, 20 foot lanes or nice. Don't think about them as being wasted space because you can also graze those lanes as an extra paddock as well. For your gates, avoid potential wet areas and that's tough to do because over time, traffic in and out of those gates, they tend to wear down and you'll get a wet spot there. But now when it is wet, it's a good time to see where your water pools or how it flows so you can avoid those spots or do some grading if needed to get those gates up and out of those wet spots. I prefer square or rectangle paddocks over to polygons. One, my forage utilization is going to be a little bit better. The other thing too is going to cost me less to fence a square or fence a rectangle in terms of materials than a polygon. Polygon is just going to have more surface area that I'm going to have to fence. If possible, make curved corners rather than square. So if your horse is out exercising himself or running down a fence line, he just makes a general curve rather than running into a corner. In general, I'm going to size my paddocks in terms of forage production and utilization of forage. I'm going to size my paddocks based on forage production and I'm going to think about forage demand and try to get my paddocks may be different sizes, but they're going to be somewhat equal in the amount of forage produced within each paddock in general. If not, kind of a general rule of thumb in terms of space for a horse, one acre per horse rule of thumb. When you're initially designing your paddocks and fencing, start with a map. That really makes things a lot easier. There's lots of places that you can get maps now. You can get a map from your NRCS or USDA service center, Google Earth, WebSoul Survey. You can use maps from. This is just kind of a general place here to start with and kind of some general ideas and how I would go about fencing this piece of land. You've got native grasses. You've got introduced grasses here. Start with a perimeter fence. I'm going to begin to separate out my native from my introduced forages. Water points are important to have a water point within each paddock. You can see some of these paddock areas here don't have water points, so I'm going to develop them. I'm initially going to use my ponds as a water point initially. Then over time, I'm using a well here and adding a water point in this paddock and another water point here. On this side of the road, I'm busting up this into three paddocks, but I'm still using the single source pond here as a water point. But I'm putting a lane in so that I can have access from these two paddocks here to walk down the lane for access to water. Then over time, I may add another water point here just to improve my utilization of my forage and general utilization of the paddock as well. Here's another way to kind of look at initial paddock design. We have an area here that, again, we have a water point, a pond water point up here, and some native introduced. Break it into where we have access to our pond here for a water point and separating our native pastures from my introduced. And all of a sudden we have a four paddock rotation going on here. We're using a lane up to this pond for water access. And then over time, we can, as we see how this system works, we can add more paddocks as needed. I've extended the lane down here to provide water for these lower paddocks here as well. And over time, you might want to put in an additional water source to supply water to these lower end paddocks here and continue to use the pond up here on the north end. This is a real simple formula for determining, well, how many paddocks do you need? And this takes into consideration grass recovery or regrowth after grazing, the length of your grazing period in terms of days. And then you add one because livestock always have to be standing somewhere. So if we think that we need during the great, during the active growing season, 30 days of recovery and we want to rotate every three days, then our paddock number, we would need 11 paddocks. Just kind of a good general rule of thumb formula and it works fairly well. If you do have ponds, you might consider limiting the access to that pond to do a couple of things. One, to maintain your pond edges to keep them from eroding from just livestock coming in and out of the pond, keeping livestock out of the pond and improving that water quality. So this is something that we've used in the past where we actually go in and we fence out the pond, but we put in a limited access point here for livestock to have access to the water. Here's what it looks like. We've got a, we have a very, this is a fairly steep ramp down into the water, into the pond here. And then this is an electric fence that's floating out here on this pond surface. So as they walk in, the idea is that we want them to walk in, get a drink of water and move back out. We really don't want this to be a point here where they can relax and stand in the water. We want them in and out. This works pretty well if initially the only access for water you have is a pond. Fence height for perimeter fencing, we would want to discourage that animal from jumping out. So the height of the fence around the width of a mature animal should be adequate. Cost, this is not actual cost, but a comparative cost index. So the lower the number, the deeper the cost is going to be relative to a higher index number here. So you can see there are high tensile fence. Single strand is going to be the cheapest to install. Our controller is not included in this cost compared to a board fence using two by six treated boards. It's going to be very expensive. Life span over here is also an important consideration. I've seen high tensile fence last even longer than 25, 30 years. If it's good quality, put in white with a good charger. I've seen high impedance fence chargers last 15, 20 years under a five gallon bucket. They're built to last pretty well. Again, you're not going to use this for perimeter fencing, but this kind of gives you an idea of comparative cost between different materials. Estimating the amount of material that you're going to need. A common measurement in fencing is a rod. So a rod is equal to 16 and a half feet. A quarter mile is 1320 or 80 rods. And you can see how this comes all the way up to a mile being 320 rods. And how many rods it would take to go around or fence a perimeter based on acreages and shapes. Again, a comment about energizers and jewels. Voltage is the is the generic energizers ability to deliver a power or shock. When we test the fence and you have a fence tester that you hang on to the fence wire, it's going to measure the amount of voltage that's in that fence or going going down that fence. A jewel is the amount of power or shock that the energizer can deliver. And it's it's the push that's going to push that voltage down the fence. So as I mentioned earlier, it's going to compare compare energizers based on output jewels. This is a really nice illustration here between two tractors. We have a long tractor here. It's putting out 2700 RPMs. They're both putting out 27 RPMs. But this one in terms of jewels would have much greater horsepower or jewels compared to this long tractor cost wise. Whoops cost.  There's going to be variations in warranties between between the energizers. So look for a dealer. That's important to who your local dealer is where you can get this equipment from who's going to stand behind it. What the warranties are, compare your output jewels and price and get what you need for the fencing that you're trying to power. Energizers. There's going to be three basic types. One is just going to be the plug in main. This is probably the most reliable one of the three here. Just a plug in fence charger works great powers a lot of fence. The MBS is going to be a main plug in with a battery backup or solar option is very versatile in application. I have not used one of these. I've used the main in plugs and I've used the solar powered energizers as well. Solar solar has really improved over time, but you still run some risk with extended periods of cloudy days. You can lose charge also help and sometimes damage these solar panels. And it's a good idea to have a battery backup for insurance materials. Here's here's a role of your high tensile wire. Typically, I really like to use the 12 and a half gauge. This is class three, so it's got the class three galvanize on it. Here's a V match role 100 to 200 feet per roll is fairly common and what you're going to get high tensile. You can you can get various links of this. A common one is 2640. This is coated high tensile wire, which is going to be a little bit more visible to the animal. It's probably going to last a little bit longer with that with that coating on it as well. I threw this up here. This is this is not Bob wire. This is smooth twisted wire. If you put barbs on it, it would be barbed wire. You can use this. I don't really recommend it. I would much prefer going with the high tensile over this 12 and a half gauge smooth wire. This is great stuff for temporary cross fencing. There's all different types that you can get here. These things will pull quite a bit of voltage. You can see that it goes everything from about a two inch three inch wide poly wire up to strands of poly wire. This stuff is as I mentioned great for cross fencing. If it's temporary cross fencing and you're using it year after year, if you keep it up out of the weather, you can use this stuff for years and years and years. Materials. This is a combination of board and high tensile. Works very well. Pipe, down in the oil producing areas, you see a lot of pipe fencing. It's attractive. It looks good. It will weather. It will rust. You've got a lot of extra labor in keeping this stuff painted. You also need somebody to come in and weld this up. The other thing about pipe fencing, if something hits it, it's not going to give very much. Wooden fence post. I really like these. A well treated, good treated wooden fence post can last a long time. I know of some treated fence posts that have been in the ground about 50 years and they're still, they're still good strong post and holding that fence up well. I don't recommend a T post because if you're going to use it, make sure you do cap these in. I have seen some real rats with animals trying to go over or scale over the top of T post and get hung up on the tips and really some pretty bad injuries can occur. I don't recommend a T post for horse fencing. Hourglass fence post. I like these a lot with high tensile fencing. I don't have to worry about insulators. I can just put a pen in here and carry my fence. If something hits this, they will give. They will come out of the ground. The problem with these is over time they will weather. You need to handle these with gloves. You need to get them pre-drilled because they're like drilling through concrete. They're very hard. If they do break, they tend to splinter, but they work very nicely in combination with the high tensile fence. These little step-in posts are very easy to use. You just step them in. They work great for cross fencing and temporary fencing.  Remember the high tensile fence is the power. It's the voltage coming down through there. It's going to keep the animals in. These are just to keep the wire up, keep it straight. Materials I really like standoffs on a perimeter fence. This is a standoff with a wooden fence here with a high tensile line going down through it. This gives a couple of things. It gives some protection to your perimeter fence, keep the animals off that perimeter fence. And then also you have a power source here that if you want to do any cross fence in with a temporary fence, you can tap right into this line coming down through here with the standoff and off you go. Candles, gate handles, and electric voltage. There's quality variations in these electric handles. They will weather and crack over time, but the more solid they are, the longer they'll last and tend not to crack. Some of the cheaper ones don't last, they'll crack, and you can get a pretty good shot going through these. This is kind of an interesting little deal here. This is just a garden hose reel, but if you're using a lot of temporary fencing like this, you can roll a lot of cross-fencing material up onto these reels and then pull it right back off. Works really well. I like these spring fencers on high-tensile. It just kind of gives you an idea of how much tension you're putting onto this. Also, you're going to need electric insulators. Here's another view of an insult tube. These work very well. A garden hose is not an insult tube. People get by with those, but they don't get by very long. Garden hose will crack, will weather, and it will short your fence out and cause you problems later on. This is a spinning genie here. I would not use high-tensile fence without a spinning genie. High-tensile fence, if you're trying to work with it without a spinning genie, it can turn into a slinky really quick, so it keeps it restrained, keeps it in a roll, makes it very easy to spool off or rewind. Additional tools here. This works very well to get an initial tension onto a high-tensile fence. This is a little tool here that you can use to take the slack out of a fence if it gets hit. I've used these in the past. I'm not hugely wild about them, because as they roll that wire up, once it's kinked in there, it's forever kinked. The other thing is that it can kind of crack to galvanize, which can be a problem later on. If you use these, get the tool that goes with it as well.  This is another tool that you absolutely have to have with an electric fence is a tester. Spicers work very well to splice fence together, splice broken wires back together. If you use these, get the tool that goes with them. This is a high-tensile fence stretcher as well. I mentioned corner post and corner bracing. This is just kind of a schematic of a very well-built corner brace. One thing I wanted to mention that if you're obtaining cost share from natural resource conservation service, it will have specifications that you have to follow in order to get your cost share money from them. Whatever specs that they have, you will need to follow. This is kind of based off of those NRCS specs. This is an inline cross brace. How many of these you need will vary based on the length of fence that you're putting in and the terrain that you're putting that fence on. Level land you can get by with fewer inline braces, curves, rough topography you're going to need more. Some variations of braces. This is kind of a dead man here, another dead man version here. This is an extremely well-constructed brace here. You can see the length of the post here. They're using the same diameter post as they're doing here. This is an extremely strong brace. This is a pipe H brace as well here. With electric fencing on these, you really have to make sure that you have tube type insulators to keep from grounding out with these posts, with the metal posts or pipe fence. I'm going to skip through this just because of time. A couple fencing systems here. This is a combination of power pipe wire and ground wires. So you can do a combination of those. You can see the grounds here down here on the bottom. This is an example where all the lines are hot. And then finally, gates. If you're using electric fence and have inline gates, you've got to get the power from one side of the gate to the other. A lot of people will go underground underneath that gate. If you do here in North Dakota, make sure that you get it deep enough that the shrink swell of the ground when it freezes, falls, it's not going to cause you issues here. Another option is to go up and over and then back down. So put the wire instead of underground above and then back down. Two examples of board fencing here. If you have trees that you want to protect, you might consider putting a small fence around those as well. High tensile with the standoffs again. Very nice perimeter fence here with the v-mesh. This is a really nice fence here with an inline gate. I like this gate because they've also put the mesh fencing over the gate as well. And I wanted you to see how well the braces are around this gate. That's a huge problem with gates, especially the longer they are, the stronger this brace needs to be to keep that gate from dragging. Some examples here. This is a woven wire or a chain link rather. I don't really care for chain link fencing at all. It's very expensive. It has some rough points on it. I don't really recommend it at all, but showing it here because you can use it. I don't recommend it. There are woven electric fences available for horses as well. A lot of this application is used with sheep, with goats, with smaller log stock. This is a good example of the insulating tube, insulating a hot wire around the post. And then finally, a list of resources here. Most of these power fence companies also have materials for sport and how to install their material. And it's all really quite good, quite good information because if you buy their products, they want you to have success with their products. So most of them do have a very good fencing guide along with their materials here. But with that, I am done. And I'll be happy to answer questions too. All right, thank you, James. We are going to move on into Mary's presentation, and we will follow up with some questions at the end. There we go. Okay. Awesome. Thanks so much, James.  And so I know Paige is managing those, and we'll get to those after a little bit. So with that, we are going to switch a little bit over to why manage manure and how does this go along with fencing? Same way it went along with grazing last week. If we're doing anything with our animals, if that's in a paddock or out on a pasture, we're going to have to manage that manure. And so why manage manure? I'm always going to start with this slide.  If the nutrients are not used and they become a pollutant, and they're essentially wasted. So I don't want to waste, but I also don't want you to pollute. So improper manure storage and land application leads to excessive soil nutrients, surface runoff, leachate, and then water contaminated with manure. And we don't want that. Okay. Reasons maybe you would rather manage your manure. External parasites such as flies, bacteria and pathogens, rodents, odors, internal parasites, weed seeds, and then of course just visual appearance. Who wants to see a pile of manure sitting there? So all of these are reasons that you might more so want to manage your manure. We went over this last week.  But just quickly, short-term and long-term. So short-term stockpiles in North Dakota may not be for more than nine months. So you can stack your manure in one place for nine months. And then you have to spread it and then move to a new location. If you're going to have a permanent stockpile, you can do that. That's for more than a year. So forever permanent. But it involves a little bit of soil investigation, some regulatory oversight. Again, that's just in North Dakota. There are rules in every state. And so if you contact your extension person in your state, they'll be able to point you to whoever that is. Of course, no matter where we're at, we're going to want to protect our surface water and our groundwater. And so we're not going to put manure in gravel pits along streams or lakes right now. We're flooding. One side of the state has five inches of snow right now. It's snowing and raining on them. The other side of the state is flooding right now. And so floodplain, don't put your manure there because we're going to lose some of our nutrients as well. All right. So I'm still going to do, I know I went over this last week, but I'm going to go over it again. Wanderer management this spring, we need to assess where manure is currently stored. If it needs to be moved before becoming inundated with flood waters, if you can still move it and you know those flood waters are coming, let's move it. It might be too late, might be too muddy. I know all of the mites. I totally understand. But if you can do it, let's do it. Turn, turn that snow bedding pile to keep it heating. Let's not leave snow in there until June. You might say, but Mary is not going to heat their snow in it. It's cold enough to keep snow. That's okay. It is heating. I guarantee if you open it up, you're going to feel some warmth in there. Let's keep it heating by turning it clear, and now get it out of your lot. And then James said this, and Paige mentioned it in the chat, and I'm going to tell you again, it doesn't matter what we're talking about, plan for next year. Look right now. Where's the snow? Where is the, where is the water flowing? If it's raining outside, where are we having issues with ground and footing? Plan for next year. So take some notes. I know it sounds crazy. Like I'm going to go out and take notes about where this mud puddles at Mary. Yes, that's what I want you to do. Plan for next year. So that next year at this time, you're not saying, well, I should have addressed that last year during the summer. Okay. So manure snow piles and safety. So they're, they're not necessarily the same thing, but I'm going to talk about them at the same slide. So on the one side, I just put the snow mountain up snow, winter mountain. So you might have a little bit of hay, some bedding, some leftover snow, just a little bit of everything. Some manure. Turn that pile. I promise you it'll heat. There's another one there on the right side of the slide. You can see my arrows randomly pointing to the sky. I did that because there's power lines up there. And while in this picture, it does not look close, but if you bring your tractor and your loader over, I can tell you from experience, it's pretty darn close once you open up the grapple. Okay. And so initially when this pile was starting to be built, it wasn't an issue. We weren't close. It was far enough away. Winter was early. It didn't matter. And now this weekend as we were clearing out corrals, we were very close to being dangerous. And so just think that if you're going out to clear to move, to turn, do any of the things that we're talking about today think, oh, where are we going to, we're going to look up. We have to look up. And then page put in the chat too. I want you to look down.  And I put this here. So managing mud on horse farms. There's a university of Minnesota has a whole webpage about this that I put this picture here to remind me to tell you about this. So this looks like just mud and water, but very likely in North Dakota underneath of that, there's a nice layer of ice. And I can also tell you that from experience. I had to wash some clothes this weekend. Cause I ended up in my hind end because of ground that looked like this. It just looks like mud and water. And then there's a nice layer of ice under a lot of this still. So when you're out there, not just for your own sake and your own non-smelliness, but also for your animals, whatever animals you're managing, let's just be careful and understand that there's probably still some footing issues there. Okay. So I said, let's turn those snow mountains. We're not going to go into a whole bedding lecture here. I have that on the next slide. I have a link to our workshop for that. So I just want to remind you, if we're going to compost, not only are we reducing the volume and managing that manner, we're also reducing weed seeds. You guys don't like weed seeds. You don't like pathogens. We for sure don't like parasites. We're reducing nutrient loss. So that's the first slide when I said, why care about managing manure? We're reducing nutrient loss by increasing our nutrient stability. So let's go ahead and turn our, our manure compost it so that we can have a lot of benefits from that. Okay. Dry lots, sacrifice areas, exercise pads. They're probably all called the same thing. And so James, again, James touched on some of this in his slides too, as far as space and footing and shelter and water, what we need to make sure that those animals have everything they need. Sometimes, where did I put it? Oh, here we go. Sometimes this is why we need a sacrifice area. Okay. So I know this doesn't look good. The horses are dirty. They're, so here's a couple of positives. Okay. One, yes, they're tearing up the ground, but it's just in that area. Okay. That's why we're calling it a sacrifice area. We're not going to go any farther out on that grass. We're going to give everything time to dry up. We're going to feed them hay. And I know it looks bad because it's on the ground. Maybe we're thinking we're wasting it, but we actually have it in a feeder and it's up off the ground. That's good for parasite management. Okay. It's going to look like this for a while, but it will get better. So I'm going to go back here. Things to consider location, soil characteristics and structure, ease of use. So when James is building his pad ox, he's going to use the hay. He's going to build his dog. That's really important for the animals. Something that he considers in there is where is it easiest for us to come back to, whether that's back to water. If we're not completely developed in all of our areas, we don't have water development in all of our areas. Where is an easy place that all of our animals can come back to. Is there a central place? Is it better on one end or the other? So ease of use. And then also ease of use for storing. If we're going to have a sacrifice area or a dry lot where And so we want to make sure that it's easy for us to access with our type of equipment that we have. And then where are we going to put that manure? Space. Is it enough space? Miner collection. Can you get in there? Can you do that? If you're going to have somebody else come in and haul your manure, if you're not taking it out of your pen yourself, can they get their equipment in?  What is your accessibility there? And then fly control. If we're taking our manure out of those areas, we're doing a better job of managing that manure. But if we're going to put it right beside our pen or sacrifice area, maybe that's not a great place to put it because of flies. If you're going to put it there, let's turn it to make sure we're killing those flies and keeping that cycle going. So I just highlighted one thing here. So compost manure rather than spreading it raw on fields where horses graze. So last week, this was a big question and we're going to skip ahead to that question. Here's some pasture drag examples. So people had asked last week, can I use a harrow? What kind of fanciness level do we need? We can go anywhere from unfancy, right? So we have a chain link fence with some bricks on top of it, just enough pressure to kind of break up some of that horse manure all the way up to something fancy. Just like what we would call a harrow. And so basically we're just trying to drag across and break that out. We want to break up the manure. It gives the manure more access or surface area to sunlight. And then that sun is going to help kill our parasites. Okay. So there are no restrictions in North Dakota as far as when to spread manure. But like I said last week, we all almost all of us have a smartphone. We have a weather app. We know when it's going to rain.  We have a clue. We totally know. Okay. And so just common sense. Don't spread it before during or after a large rain event and don't spread where water quality is going to be compromised. So we have those valuable nutrients. Let's not waste them. Composting helps us reduce weed seeds, pathogens, parasites. It also reduces the volume, which is important for us. The goal of a dry lot areas to sacrifice the small part so that we can have more healthy productive land over here. So we're going to sacrifice a small part of land to make sure that everything else is really good. And especially in a year like this, when we feel like we're a month behind. Let's, let's make sure that we're not, we're going to be really excited when the sun comes out next week and we're going to try to turn stuff out. Let's just hold off a little bit and have that, that one smaller area that we're messing up for a bigger area that'll be better later. Proper management can lead to parasite load reduction and then just use common sense when you're spreading. With that, I'm going to turn it over to Paige. Paige is going to go through the questions that are in the chat.  We'll just verbalize some of that for you who are not reading it. Alright, so now's a good time to throw those questions in. I'm going to scroll back up to the top and ask some questions of James verbally. James, one of the questions was how deep do our fence posts need to go into the ground, both our brace fence posts and our line fence posts. I believe I had a slide on that and I believe it was at least two and a half to four feet. Yeah, that's what I remember seeing as well. So four feet for those corners and two, two and a half for the others. Also, if they're getting cost share, NRCS does have depth recommendations that you have to go by. Yes, correct. One of the questions was what if you're having horses in with other livestock like our small animals like goats or sheep, what would be some good options that would work for both species? The mesh fence, the woven fence would work well. That's probably going to be the best option for sheep. Goats as well. It's probably about as good as you can do with a goat. How about some considerations for what type of fencers to use to keep predators out? I would definitely lean toward the low impedance high voltage fencers like we talked about that would go with the high tensile fence. I would probably also use that in combination with some type of a mesh fence as well. So I might run the mesh fence, but underneath the mesh probably run a high tensile electric wire. So if they try to tunnel underneath that mesh wire, they're going to get hit. Thank you. I think that kind of wraps up the questions. There was some discussion about our tough winters here in North Dakota and just keeping that in mind when you're selecting your fencing materials as well as where you're putting your fences. Some of those materials are maybe a little less winter hardy than others. So keep that in mind when you are planning your horse fences for sure. And then just another reminder about calling 811 before you plan to install your fence. It is the law as well as staying safe and avoiding any buried utility lines. All right. I'm not seeing anything else pop up into the chat. So with that, we are going to wrap up this webinar. Thank you all for joining today. There's a poll popping up on the screen. If you fill that out for us, we'd appreciate it. Also as a reminder, you'll be getting an email with a recording for this week's webinar, as well as a link to the resources that we referenced today. I also want to invite you to join us next week where we're going to talk about traveling with your horse. So whether you're going an hour or two down the road or you're hauling across the nation, what are some things to consider both from, you know, the paperwork and legal side. What about overnighting? How often do we load and unload? How do we plan for those long trips and those short trips? Just how to best manage your travel with your horse. And then also in two weeks, we are going to cover a footing topic. So arena footings have popped up as something we wanted to visit about from past webinars, as well as footing around the horse facilities. So we mentioned a little bit today about those dry lots. How do we make those drain correctly? How do we deal with footing in high traffic areas? So just a little heads up on what's coming up over the next couple of weeks. Also want to invite you to give us suggestions on what you want to hear about in the future as well. So we are basing these webinars off of what previous attendees have told us that they want to learn a little bit more about. So we're always looking for feedback for suggestions for the future as well. All right, I don't see any more chat questions or pop questions popping up into the chat. Mary, do you have any last final words before we log off? No, I think that's it. You covered it all and so we'll see you all next week. All right. Thanks everybody for joining us today.


As-salamu alaikum. Welcome to lecture number 45 of the course on statistics and probability. Yes, the last lecture of this particular course. Students, you will recall that in the last lecture I discussed with you the chi-square test of goodness or fit. And we applied this concept to the example in which we fitted the binomial distribution to real data. In today's lecture, I will be applying this concept to the case of the Poisson distribution. And as you now see on the slide, the platform manager of an airlines terminal ticket counter wants to determine whether customer arrivals can be modeled by using a Poisson distribution. The manager is especially interested in late night traffic. Accordingly, data for the time period of interest have been collected as follows. Number of arrivals per minute are 0, 1, 2, 3 and so on. And the frequencies are 84, 114, 70 and so on. And the sum of the frequency column is 400. The question is, is the distribution Poisson students, let us try to understand this problem. That is the number of arrivals per minute. That means 400 minutes and they should have been selected at random if this distribution, if this whole method is to be properly applied. Now, the 400 minutes we have randomly selected, we have seen per minute how many people are arriving and that data is in front of you. As you once again see on the screen, corresponding to the number of arrivals per minute equal to 0, the frequency is 84. What does it mean? It means 84 minutes is kisham ke the, when we did not have even one person arriving. Similarly, 114 minutes were searched when only one person arrived at the counter and 70 was searched when there were two persons arriving in that one minute. Students, the question is, what is this distribution Poisson? You remember when I had discussed the Poisson process, how was it defined? A process where events occur randomly, either over a time scale or over a distance scale. Obviously, we are talking about the time scale and one minute is being regarded as one unit of time. So, we are understanding that customer arrival is a random process. So, we would like to think that this data should be fitted by a Poisson distribution. If I do want to fit a Poisson distribution to this data, students, what will be the procedure? You remember that in the case of binomial, we first computed x bar and then we equated x bar to n p. n was already known and so p was found using this equation. Now, in this particular situation, students, the one lone parameter of the Poisson distribution is mu, where mu is the mean itself of the Poisson distribution. We will have to replace it by its estimate and that of course, is x bar. So, as you now see on the screen, in order to compute x bar, we construct the column of f x, which is the product obviously of the x column with the f column. Sigma f x is equal to 800 and dividing by 400, x bar comes out to be equal to 2. Now, the formula of the Poisson probabilities is e raise to minus mu, mu raise to x over x factorial, but since we have to replace mu by x bar, our formula becomes e raise to minus 2, 2 raise to x over x factorial. Applying this formula, students, we obtain the Poisson probabilities as you now see in the third column on the slide in front of you. You will put the values of the Zahire 0, 1, 2, 3, 4 and so on and you obtain probabilities as 0.1353, 0.2707 and so on such that the sum of the probabilities is 1. Multiplying the probabilities by 400, the sample size we obtain the expected frequencies which are 54.12, 108.28 and so on. Students, now that we have computed the expected frequencies, of course, now is the time for us to apply the chi-square test of goodness of fit. So, as you once again see on the slide, given the column of the observed frequencies O i and the column of the expected frequencies E i, we will be constructing and the column of O i minus E i whole square divided by E i. Adding this last column, chi-square comes out to be 78.88, but students, please before you go on with the thing, please note that we have combined the last four values, the last four categories that we have. If I add 0.08 to 0.36, the sum is still less than 5. When I add 1.36, the sum is not yet 5 and then when I add 4.80 to the sum of the first three, I obtain 6.60 as the expected frequency. Since I have combined the expected frequencies, I must also combine the observed ones and thus I obtain 24 corresponding to 6.60 and students, these are the two numbers that will be used in order to find O i minus E i and the other quantities in that particular part of the table. This sorry bath, you may have to start key students, I hope you remember that even in the last lecture I mentioned to you, that this is a very basic requirement of this particular test, the chi-square test of goodness of it, that none of the expected frequencies should be less than 5 and therefore, if we do obtain sum which are less than 5, then we will add in the manner that I just explained, so that the thing that we get ultimately is 5 or more. Let us have a look at the slide one more time. The sum of the last column is 78.88 and this particular chi-square statistic follows the chi-square distribution having k minus 1 minus r degrees of freedom, where k is the number of categories that we have after having combined any classes that we needed to combine. Therefore, if you count now after having combined the last four, we have 7 classes and not 10. You will recall from the last lecture that r is the number of parameters that you estimate from the sample data. As I just said there is only one parameter of the Poisson distribution mu and we are estimating it by x bar, therefore r is equal to 1. So, the degrees of freedom for this particular statistic are k minus 1 minus r that is 7 minus 1 minus 1 that is 5 degrees of freedom. And students, this is a right tail test, the entire level of significance, the entire area has to be in the right tail. So, if we set alpha equal to 0.05, then we will look in the chi-square table under 0.05 against 5 degrees of freedom. And as you now see on the slide, the critical value is 11.07. Since our computed value 78.88 is much, much larger than 11.07, therefore it is evident that in this particular problem, we reject the null hypothesis which said that the fit is good and we conclude that it is not a good fit. In fact, students, here 78.88, which is our result, it is so much larger than 11.07 that we are inclined to think that it might even be highly significant. You remember that if your level of significance is 5 percent, if it is only 1 percent or if your computed value is further than that, then of course we say that it is not only just significant but highly significant. So, I would like to encourage you to have a look at the chi-square table for 5 degrees of freedom under 1 percent alpha and then find out for yourself. The next point I would like to convey to you is, what is the rationale of this point that I said? So, it will always be the right tail test students. OI minus EIE whole square over EIE and then you add these up. It is the difference or the distance between the observed and the expected frequency. If those distances are small, then their squares will be small and when you add all those quantities, whatever you do after that when you add that thing, that will also be relatively big. In fact, do not you see students that if your expected frequencies, your observed frequencies with exact tally, then OI minus EIE will be equal to 0 or if they are all 0, what will be chi-square equal to? Obviously, 0. So, this is an ideal situation that your expected frequencies are exactly matching with your observed frequencies. Of course, we cannot expect such an ideal situation in the real life world where we have so much randomness, but I hope that you have got the essence of the whole discussion that the smaller the value of chi-square, the more evidence there is that it is a good fit. If it is so large that it falls in the tail, the right tail of the distribution, then we say that this is not a good fit and we should reject H naught. Then we say that this is not a good fit and we should reject H naught. Students, now let us start the discussion of the chi-square test of independence. You are interested in the formula that is quite similar to the one that we just had and actually this can also be regarded as a kind of chi-square test of goodness of fit. Let me explain this concept with the help of an example. A random sample of 250 men and 250 women were polled as to their desire concerning the ownership of personal computers. The following data resulted. In the top row, you have the men and the women and in the first column, you have the two situations, either they want to own a PC or they do not want a PC. Now, the figures in the body of the table are 120, 130, 80 and 170. Now, the figures in the body of the table are 120, 130, 80 and 170. You can see that out of 250 men and 120 women, they want to own a personal computer, whereas the women we chose 250 men and 80 women are of this type. We would like to test the hypothesis that the desire to own a PC is independent of the sex of the person and we would like to conduct this test at the 5 percent level of significance. This test involves exactly the same kind of a pattern as we have had before. The first step is the formulation of the null and the alternative hypotheses and the null hypothesis students in any such situation will always be that the two variables of classification are independent, whereas the alternative hypothesis would always be that the two variables of classification are not independent. The underlying mathematics of this procedure is such that as you know, we always begin by assuming that H naught is true, that the two variables of classification are independent. Then, we conduct all the steps under this hypothesis and in that case, our expected frequencies and our chi-square will be computed in a way in which the testing can be done. After having set the hypotheses, of course, the second step is the level of significance and we can as usual set it at 5 percent. The third step is the test statistic and as you now see on the screen, the test statistic in this situation is chi-square is equal to sigma sigma oij minus eij whole square over eij and please note that the first summation is over i and the second is over j. Students, I had told you that the pattern is very similar to the one that we had before. The difference is that we are saying sigma sigma oij minus eij, but I am not saying oij in Punjabi. I am saying it in English and why is it that we have a double summation students? The reason is that as you know, in this case, we have a bivariate table. So, I represents rows and j represents the columns. That is over all the values and therefore, we have the double summation. What is the next step? Of course, the fourth step is to compute our test statistic. Until here, students, pehle hame eij joke is particular problem, there are four e11, e12, e21 and e22. These have to be computed and as you now see on the slide, e11 that is the expected frequency of the first cell. The cell lying in the first row as well as the first column, students, this expected frequency e11 is obtained by multiplying the marginal total directly to the right of this cell by the marginal total directly below this cell and dividing this product by the grand total 500. Hence, e11 is equal to 200 into 250 over 500 and that is equal to 100. Students, ejo formula henna, this is according to what you learnt quite a few lectures ago, that if two events A and B are independent, then the probability of A and B is equal to the probability of A into the probability of B. Usi ki ek modified form abhi istimal ki and this is the way you will compute all the expected frequencies. Me pehle repeat karthi ho jis cell ki expected frequency chahiye uske saamne wala total uske niche wale total se multiply karthi je and divide this product by the grand total of all the observations. Next, of course, we will construct the columns of oij minus eij and the square and so on. And hence, as you now see on the slide, constructing all those columns, chi square comes out to be 13.33. Students, ye jo columns humne banai hai ki ab aap ki apni marzi hai ke aap kis order me oij ki values iss column me likhe, but the only thing that you have to keep in mind is that whichever order you choose for the oij, the same order must be maintained for eij. Students, what is the fifth step of any hypothesis test in procedure? The critical region. Now, in this particular situation, it can be mathematically shown that this particular statistic chi square follows the chi square distribution having r minus 1 into c minus 1 degrees of freedom, where r is the number of rows and c is the number of columns. This particular example may r is equal to 2 and c is also equal to 2. Therefore, we have 2 minus 1 into 2 minus 1 degrees of freedom, that is 1 degree of freedom. So, looking at the chi square table, against 1 degree of freedom under 0.05, we obtain, as you now see on the slide, 3.84 as the critical value. The last step is the conclusion and since our computed value 13.33 is larger than 3.84, therefore, we reject H naught. Students, our null hypothesis is that the two variables of classification are independent, that is, the sex of the person and the desire to own a personal computer, these are independent. So, we have rejected this hypothesis. That means that we are saying that the two variables of classification are associated. So, naturally, the question arises, which gender is there in which it is a higher proportion of persons who would like to have a personal computer? If there is a relationship with sex of this thing, then we should study it, that in which category this thing is more prevalent. So, let us have a look at the data once again. As you now see on the slide, out of the 250 men, 120 are such who would like to have a PC whereas, out of the 250 women, only 80 are of this type. Now, since our chi-square has come out to be significant, therefore, we can say that this difference that we can see in the data, this indicates a significantly higher proportion of men wanting to have a personal computer as compared with women. Students, with reference to the chi-square, always keep in mind the very important point that for both the chi-square test of goodness of fit and the chi-square test of independence, the number of observations in our sample should be at least 50. Otherwise, this procedure of hypothesis testing will not be suitable. Students, this example we have done, this was the simplest case when our bivariate table, which is technically called a contingency table, it was only a 2 by 2 table, instead of the rows or the columns. But of course, we can apply this procedure when we have more than 2 rows or more than 2 columns. So, let us consider one more example as you now see on the screen. A national survey was conducted in a country to obtain information regarding the smoking patterns of the adult males by marital status. A random sample of 1772 citizens, 18 years old and over, yielded the following data. Now, students, you can see that we have 4 categories for marital status, single, married, widowed and divorced and we have 3 categories for smoking pattern. Total abstinence, only at times, and regular smoker. We would like to use this data to decide whether there is an association between marital status and smoking pattern. Students, I would like to throw this as a challenge to you. The steps are absolutely similar to the ones that we just did. So, please conduct it yourself and find out whether there is an association between the marital status and the smoking pattern or whether you do not find such an association and you accept the null hypothesis that the 2 variables of classification in this case are independent students. This is to determine whether or not there is an association between 2 qualitative variables. Jase, sex of the person and whether or not he or she wants to own a personal computer. Obviously, these are qualitative variables. If we want to know the relationship between quantitative variables, then what we will do? You will recall lecture number 15, in which I discussed with you in detail the concepts of regression and correlation. Students, where we talked about R or regression line fit, that was in the area of descriptive statistics, but then we also have methods by which we can do inferential statistics regarding regression and correlation. We can test hypotheses about rho, the population correlation coefficient and beta, the population regression coefficient. Which we have R or B, they are of course the sample values, but we would not have a chance to discuss those in detail in this particular course, QK students. If we look at it this way, there are many more things which could be discussed, but the basic purpose of the course is to give you an insight into the fundamentals of the subject. I think that what we have done up till now would suffice for that, inshallah. This brings us to the end of the series of topics that I wanted to convey to you in some detail in this course. For the remaining part of today's lecture, students, I will be conveying to you a few very interesting points. The very first of which is the degrees of freedom. I said to you that the parameters that exist in the mathematical equations of those distributions, they are called degrees of freedom. But the question is why on earth are they called degrees of freedom? What do we mean by that? Let me try to explain this to you with the help of an illustration. This is the line segment. This thing is still free to move in this plane. It can be rotated and otherwise it can move. So, we have one degree of freedom. Students, if I fix both the points, now I do not have any degree of freedom for the movement of this pencil or the line segment. But if we step into the three-dimensional world, the one that you and I live in, then you will note that this plane, this whole plane is fixed here. This plane itself is able to move in this three-dimensional world. And in the three dimensions, I do have one degree of freedom. So, this is the kind of concept that is involved in this discussion. Now, coming to our statistical work. You remember that when we tested H naught mu is equal to something against H alternative that mu is not equal to that quantity. And if we were using the t test, we said that the t statistic x bar minus mu naught over s over square root of n follows the t distribution having n minus 1 degrees of freedom. So, this is n minus 1 q. Let me explain this with the help of an example, which you now see on the slide. Suppose that we have a sample of size n equal to 6 and suppose that the sum of the sample values is 20.  If you freely choose the first 5 values and the sum that comes out to be 17, then Lamohala sixth value has to be equal to 3. Hence, we do not have 6 degrees of freedom to choose values, but we do have 6 minus 1, that is 5 degrees of freedom. To dekhah aapne students, it is quite an interesting concept. Ishi baatko ek aur tarasi sumjne ki koshish karte. I have 6 observations. I find their sum and then I throw away one of those 6. Students, I can regenerate that one that I have thrown away, because I already know the sum and I know I have the other 5 values. To degrees of freedom jahena, they are 6 minus 1 wo ek jo mai regenerate kar sati hu, that is the one for which I do not have the degree of freedom, that that can also be chosen freely. Wo to un baaki 5 values se hi regenerate ho jaegi. So, this is the kind of concept why we say that we are dealing with degrees of freedom. Coming back to the T distribution, the chi-square distribution, the F distribution students. Generally, we can define degrees of freedom in this manner that the degrees of freedom are the total number of observations in our sample minus the number of parameters that you estimate from the sample data. Ye jo T distribution ki baat hai hi kare lete sab se pehle. As you now see on the slide, in testing H naught mu is equal to mu naught against H alternative that mu is not equal to mu naught, our statistic X bar minus mu naught over s over square root of n follows the T distribution having n minus 1 degrees of freedom. Here, the point to be noted is that s is the estimate of sigma, the population standard deviation. Goya hum ek parameter jo hai population ka, that is the standard deviation usko hum estimate kare hain by the sample standard deviation small s. Chuke hum ek parameter chahe aap usko standard deviation kehle je, yaa population variance kehle je. Chuke ye ek parameter estimate ki aaja raha hai from the sample data therefore the degrees of freedom are n minus 1. Ishi tara f statistic ki baat kare lete hai you will recall that when we were trying to test H naught sigma 1 square is equal to sigma 2 square, our test statistic was f is equal to s 1 square over s 2 square aur hum ne kaha tha ye statistic n 1 minus 1 comma n 2 minus 1 degrees of freedom vali f distribution ko follow karte hai. Why do we have n 1 minus 1? Because s 1 square that we have in the numerator is an estimator of sigma 1 square to hum ne goya ek parameter estimate kia and hence n 1 minus 1 degrees of freedom for the numerator and a similar situation holds for the denominator aur jo abhi hum ne thori der pehle chi square test of independence conduct kia usme to bohati interesting situation hai. I said to you that we have r minus 1 into c minus 1 degrees of freedom to ye kistrase ho gaya boh jo example tha ke jisme either man or woman and either you want a p c or you do not want a p c to usme 2 rows and 2 columns and therefore we have 2 minus 1 into 2 minus 1 that is 1 into 1 and that is 1 degree of freedom. Ye kaisin hua? Let us look at that table once again. Now I am presenting to you the same table in such a way that I do have information regarding the marginal totals and the grand total, but I have removed the four data values that I had in the body of the table. For example, suppose that the value of the first cell is 100 then automatically the value in the second cell of the top row will also be 100 because the sum of the row is 200 and the value in the second cell of the first column will be 150 because the sum of that column is 250 and then of course we can also find the value in the last cell that is the cell in the second row and the second column. To lubbe lubab ye students that you can choose only one value freely. Baak ki tino ki tino juhena wo to un marginal totals ki wajah se determine ho jayengi. Aap freely choose nei kar satte. The only one that you can choose freely that may be in the first cell or in the second or third or fourth usse farak nei patta. The point to note is that you have only one degree of freedom and one can be written as 1 into 1 and that can be written as 2 minus 1 into 2 minus 1 in other words R minus 1 into C minus 1. Of course, if you have a 2 by 3 table or a 3 by 4 table to ye hi baat jo me ne present ki it will be in that context aur me aap ko encourage karungi ki aap ek 2 by 3 table banaye aur marginal total likh le aur uske baat khud iss pe work karein ke aaya ye jo formula R minus 1 into C minus 1 wala hai jo ke uske iss me 2 minus 1 into 3 minus 1, yehani 1 into 2, yehani 2 degrees of freedom ke mutta radif ho ga kya waak hi uss table me which has 2 rows and 3 columns. Do you really have only 2 degrees of freedom iss pe aap khud work ki jay. Students, the next concept is that of the P value aap ne dekha ke humne bohat saare hypothesis test kye we discussed various procedures, but in every situation what we were doing was that we determined the critical region and we decided whether our computed value falls in the acceptance region or in the critical region, but students we can also draw a conclusion regarding our hypothesis by means of what is called the P value. As you now see on the slide the probability of observing a sample value as extreme as or more extreme than the value that has been observed, given that the null hypothesis is true, this probability is called the P value. Students I would like to explain this to you with the help of the example in which we considered the hourly wages of computer analysts and registered nurses. As you will recall the statement of the example was a survey conducted by a market research organization 5 years ago showed that the estimated hourly wage for temporary computer analysts was essentially the same as the hourly wage for registered nurses. This year a random sample of 32 temporary computer analysts from across the country is taken. The analysts are contacted by telephone and asked what rates they are currently able to obtain in the marketplace. A similar random sample of 34 registered nurses is taken and the resulting wage figures are listed in the following table. As you can see the wages are 24.10, 23.75 and so on for the computer analysts and similar figures for registered nurses. We would like to conduct a hypothesis test at the 2 percent level of significance to determine whether the hourly wages of the computer analysts are still the same as those of the registered nurses. Students you will recall that our null hypothesis was mu 1 minus mu 2 is equal to 0 against the alternative that mu 1 minus mu 2 is not equal to 0. We carried out various steps and the computed value of our test statistic came out to be 3.43 which was greater than the tabulated value, the critical value 2.33. Hence we rejected H naught. The figure that you now see on the screen illustrates this point Abdeh Krenke, our value 3.43 johair that is lying in the right tail of the sampling distribution and it is to the right of 2.33 and therefore we reject the null hypothesis. Students yeh joh result humne obtain kia we could also have got the same result by the p-value method. Ab p-value kya thise? Subse pehla step johair wo yeh hai ke aap compute keejiye how much area lies to the right of our computed value. So, as you now see on the slide the value 3.43 in the right tail of the standard normal distribution is in front of you and as you can see the area beyond this point is 0.0004. Of course, we get this area by looking up the area table of the standard normal distribution. Now, since this is a two-tailed test therefore we are interested not only in this area which is on the right tail, but also the area which is to the left of minus 3.43 and obviously that is also equal to 0.0004 students in this situation the p-value is given by 0.0004 plus 0.0004 and that is equal to 0.0008. That means, we have computed the probability of this point students that our value actually is greater than that of the p-value or greater than that of the p-value and since this is a two-tailed test, we will also say that the negative counterpart of the p-value is greater than that of the p-value or greater than that of the p-value. We have found that it is only 0.0008. Now, since this is extremely small and much smaller than the level of significance that is 2 percent in this problem therefore we reject H naught. Students we always begin by assuming that H naught is true. Point is that if H naught is true, then what is the probability of extreme value? 0.0008 is highly improbable. So, if we have such a low probability for getting this value that we have actually got then we say that there is something wrong and we cannot accept the null hypothesis in this situation. So, therefore, we will reject it. I would like to encourage you to work on this problem on your own. You will simply take the area in that one tail. You will compute the area beyond that value which you have actually got as your computed value of your test statistic and once you have computed it, you will see that is that large or is it small? As you can now see on the slide, the p-value is a property of your data and it indicates how improbable your obtained result really is. So, if it is a very very small figure then we say that this result is very improbable under the null hypothesis and therefore, we reject H naught. A very simple way of understanding what interpretation to make is that if your p-value is less than the level of significance, then you will reject H naught and if it is greater then you would accept H naught. But I would like to encourage you to work on this problem on your own and to study quite a few text books and other books in this regard. What I would like to now discuss with you students is the relationship between interval estimation and hypothesis testing. Students, that there seems to be a relationship between interval estimation and hypothesis testing. After all, do you not remember that when we derived the confidence interval for mu, dharmyanme area rakhata 1 minus alpha is side way alpha by 2 or dhusri side may be alpha by 2 and when we did hypothesis testing, in case of a two tailed test, if our level of significance is alpha, then we have alpha by 2 area on this side and alpha by 2 on the other side. They definitely does seem to be some kind of a relationship between the two. So, let me throw a proposition to you. As you now see on the slide, let L comma u be the 95 percent confidence interval for the parameter theta, where L stands for the lower limit and u for the upper limit of the interval. Then we will accept the null hypothesis H naught theta is equal to theta naught against H 1 theta is unequal to theta naught at 5 percent level of significance. If theta naught falls inside the confidence interval, but if theta naught falls outside the interval L u, then we will reject H naught. In the language of hypothesis testing, the 95 percent confidence interval is known as the acceptance region and the region outside the confidence interval is called the rejection region. Also, in this way we can say that the end points of the confidence interval represent the critical values. Students, I would like you to work on this on your own. You might arrive at the conclusion that what kind of a relationship exists between interval estimation and hypothesis testing. You might arrive at the conclusion that what was just presented is correct or you might arrive at something slightly different. So, that is a challenge for you. Students, we are drawing close to the end of today's lecture and to the end of this course. But before I close, I would like to convey to you that statistics is a vast discipline. In this course, I have presented to you some of the very basic and fundamental concepts. But of course, there are numerous other concepts that could be discussed. For example, with reference to experimental design, we could talk about the Latin square design, factorial designs and so many other very interesting and important designs. With reference to statistical inference students, we could have talked about non-parametric statistics, non-parametric inference and in a similar way, there are so many other topics that we could talk about. This particular course should be regarded as the beginning of a long journey in the science of statistics. And if you are interested, you would definitely want to pursue studies in this area at some stage or the other. Another very interesting and important point, students, that I would like to convey to you is that I have presented so many different problems and so many different techniques and most of the time I have been presenting to you the solutions of the problems step by step in a lot of detail. The purpose was to give you an idea of the fundamental concept involved in that particular technique. But having said this, students, I would now like to convey to you that there are numerous computer packages that you can use and you can arrive at the solution within seconds. You are a computer science student. Who can know this better than you? And the packages are numerous, as I said, SPSS, SAS, Stata, S plus, Minitab and so on and so forth. So, I would like to encourage you to attempt quite a few problems on these packages. You will simply be clicking a button and you will be arriving at all your regression coefficients and all kinds of things that you would like to do. And speaking of latest trends, students, I would now like to give to you one of the latest definitions of statistics. Statistics is a science of decision making for governing the state affairs. It collects, analyzes, manages, monitors, interprets, evaluates and validates information. Statistics is information science and information science is statistics. It is an applicable science as its tools are applied to all sciences including humanities and social sciences. And students, at this point, I would like to repeat what I said in the very first lecture of this course. It has been said that statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write. Yes, this is the importance of statistics. Education is a wholesome process. The essence of effective teaching is effective and inspiring communication on the part of the teacher and concentrated attention and effort on the part of the student. As far as the teacher is concerned, the teacher is concerned, as far as students' responsibility is concerned, as I said, it is concentrated attention and a lot of effort. And as far as students' responsibility is concerned, as I said, it is concentrated attention and a lot of effort. Students, if this course has been able to inculcate in you the fundamentals of statistical and probabilistic thinking, then it has served its purpose. I wish you the very best in your pursuit of knowledge. May Allah bless you with the wealth of knowledge and with the capability to utilize this knowledge for the betterment of humanity. Salih al-Nakhmi. Ejazat dichi. Allah hafiz.