Good afternoon, I'm Austin Davis director of government affairs with the Lake Champlain Chamber your friendly local regional chamber of commerce with a knack for nurturing prosperity in our local economy and This particular live at 25 525 broadcast. We're gonna be talking about housing and To talk about housing. I've brought on Troy Bachman who is the director of the Northwest Vermont Realtors Association and She is really just if you want to talk for an expert on Our region's housing you could find no better person at this point in time But you know, I don't think unless you're living under a rock It's not hard to see that there's some issues around housing in our region and at the chamber we speak with many businesses every day and it's really noticeable that it's the limiting factor and most of our economic endeavors and It really every pursuit You know, it's it's issues with staffing in our health care systems and other essential services it's preventing growth of Some businesses and you know, we're seeing a lot of incredible things in our regional economy despite this However, we could do much better and we need to do much better So I'm really happy to have Troy on the program here today to talk a little bit about You know the state of play and some things we can look forward to but thanks for joining Why don't you tell us a little bit about yourself and the organization? Sure. So thanks for having me on As Austin mentioned, my name is Troy Bachman. I am the CEO of the Northwest Vermont Realtor Association We are a trade association representing realtors. We've got about 700 members and We our service area is Chittenden Franklin and Grand Isle County. So most of our realtors Are serving that area of course many of them do business throughout the state And we are a part of a broader coalition of realtors across the nation So our focus is really Training for our members education and allowing them to provide the best service to clients And with that that requires some understanding of our market and what's happening with that So glad to talk dive in and talk about, you know a little bit. What's happening if you want to go into that Yeah, you sent me a really enlightening report day or a couple days ago when we decided we're gonna set this up and Yeah, just tell me a little bit about that And this is something you're doing on a regular basis and yeah provide some great insight into what's happening in our market Sure, so I'm gonna plug the Vermont Association of Realtors They provide amazing market statistics for us through a firm called domus analytics And this is available for all of our realtor members so one of the reports that we see on Is about the median sales price and we're always kind of looking at that and we all know Anecdotally that housing prices are going up Wages maybe not matching that and that can be really challenging So right now we see the median sales price across Vermont for all types of properties So condos land single family all types of Residential properties is about three hundred seventy five thousand dollars now when we zoom in on just Chittenden County That number goes up Which is not surprising like we're one of the biggest Metropolitan areas in Vermont and the median sales price in August 23 was five hundred thirty four thousand dollars And that was just for single family homes so We can see and and the graph you're seeing on our screen is tracking the median sales price Over time so you can see like there are some dips every now and then and that's natural with a any type of free market But we can see that that's going up and up and up Which has led to a lot of concern about how Vermonters are going to afford housing Certainly, and it's not as if it's an easy time and we're all seeing inflation across the board not just in housing either way at wages and We're at a time when student loans are resuming payment You know, what are some other factors at play for folks? Yeah, I mean the pandemic caused sort of a nominal influx of people into Vermont other things at play is a historic shortage of building supply And that's not just from the most recent sort of pandemic issues But that stems all the way back to 2008 like the initial great recession the initial great recession The recession as if there were only one the one in 2008 You know the housing industry kind of screeched to a halt for a while in terms of buildings, so we're already working from a Stunted supply And so I you know, it's not just a Vermont issue. It's a nationwide issue So yeah, we've got the inflation. We've got student loans a lack of supply and then Contributing to our supply issues of courses act 250. Yeah. Yeah, I was gonna say so there's there's obviously some things that are specific to Vermont too. We have a 50 year old land use policy that hasn't seen much in the way of updates throughout its time It was created at a time when we didn't have an agency of natural resources and very little local planning and zoning So it had a certainly had a purpose at that time. We've only added to it ever since and so I think that's something We'll probably want to get into a little bit about some chances to reform it We also have a really old housing stock in this state Can you talk a little bit more about that? I certainly live in one of those old houses Yeah, I mean I I live in a house from what 1905 and again, that kind of stems from the challenges in development the labor shortages of like finding people to work on your house if you can't do it yourself Yeah, we have it definitely have an aging housing stock and again, we can kind of look at some of the regulations and you know That's not the only reason but it certainly doesn't help in trying to generate new supply And then another factor Again, we see this nationwide, but specifically in Vermont is we have an old older population And people are living longer more independently Staying in their home staying in their homes longer And especially if you've locked in a 2.5 interest rate or maybe you've paid off your house Maybe you you don't have a mortgage payment at all. There's not a lot of incentive to move right now and downsize Because that's an issue that we see as well of like we do have an older person maybe looking to downsize But where are they going? They don't have properties to downsize to yeah, we really do have an issue with Housing at the entry and exit point of the market You know there it's there folks who are older who are in Larger homes that could probably be more suitable to young families. They might be empty nesters at this point. They're in retirement and you know there as you said probably paid off the margin but not but It's hard to find something that actually makes sense for them if they're gonna sell this home to then put what equity They have in that and have some extra for retirement You know that price that you put on the screen there 425 sure they could sell their home But where are they gonna move to where are they gonna move to yeah? That is the question on everybody's mind and I know for our members a lot of people are having to Kind of get creative with the housing options that they're helping people find like more and more we're seeing people young families by multi-family Like duplexes and quadplexes when they can find them and renting out part of that And that's a wonderful option and that middle housing point of like It's not necessarily a single-family home But it's not a big apartment complex is something that I think our market could really use more of And it's all it's already all through the downtown area like we already see it And I think sometimes there's like resistance to the idea of let's put some some duplexes or some quadplexes But that already exists. It's just old And there's nothing wrong. I don't think was building more of that Well, it's a great place for a great opportunity for the young professionals in the state to start building some equity and cementing some roots here two things that I think are very difficult for this generation, especially with the Older population or state Yeah Yeah, I mean I If I'm sure it's no surprise to anybody watching this. I'm not from around here. I'm from the south I was I moved up last summer And part of the conversation you alluded to this in that your opening was like my association had we I had to work remotely for a while while I tried to find housing and with an aging population and Inflation increasing and causing the dollar to go further Taxpayers are often on the hook for that. So our nice all the nice services we get for our taxpayer dollars We need more people here Yeah, in order to keep the tax burden where it is as opposed to increasing that and I think that's an important point I mean, I think much of our conversation around Development of new housing in our state has consistently been one about growth and an underlying idea that growth is bad And I think we're at a point where especially because we've suppressed growth for so many years You know, we need we absolutely need growth without growth. We're gonna collapse We're not, you know, right now one in five or mantras is over 65 coming soon to one in four and You know, you're not gonna have the The hospital staff the emergency staff that you know, the folks who can work on these old houses If we don't do something and do something soon And then also we're not gonna have the tax base to support all the services these older folks need so growth is necessary and You know, we're not seeing new units come online. I think you had some data on that too Or are you gonna or no? Um, well, oh, yeah, I could talk about the month of supply. Sorry. Um, so right now our market has About two point one month supply of inventory, which means it would take two two point one months for Us to sell anything residential that is on the market. Um, that doesn't account for new construction. That's like what is existing but the Sort of thought of a healthy market depending on like where you are in the nation is anywhere from like a healthy housing market has four to six months at least on the market and The numbers aren't lying like we've only got two months Which can be kind of scary, right?  Well, this is what we have There's probably not anything new coming on in the next, you know, 24 hours or whatever. So there's it the pressure on first-time home buyers The pressure on people looking to find Something a little different for their housing needs is really high and it really is all housing needs, too You know really what we're talking about is when we're at that level of inventory Just slowing down all turnover and all churn in the market, right? And so, you know, we're even talking about our folks who are unhoused in our state And I think actually our state treasurer. I'll give them credit where credit's due He had this fantastic webinar not long ago and the title of it was based off of a book whose author he came on in You know homelessness is a housing problem, which I think is is something that's Extremely lost on a lot of individuals in this current debate. Yeah I mean and we can't forget the connection that homelessness has with like general poverty and again, we're seeing This is a wage issue as well of like We've got a shortage of people to field jobs. We've got a shortage of Again it just all comes back to like we do need people here, but the problem is we have nowhere to put them Yes, and the big question is how are we gonna solve that issue? Which you know you and I talk a lot about this but like we're not gonna build our way out of it There's gonna have to be some other reforms to support the market Other than just like the solution is not Build new housing like that is not the end all be all solution. It has to be one piece of the puzzle Yeah, so let's talk. Let's talk about Reforms of our our housing. I mean, I think you know we touched on act 250 a while back and in the earlier in the conversation and what we found is you know, this is a land use law that has Stymie growth for 50 years and to a large extent has kind of Accomplish what it's out to not do which is create sprawl And I think you know for me when I think about the epitome of that It comes down to what's called the 10-5-5 rule. Can you tell us a little bit more about that? Sure so the 10-5-5 rule is a portion of The legislation that says you can't you can only build 10 houses within five years within five miles of each other So when you think about like a traditional suburban neighborhood, that's typically more than 10 houses and We don't have a lot of those here, and it's not necessarily houses. It's like 10 units So I know there was a proposal to bump that number up to 25 and I don't know if you want to talk a little bit about how that played out But you know, I think that was really scary for a lot of folks, you know and I Think we we ended up, you know getting something across the finish line lads that are processes last session that Allowed to go to 25 or 24 in limited areas we're talking about 0.5 percent of the states overall landmass and I think we need to go much Bigger and bolder than that if we're gonna start hitting our targets You know and you think about that from an incentive perspective and when you say You can't do more than 10 units within five years within five miles You know unless and you're automatically going into active 50 if you don't and you know active 50 is notorious for Adding additional time energy and cost So I was looking into this a little bit as someone sort of new to the area within the last year or so And I found There was a number granted. This is from 2017. So many years old at this point in terms of the rate of inflation But it said that an active 50 review process could add up to if not more than $50,000 to the cost of the home And that prices out So many people, you know like especially with the interest rates that we're seeing right now like the 50,000 extra dollars is the difference between I Mean I'm not I'm not the mathematician But that is that's like a significant difference in your monthly mortgage and your down payment like in your tax bill You know, I think anyone has ever bought a home can tell you the difference between, you know What that $50,000 can make yeah, and you know that certainly just gets rolled into the overall long-term price of Rental units too. And so all of it, you know does go back to this But back to my incentives point, you know, if you know that you might be adding $50,000 per unit On this project, you're gonna try to avoid that you're gonna try to get it look that market rate lower And so what you're telling a developer is Tell you what build nine units and Then maybe wait five years and we'll talk and we'll talk again or Go five miles down the road So, you know what we've really seen and I think you know, you talked to a lot of I remember I was working this at the Center for Research of Vermont many years ago and One study they did was they asked for monitors, you know describe sprawl I was gonna ask you is like I feel like we need to define what that is because we know what that is But for the purposes of this conversation like put a definition on it Yeah, and the funny thing was a lot of folks once they define sprawl They'd be us is sprawl is your community sprawling and a lot of the folks in these are our local communities would say no And then they'd kind of go back and say well by your definition We do have sprawl and I think I think that it's these little Pockets of acreage that provide, you know, the agrarian feeling lifestyle that folks move to Vermont for many folks do That doesn't necessarily feel like sprawl But when you take a larger step back you realize, you know, my My home on a dirt road the end of another dirt road is is contributing to the sprawl at the end of the day Well, and an important factor of all this too is like act 250 My understanding correct me if I'm wrong was designed with some environmental concerns in mind and Sprawl is not environmentally friendly, right? Like if you're in a community if you live at the end of a dirt road, you have to get in a car and Drive wherever you're going. You're not necessarily taking public transportation You're not walking to the grocery store. That's 15 miles away, right? So you're Contributing to your carbon footprint goes up and I mean I would argue that sprawl Has a big factor in carbon emissions, right because we you you got to have a car Yeah, and I think it goes back to you know, and then on the other side of things it's expensive develop in our downtown Sure, that's where the majority of the regulatory burden is A lot of the sites in our downtowns are need to be some kind of remediated They might they're brownfields, which means at one point in time There was some kind of contamination on that soil. So you need to mitigate that before you ever start and It is difficult to actually build our downtowns and that leads back to each of our organizations respectively long-term goal, which is removing some of these downtowns From Act 250 jurisdiction the ones that have shown that they have rigorous Regulatory environments that can handle the work of Act 250 the work that wasn't done by any municipality when Act 250 is created they can handle it and they can do it at a local level which Ironically is a bit of a Vermont value, but we don't allow it often So, yeah, I mean I think I'm excited to see a couple of studies that are going on there's And I'm curious what your thoughts of walking at the progression, but currently right now due to a bill that passed last year Act 47 there are four separate studies going on and I Orage anyone watching to to plug into these the Natural Resources Board is undertaking a study and they're looking at So many factors. I don't think it's worth listing But what they're coming out to is in what it might look like is a tiered system where communities based off the place based off of their regulation our municipalities are there are Gonna see different levels of review we might actually be able to see Downtowns like Burlington be able to do their own review and operate on their own Still acknowledging all the criteria and whatnot of Act 250, but doing it as their local level And then I think there's gonna be some areas in the state that will likely look a lot like Act 250 currently is with some revisions and then I think there will be some areas in the state that folks will actually increase jurisdictional thresholds and Ironically those are areas that Act 250 was always meant to protect and maybe didn't do the best job protecting You know some of these places and more vulnerable areas. We're talking about really about 18% of the state So that's a really exciting kind of component of how that the natural resources boards using their time and then There's a designation study which if I believe the website is Designation2050.org you want to take a look at that we can post that in the description of this video. There's a Summit next week on Tuesday if and there's some other opportunities to engage, but this is about the designation programs for different cities and towns across the state that allow for revitalization and tax credits for things like you know mixed-use development and flood mitigation which Is very very on the tip of our tongues this this session. I mean this summer and you know They're looking at how do they make those designations have a little bit more of How can we look at places like Burlington and say okay, you know actually the fact that they've achieved this designation Means that maybe they don't need Act 250 or places like Winooski. That's had Tremendous fantastic smart growth over the last few years. It's a really opportunity city for Vermont and then You know the other two things Kind of stopping so long-winded about it, but it's just some advanced planning I was gonna you know, I'll jump in and say that while I don't think a Lot of the consequences of the regulation that we have currently were intentional We are gonna have to be Extremely intentional with how we rectify our issues. Yeah We're gonna have to take these studies and do something with the information Because you see I mean you see this all the time of like, okay. Well, like let's get some information Let's get some data on that great and then it gets kind of pushed down Right and I think that's it's so important to like take okay. Yes, we know this is an issue We now have the numbers to prove that this is an issue Let's be very intentional with how we do this in an inclusive way That provides more access for people who've traditionally been knocked out of the housing market and I think that's what's exciting about some of the work that's been mandated this last session for the Vermont Association of Planners and Development agencies and What they're looking at is mapping the state which you know, I think is always kind of one of the original Goals under activity that just kind of got forgotten, you know, it was one of those like a big piece of legislation easy to forget Yeah, I mean, it was one of those things was is always the intention But it should be easy to look at a map somewhere and say Okay, these are volatile areas. These are sensitive ecological areas don't touch them these are areas that have stormwater and sewer water and the and Fantastic electrical utilities and public transit these are places we should put development and once we have some of that like, you know inventorying of our resources of state my hope is We can have more educated conversations and we can actually make much of this conversation Place oriented and realistic and its expectations So you can kind of see all these three studies that I mentioned are all kind of braiding together and the hope is that you know Before December even we have a nice idea of how they've all braided together And we can use the work of the planners and developing agencies And we can use the work of the existing designation programs and how folks want to re-envision that And then put that into How the Natural Resources Board wants to handle some Locations and we can start to have it so that we can build more housing in the places that it needs to be Yeah, and I think it's important to remember and to remind our bike ourselves and everyone Nobody wants us to build Lockstock and barrel no holds barred wherever you want to build you can build like nobody wants that right? Well somebody probably does want that but there are people they're not on this show But I think it's important to like the concept of there's gonna be new housing near my house. What does that mean for traffic? What does that mean for? The crowding at my grocery store. What does that mean for my daily life? What does that mean for my backyard? What does that mean for my backyard? And I think we need more people not to be a total cornball But like we need more people to say yes, I want this here Because it does Have benefit for everyone and living in that community living in a community that is more connected is is good for people We our association did a polling project and it wasn't in Chittenden County. It was in Franklin and Grand Isle where we asked registered voters about their preferences and you know, what type of housing do you want to see developed and It was clear that there was there's tension between like Yes, I see that housing cost and housing availability is an issue, but I don't want it here. Yeah, and I think that's a big barrier Especially in the active 50 process because of the public comment and public review Yeah, and really, you know, I think the thing we even hit on here is the active 50 does create this Just war of attrition. It's about how long can you drag it out? But the longer you drag it out the more you're out in costs.  Well, I think I think the issue with active 50 when we talk to legislators is There's there is that question of like what point do you give up? And it's you know, the people point to their oh, we don't see many people give up most active 50 permits go through to completion and it's like At what cost literally that was the thing I was going to say it was at what cost the second side of that is Who never tried because they've seen that cost They've seen that time and energy expended and they said not worth it And if you're a developer, I think another you know speaking of cost I don't have the hard data on this but I would assume that if you were a developer Looking at okay, I can build 24. Let's just say we bump it up to 2555 I can build 24 units That's gonna change how you cost your materials. It's gonna change how you cost your contractors like it does The buy-in book mentality is not always the best, but it does have I mean it is it is part of the conversation you know of Okay, but if I had to start at nine take a pause Do nine again like it's yeah, I think that's a well We're coming we're coming up at two minutes And so I want to get your final thoughts, but I want to put a finer point on what we were kind of just getting at and that is You know what we need is Vermonters to start looking at the current housing situation with a less myopic lens and what we need folks to look at is That we do need some growth. We have Push back growth to try to stay to our agrarian bucolic roots as long as possible However, it's where the point of emergency We're not gonna have the nurses and doctors or the EMTs or firefighters the teachers or the teachers And the baristas and or the servers or anybody we're not gonna have them if we don't do something and do something soon and you know, we need everyone to start looking at it that way and You know, if you would like to plug into these efforts My email is just austinvermott.org and I don't know if you want to share yours Sure, mine is at Troy at nwbt realtor.org and Certainly want to plug you in reach out to your legislators. Let them know that housing is an issue of concern for you And that you want to be a welcoming state a state that says come here do great things Join our community Because to say no to housing is to say no to new Vermonters at the end of the day So anything else you want to leave us with? It's been a fantastic conversation. Well that really you said it better than better than I can but yeah, I'll just give a plug for Definitely it is important to be engaged with all aspects of your community and at this point Broad housing goals is what builds our community Right. We want people to feel included here and you know, that's something I certainly love about vermont That's something that made vermont really attractive to me and my family was It seems like a welcoming place and so let's make that true with our housing And you've got to engage with your legislators and kind of keep keep an eye on that And uh, that's that's probably what I would say nothing says you're welcome here like a healthy housing You know, that's exactly right. Let's change our slogan Let's leave it at that for today. Again, I'm austin davis with the lake champaign chamber And thank you trolly for joining for this And uh, have a great evening folks


Welcome to your sixth session of statistical inference literacies and today we're going to look at how we employ the different non-parametric tests. Today is the 17th, we're dealing with the non-parametric tests and then on the 24th we will be looking at how we solve simple linear regression and correlation and then the last session will be on the day first, we'll look at the time series and then we would have concluded everything relating to content for STA 1502 and then we can look at exam preparations as well. So there's not a lot to go on, only the last two sessions then we will be done. So if by tomorrow or by next week there are not a lot of people also joining then I don't think we will have the exam preparation, this will be the 31st of May will be the last session that I will host and then if you just need any help then we can discuss that outside of the sessions. Okay, any question or comment or query before we start with today's session? Nothing is it. So the first one that we're going to look at is the Wilkinson ranks some tests for difference in two means or medians.  By the end of the session you need to or you will learn how to use Wilkinson ranks some tests for two population medians and also how to present the Wilkinson sign ranks test for comparing two page samples. So with Wilkinson ranks some tests, there are several things that you need to be aware of. The test has to be of two independent population medians. The populations needs to be normally distributed and the distribution has to be free procedure distribution and we use it only when there is rank data available and we must always use normal approximation if either of the sample size is larger than 10. Some of the things that we need to be aware of is that we need to use it when both your sample size 1 and sample size 2 are less than 10 and we assign the ranks to determine which one will be your N1 and N2 samples observation and if unequal samples then N1 which will have the smaller size sample will be referred to or the samples that will have the smaller size sample size will be your N1 and that will be our smallest rank value of 1 and our largest value rank will be N plus 2. Then when there are ties, let's say there are two position ranks, the values that are the same, let's say it's 30 and 30 they follow one another, what we do in terms of that then we take the rank ties so it means you're going to say that other one is 3.5 and the other one is 4.5 something like that so it will take the rank ties and the sum of the ranks for each sample so you will have sample 1 and sample 2 so for each sample you will also label them as T1 and T2 so those will be your ranks and our test statistic we will refer to the T1 which will be the ranks calculated from the small sample, you must always remember that our N1 will always represent the small sample so if we have two samples just if we have two samples, sample sizes, let's say we have the first sample size N1, if N1 is equals to 11 and N2, sample size N2 is equals to 9, what just give me a second, I just need to just hold on, sorry about that it was my mother I thought it's something agent I must apologize for that so what I'm saying here is if our sample size is N1 is equals to 11 and N2 is equals to 9 according to the Wilkinson it means then we need to reverse them because your N1 needs to be the sample size with the smallest values so this will be our N1 and this will be our N2 and that's how you will assign your sample size numbers and then when you calculate the rank from those values therefore it means your T1 that we will calculate the sum of the ranks that you will calculate from the sample size N1 that will be our test statistics and that will be the value that we use to make a decision so in terms of checking the ranks we use this formula because it is the sum of the rankings and it must satisfy this formula before below action and we take your ranks for T1 plus the ranks for T2 should be the same as your sample size both for sample size 1 plus sample size 2 times the sample size 1 plus sample size 2 plus 1 divided by 2 and that is why you don't have to repeat N plus 1 N plus 1 twice so we can represent them by the sample size N and they need to be the same so if I add the ranks they should be the same as if I add the sample sizes on the other side and my electricity is back so in terms of when we go and make a decision because it's hypothesis testing then there are a couple of things that you need to be aware of our median will be from position one or population one and population two so we will calculate the median or they will give you the median to calculate that and some of these things remember like I said 1502 is a second module to 1501 so most of the basic things that you needed to learn you lend them in 1501 and then you come and apply them in 1502 so for those who are doing 1501 and 1502 at the same time it means you're learning this concept at the same time which makes things a little bit difficult and different because then some of this concept you lend them later in the in the course and then some of them you lend them in the beginning of the course so you need to just make sure that you adjust how you lend content relating to 1501 and content relating to 1502 so in terms of calculating the test statistic we know that we take the sum the sum of the ranks from the sample size and that is our T1 now if we want to make a decision to either reject the null hypothesis or to accept the alternative or reject or accept the null hypothesis then we need to do a hypothesis testing and the first step of hypothesis testing is to state your null hypothesis and also so that you are able to determine where your original rejection will be when you go find the critical value so in terms of state in your null hypothesis if it's a two-tailed test you will say the median the median for population one is the same as the median for population two alternative will state that they are not the same so you can either use an equal sign m1 equal m2 or in your alternative hypothesis you will state that m1 is not equals to m2 therefore it means there are some differences then you will need to go to the table to go find the critical values now on the hypothesis testing when it is two tests or two sided tests or two tailed tests then it means there are two regions of rejection so you're going to go and find the upper t1 critical value and the lower t1 critical value so there will be two critical values on the table and we'll look at the table so that you can see the values that I am referring to so the the critical value once you have them then they will create two regions of rejection so as you can see here if it's below this t1 lower limit you will reject the null hypothesis if it's above t1 upper limit you will reject the null hypothesis otherwise if it falls between in between those two values you do not reject the null hypothesis for a left tail for a left tail your alternative hypothesis will state that your mean one is less than your mean two and you will only have one region of rejection because you will only use your lower limit tail so sorry because I'm getting switched over to the low shading right and for the right tail test we use the alternative will be your mean median for population one is greater than the median of population two and you will only have one region of rejection the other thing I forgot to mention is please make sure that you complete the register I can see that Jacques has posted it okay so let's look at an example of how we apply all that knowledge that we just learned so a sample data are collected on the capacity rate and this is the percentage of capacity for two factories and we need to test at the median operating rate for the two factories the same so that is the hypothesis that we need to test that at the two medians of those two factories factory A and factory B are they the same and we are given here our factory A the rates 71, 82, 77, 94 and 88 and factory B the rates 85, 82, 92, 97 we need to test for the equality of the population median at 0.05 and because they told us at the equality so it means we are doing a two tail test the first thing that we do is we put the capacity in order so that we are able to rank them so we start putting the factory A and factory B values together and that one will be two and we have 82 and 82 remember with 82 and 82 they are the same so it means we need to take the average and put the average value or average rank for those two so it will be because they are on position three and four we take three plus four and divide by two and that gives us 3.5 so they both going to have a weight or a rank of 3.5 3.5 and go in 85 88 it's on position six 92 on position seven 94 on position eight and 97 on position nine and once we are done with the ranking them then we add all the ranks for factory A all the ranks for factory B once we are done with the ranks then we go and select our test statistics so because now we need to also determine which one will be our N1 remember N1 is from the sample size with the smallest value so factory A has one two three four five values and factory B has one two three four has four values so therefore it means factory B will be our N1 and factory A will be our N2 and then our T1 will be the sum of the ranks from our N1 which is factory B so it will be 24.5 so that will be our test statistic so we know now we have our test statistic of T1 X 24.5 and we have our N1 being four values of the sample size value four and N2 with the sample size value of five and we need to test this at alpha zero comma zero five so now we need to go and do our hypothesis testing but before we go and do the hypothesis testing let's go find the critical value so now our critical value we use our alpha that's the other thing we use our alpha and the N so going to look at where our alpha value because our alpha value is zero comma zero five so the table is broken down into a one tail and a two tail alpha value so this is the one tail zero comma zero two five and a two tail zero comma zero five so because we're doing an equality and therefore it means we're doing a two tail so that is why we go to the two tail alpha value of zero comma zero five and then we're going to look at N1 you will find the value of N1 at the top and the value of your N2 will be at the bottom or on the left so our N1 we said it's four and our N2 is five and where they meet we will have our T level of lower level or lower limit and T upper limit and we're going to use 12 and 28 as our critical value or our limits so that it creates two regions of rejection okay so now let's do the hypothesis testing let me write down the values we had 12 and 28 so now let's do the hypothesis testing the first type of hypothesis testing is to state the null hypothesis and the alternative hypothesis so we know that our null hypothesis was the mean of one the mean of population one or factory one will be the same as the mean of median of the factory two and the alternative will state that they are not the same and then what else are we given we are given the value of alpha we are given the value of N and we found the region of rejection so our lower limit was 12 and our upper limit was 28 so we need to take our T1 T1 we found that it was 24.5 and we need to look at where does it fall does it fall in the rejection area or outside of the rejection area it falls between 12 and 28 because it's 24.5 so it falls in the do not reject area and therefore we do not reject at alpha of 0.05 and we can conclude that there is not enough evidence to prove that the medians are not equal and that's how we do the hypothesis testing let's look at how the questions are asked in the exam or in your assignment so given the test statistic of T1 is equals to 242 so here they have given us the sum rank of our smallest sample size and N1 is equals to 6 and T2 is equals to 36 and N2 is equals to 9 use the Wilkinson ransom test to determine at 5 percent level of significance whether the location of the population one is to the right of the location of population two which one of the following statement is incorrect so one is the question asking us they want us to test the hypothesis that our population one is to the right of population two is to the right of population two so therefore it means it will be greater than so it's a one tail test so they are null hypothesis they are stating that the median of population one is greater than oh it's I can put equal don't have to even put that it's equal to the median of population two and our alternative which is H1 will say the median of population one the median of population one is greater than the median of population three that is the hypothesis that we want to test they also gave us at alpha of 0.05 so therefore it means we're going to have to use the table and only go to alpha a one tail test because this is a one tail test so this will be a one tail test so we'll we'll go to one tail test of 0.05 they also gave us our n1 is 6 and our t1 is 42 and 2 is 9 and t2 is 36 so therefore it means this is our test statistic our test statistic because it comes from the sample size n1 of 6 which is lower than or smaller than six or smaller than nine okay the question is also asking us or the other key thing that they gave you here was to tell you that you need to use a well-concentrated some test but we need to find which statement is incorrect in a way so let's after analyzing what I'm given and understanding what the question is asking me that I must identify the incorrect statement now let's go and answer the question option one it states that the hypothesis the null hypothesis the population location are the same because it's a it is a hypothesis testing it doesn't really matter what we put on the null hypothesis because always there is an equal sign so the null hypothesis would be correct the alternative hypothesis states that the location of population one is to the right but that is what they said so is to the right so they gave us that statement so that is correct number three it says the region of rejection it's between the lower limit tl of the three and tu of modern of modern 63 so we need to go and find the critical value so if you have tables in front of you just gonna open my table stop sharing from that side and share my entire screen so I can show you the table so now this is the tables we're gonna go and look for wilkinson there we go so here we need to look for one tail test 0.05 there we go there is our table that we are looking for I can make it bigger so you can see it so we're looking for alpha of zero comma zero four and we know that our n one is six and our n two is nine so we need to go at the top and look for six and yeah the bottom we look for nine so we're both meet that is where we want to be steady three and 63 okay so that is the three and 63 so the region of rejection is correct our t test or our test statistic it says it is 42 so our t test statistic we said is 42 that is correct now in conclusion the two population locations are not the same and we reject the null hypothesis is that true let's test that because we know in terms of our critical value we have two regions of rejection we have 33 and we have 63 if it falls somewhere in between we do not reject remember do not reject if it falls this side we reject we reject the null hypothesis so let's see our 42 falls where falls in between 33 and 63 so therefore we do not reject so in conclusion the population location are not the same we reject the null hypothesis that will be the incorrect statement and that's how you do the hypothesis testing and you answer the questions when asked any question any comments any query before we move on into the last bit so there are no questions now let's look at Wilkinson ranked sum test for large samples so far we looked at when there is small sample now we're going to look at when there is large samples so when they are large samples sorry my thing is bring all of them all at once for large samples the test statistic t1 is approximately normal with the mean of t1 and the standard deviation of t2 so since we are approximating the values therefore it means we calculate the mean by using your sample size one plus times n plus one plus n plus two plus one divide by two remember your n will be n plus one plus two that will give us your mean of your large samples and your standard deviation is given by the square root of your n1 times your n2 times n plus one divide by two remember n is n plus one n plus n1 plus n2 that's the value of your n we must use the normal approximation if either your n1 or your n2 is greater than 10 as you can see that the value of n should be greater than 10 the previous one the value of n should be less than 10 so if it's greater than 10 then we use this approximation so it means we're going to be using the z table assign your n1 to your smallest the smaller of the two sample size so the same criteria that we used the previously we still continue using that and we can then use the normal approximation for the small sample size so we're going to use in when we calculate our test statistic which will be different through the test statistics from the rank sum here we with the small sample with the large sample we use the z test so it means we're going to find the critical values on the z table our z test is given by our our mean i'm going to call it sample mean which will be our t1 minus our population t1 divided by the standard deviation t1 which is the same as s t1 minus your n1 times n plus 1 divided by 2 everything divided by the square root of n1 times n2 times n plus 1 divided by 2 oh divided by 12 not 2 sorry I forgot for standard deviation we divide by 12 and that will be our test statistic so let's continue and like I said we're going to use the z table to go find the critical value and we will look at that when we get to the exercise now let's look at an example use the setting of the prior example which was the first one that we use the sample size where n is equals to 4 and n2 is equals to 5 and alpha of level of significance of less than of 0 comma 0 5 the test statistic was t1 is equals to 24.4 so now here what it does is because our n1 and our n2 and less than 10 we're going to use the z to approximate those sample sizes to a normal distribution so we can use we can still continue and use the same because we're going to use the normal approximation so our t1 which will be our sample size t t t1 mean for the test statistics we know that we got 24.5 so now we need to calculate the mean for the population so that we are able to approximate it and the standard deviation your n1 we remember n1 was 4 so it's n1 times n which is 4 plus 5 which is equals to 9 plus 1 divided by 2 which gives us 20 and our standard deviation will be given by the square root of n1 which is 4 times 5 and 2 times 9 plus 1 divided by 12 which gives us 4 comma 0 8 2 4 comma 0 8 2 to calculate the test statistic then we just substitute the values that we were given t1 we know that it was 24.5 which we calculated previously which is the sum ranks and our mean population is 20 we just calculated it now and our standard deviation is 4 comma 0 8 2 which we just also calculated which is the square root of that values and the answer we get from calculating all this is 1 comma 1 0 our z test is not greater than the critical value of 1 comma 9 6 because then we need to go and find the critical value at alpha of 0 comma 0 5 to find the critical value let me just go show you how you find the critical value to find the critical value we need to go to the z table usually on your thing you can also use this as your z table um because we're doing a 1 tail then it means this will be our critical value so are we doing a 1 tail or a 2 tail so it seems as if we're doing a 2 tail and since we're doing a 2 tail therefore it means we're going to find the critical value under alpha divided by 2 so let's go back to our so if we have our z of 0 comma 0 5 and we find in the critical value and it is a 2 sided a 2 tail therefore we're going to find the z value on 0 comma 0 2 5 so alpha divided by 2 of 0 comma 0 2 5 and that will be our critical value you can find it that way or you can use your table so if we use the z table i'm going to give you both options of how you find the z values where are my tables now so here is the table so you'll come to the negative side of the table that's where you will find most of this value so we're looking for 0 comma 0 2 5 0 if there are four decimals so let's look for 0 comma 0 2 let's see 0 comma 0 2 5 0 there it is and you can go out it relates to 1 comma 9 ignore the negative and you go up and you take the last digit which is 6 it's 1 comma 9 6 so you can use this method to find the z value or you can use the table at the end to go find the z value so you just need to be very careful when you use this table at the bottom especially when you are giving only one tail test because if it's one tail you will notice that even though this says alpha of alpha of alpha divided by 2 you just need to pay attention that this you can also convert it to the alpha values so for example if it's one tail and it is 0 comma 1 you just go to one tail of 0 comma 1 by using by looking at that to cross point to them and a two tail of 0 comma 1 will be 0 comma 0 5 which will be this so you just need to make sure that you pay attention to that okay so since our critical value is 1 comma 9 6 it is greater than our critical value is greater than that the z test so we do not reject the null hypothesis because the value is below 1 comma 9 6 so therefore there is sufficient there is not sufficient evidence that the means are not equal and that's how you do the hypothesis testing let's look at the example and then we're going to look at the last the last type of the Wilkinson or the non parametric test hmm an independent random sample are selected from two populations the data are shown below are shown in the table below here we have sample one which has all those values and sample two which has all those values so you can see that sample one has fewer values than sample two so therefore it means our n1 here will be sample one so we can automatically from here state that this will be sample one use the Wilkinson ranked sum test to determine whether the data provided sufficient evidence to indicate a left shift in the location of the probability distribution of the sample the test is using alpha of 0 comma 0 5 the test statistic z here is the thing that you also need to pay attention when you look at the question so it says the test statistic z for a one tail test is so it means the first thing that we need to do is to order the data that we have from lowest to highest and then rank them so I'm gonna do that so our sample one and our sample two and I'm just gonna put a line in between so that we can have both of them so let's start with let's we'll do both of them so the first value here is from sample two which has four which is the lowest value so we can start with four I'm gonna crash it out so that I don't get confused and the next one will be five and five so they have two fives five and five what else is there five six seven eight we have two eight so therefore it means we'll have eight here and we will have eight there as well and the next one is nine so both of them are from sample two so there will be two nines and what else do we have nine and ten so we have two tens so I'll start there ten there and ten there so this will be the tens and we have twelfths right ten eleven twelve two twelfths so we have a twelve there and a twelve there what else eighteen on this side what else fifteen and sixteen and sixteen so now we need to put some ranks to it so I'm just gonna do this as well rank one and rank two okay so let's rank them so this will be one two three there are four three one two three four five that will be four point five four point five four point five four point five because we take the average of the two um and then oh sure I need to split them not have them on the same so this okay my bet let's change this around this will be two this will be one because I'm starting on this side so that will be four point five and we continue five this will be six uh there are two so six seven it will be six point five six point five and seven oh there are two so it will be seven and eight will be seven point five seven point five and seven point seven point five and we also have eight and nine so it will be eight no man I'm not counting right I'm not counting right I'm not counting right um so this one is four five six seven seven point five so seven eight so this should be eight not seven so this should be eight point five eight point five and then because this is eight nine ten so this should be 10.5 10.5 because then that will be 11 then 12 then 13 14 let's just count that again how many values are here two four six eight 10 12 14 so there should be 14 of them so one two three four five four point five six seven six point five six seven eight nine eight point five ten eleven ten point five twelve thirteen fourteen there we go so we have all of the values so we just need to add them now uh which one has the smallest it was sample one right which is this side so we just need to add all of this four point five we just add four point five five plus eight point five plus 10.5 plus 12 plus 18 plus 14 equals because we add the ranks 62.5 do you also get that 62.5 so we know we now know that our t1 is 62.5 so we have that but that is not the end because we are going to be calculating the z and we know the formula for z is z is equals to or z-state is equals to your t1 minus your mean t1 divide by the standard deviation of t1 and we know that we can calculate our mean t1 because that it is given by your sample size one times n plus one divide by two so let's calculate that and there are one two three four five six there are six and we know we have already counted them they are 14 so plus 14 because it's six plus one two three four five six seven eight six plus eight is 14 plus one divide by two which will be 15 times six 15 times six is 90 divide by two you get 45 yes you get 45 so now we have our mean two so we can also substitute into the formula we know that we have 62.5 minus 45 now we need to calculate what our sigma t is so our standard deviation t1 is given by the square root of let's see if I still remember the formula it's given by the square root of n1 times n2 times n plus one divide by 12 so let's see the square root of n1 there are six of them times and two there are eight times 14 plus one divide by 12 let me see if I open my I need to fix 60 and the square root you say the square root is 7.7 we need to keep all the decimals we can round off when we're done oh gosh it requires me to put the license okay I should have dealt with it before the session so we need to put all the values so it's seven 7.7 I think there are more than that but I think those should be enough so you're just going to divide by 7.74 5967 I just want to do my stop right now and then just want to put in the license on the mere second I should be done and fix okay I can share my screen again because now I have got my calculator open let me put it this way our thingy it's 62.5 subtract 45 divide by 7.745967 and if I say equal I should get my test statistics so any of those values should be equals to that which is option number one did you also get the same is yeah that will be option number one okay which is 2.2592 right and that's how you answer the question relating to Wilkinson rank sum I'm not gonna ask you to do the next one we're gonna do it we're gonna look at how we answer the question relating to the other and then if we have enough time later on then we can come back to this question 20 which is our exercise two so let's let's now look at Wilkinson sign rank so we already covered the two the two Wilkinson when the sample sizes are small and when the sample sizes are large right now we're going to look at how do we do a sign rank type test similar you need the statistical tables you need the formulas and the calculator now with the sign ranks we're also going to introduce another table so Wilkinson rank sum test is a non-parametric test for two related populations and these are the steps that you're going to follow you know therefore you to be able to answer the questions or if you want to do the hypothesis testing step one for each of the sample items we need to compute the difference because they come from the two samples so it means we need to find the difference between the two values the two measurements it's like doing a test on before and after and then we just need to check the differences between the two when we calculate the difference we and when we go in to calculate the test statistics and all that we're going to ignore the plus and the minus sign to find the absolute value of the differences so that's what we're going to do and we're going to omit the differences where they are equals to zero for any of the sample sizes and we're going to assign rank ti from one to n and if there are ties we're going to do the difference so we're still going to do the same thing as what we have been doing but now we're only going to be working with one column and no two two columns so we're going to just assign the ranks based on that column of the differences then we need to reassign back the sign the plus and the minus so that we can calculate or find the sign rank sum t1 and after that then we can compute the Wilkinson test statistic which will be our t as the sum of the positive ranks so let's look at that example so what we're going to be doing in terms of calculating the test statistic or what we call the Wilkinson sign rank test statistic which is the sum of only the positive ranks so it means we're going to add all the positive ranks together and in order for us to find out whether we're going to reject the null hypothesis or not to reject the null hypothesis we need to find the critical value and finding the critical values we need to go to table seven which is critical values of Wilkinson sign rank test okay so other things that we will need is we need to be able to calculate the expected values for the Wilkinson rank sum sign ranks and not rank sum the signed ranks we need to calculate the expected values which is given by n times n plus one divided by four and here we assume that the sample size needs to be greater than 20 and the weighting is approximately normally distributed so our expected value will be that and our standard deviation which will be the square root of your sample size times n plus one times two n plus one divided by 24 so it's double the size okay so to calculate the test statistic we're going to use the z test and our z test is given by z of your t test which are your positive sum ranks remember those positive summons minus the expected value which is your mean divided by the standard deviation which is standard deviation of the ranks then we need to test for the difference in the pairs the null hypothesis will state that the population difference the population means the same and the alternative false state that they are different okay and for that case we're going to use a critical value we're going to find the critical value on the z test so with the sign ranks there are two ways you can make a decision so based on your rank positive value the sum of the rank positive values you use table seven to find the critical value so that you are able to use that to make a decision for the sign rank when your sample size is greater than 20 which is big which we approximated to normal distribution then we use the z test so it means your critical values will be from the z table and that will use table eight if possible okay so let's look at this example from one of the past exam paper says it will concern sign rank sum in I think this is from the sum rank sum it's not from yes it's from the sum rank sum yes and the test statistic is calculated as t is equals to 91 so they have calculated on ready the difference and they do have the answer that the different the sum of the positive differences is 91 and they say there are 18 observation pairs of which three have zero differences and the the two test is performed at five percent level of significance so now they have calculated the t which is our sum rank in this instance of the positive sum ranks and they have given us our n which is our sample size 18 so they are 18 there were 18 observations so n is 18 and they also tell us that there were three values that has a zero difference so it means we ignore those ones anyway and the test statistic we need to test it will be a two-tailed test and it is performed at alpha of zero comma zero five so we have all the information we require so this will be alpha of zero comma zero five choose the correct option so that is the question so we need to find the correct answer now the first one says the critical cut of value r and we need to pay attention to this it says it's greater than because it's for the upper limit it's 90 and the lower limit is state and we can also do the same with number one and number two let's go find out if that is correct so remember we need to use table seven so let's go to the table let's go to the table we need table seven there's this table seven we're doing a two-tailed and the alpha is zero comma zero five two-tailed two-tailed zero comma zero five we found our table so this is our table what else do we know we also know that our n is 18 so if n is 18 those are the two values so the values are 40 and hundred and 31 so those are the values 40 and 131 so going back let's see which one of the two are correct so we know that it's 40 now we did i write that 40 and 131 so the lower limit lower limit is 40 and the upper limit is 131 so this says upper limit is 90 upper limit is 131 so this one is correct this one is incorrect lower limit is 30 lower limit is 40 this one is incorrect this one is correct so this one is the correct one then therefore the other statements don't even are not even necessary for us to look at but anyway we can also look at that and say are we going to reject the null hypothesis or are we not rejecting the null hypothesis as well looks like because the t one is between 40 and 90 so we do not reject so that unless this question was asking which one is incorrect or something like that but yeah we've got oh no the the next one it says the null hypothesis is rejected no we do not reject the null hypothesis because it falls between so that one will also be not will not be correct and if we are rejecting the null hypothesis no the null hypothesis will be rejected will not be rejected but we do not say that we say the null hypothesis we do not reject the null hypothesis so that won't be right the test results are inconclusive no because we do have all information we need so that won't be correct as well so the only correct answer is number two let's look at the next exercise a pair different experiment with n is equals to 30 pairs and with our t which is our positive t of 359 using the wilkinson wilkinson sign rank sum test to determine whether we can infer at five percent level of significance if the two populations differ which one of the following statement is incorrect so now let's go back to our statement so that I can understand exactly what it's given a pair difference with experiment with n of 30 so they have given us the sample size of 30 yielded a positive sum ranks of 349 so they have given us that positive sum ranks and we know that we're doing a sign ranked sum test and we need to determine it at alpha of zero comma zero five so alpha of zero comma zero five so we need to test whether the two population differ so therefore it means we're going to be doing a two-tailed test because they just said it differs they didn't say it's less than or it's greater than or things like that so it is a two-tailed test a two-tailed test because of the weight differ okay so now we can go and ask answer the question which one of the following statement is incorrect number one it states that the null hypothesis states that the two populations are the same is that correct always remember in the null hypothesis it always says it always even equal or it will say they are the same so that is correct we're looking for the incorrect one the alternative hypothesis now remember the the test days they differ so we just need to check if also it says something like they are not equal or they are different so the alternative hypothesis states that the location of population one is different from the location of population two for the fact that they mention weights like different therefore it means they differ then it is correct number three it says the critical value is one comma nine six so remember we're using the z test for that therefore it means we need to go to the z critical value of alpha over two which is z over zero comma zero five which is the same as z which is the same as z of zero comma zero two five zero and remember you can go to table eight and table eight has all the values that we need which then gives us one comma nine six we did do this but I'm just showing you again if you forgot about it so we do go and look at one comma nine six which is correct so now what are we left with we are left with option four or option five to determine whether either one of them are correct so in terms of option four it says the region the region is in absolute value will be greater than of the test statistic will be greater than zero comma zero five it will be equals to one comma nine six so it says in the positive absolute value the value of your critical value will be one comma nine six yes because it will be positive if it was in the if it was in the lower side if we will have two regions of rejection because we're doing a two tail test so this side will be positive one comma nine six and this side will be negative one comma nine six so if it's in the positive side in the greater than side therefore it will be a positive one comma nine six if it was in the negative side of things it will be if it was less than it will be negative one comma nine six so this statement as well is correct so which leaves us with with one so it says the p value is zero comma zero six two so either we need to go and calculate what the p value is the p value you will calculate it after you have calculated your z test so now let's see are we able to calculate the z test we can calculate the z because then we need to calculate z stat which is given by your t positive minus your expected divide by your standard deviation now you will ask me when we calculate all this or when we get all this we do have some of that information because we do know that t plus is 354 minus how do we get the expected so to get the expected value remember is the formula is n times n plus 1 divide by 4 our n there are 30 times 30 plus 1 divide by 4 and that will be oh i just lost my calculator now do you have an answer so 30 they say 2 times it's a 2 comma 30 times 31 divide by 4 it's 232 comma 5 232 is 232 comma 5 that's what you are telling me 232.5 and now we need to calculate the standard deviation which is given by the square root so how do we do that t is the square root of your n times n plus 1 times 2n plus 1 divide by 24 which is the square root of 30 times 30 plus 1 times 2 times 30 plus 1 divide by 24 so 60 plus 1 is 61 so it's 61 times 31 times 30 which is 256 730 divide by 24 which is 23 63.75 2363.75 and taking the square root of the answer what do you get 48 comma 618 48 comma 61618 now now this is our z value so because it's our z value we need to go to the z table so we need to keep only 2 decimals so it means yeah we can roughly say our answer sorry i am doing something very wrong yeah this is our entire year which is 48.618412 so we need to keep all the values so 48.61 this is where we substitute 618412 right that is the answer for our z our standard deviation so we still need to calculate our z so let's calculate our z our z value our z is 354 minus 232.5 equals divide by 48.61 48.412 equals and the answer we get is 2.4991 4991 we need to keep only two decimals so we can say it's 2.50 so we need to go to the table to the z table and go look for this z value so that we can find the probability the p value and we need to be very careful because it's a two sided test so let's go the so and because it's positive then we're gonna go to the positive side of the table so we need to go to the z table remember our z table with the negative values that I showed you now we need to come here because our z value is 3.50 so it means on the left side we need to look for 2.5 at the top we need to look for 0 comma 00 okay right because we're looking for the last digit at the top so let's go to 2.5 or 2.5 is here and at the top we're looking for come on for 00 which is the first column right so we can go to the bottom again and that is the answer that we have which is 0 comma 9938 now we need to be very careful here sir we need to say 1 minus 0.9938 equals so we need to go back to our question let's go there so in order for us to find the p value so yeah we know that the p value they said it was that so but however since the value here it's positive and we're doing a two tail so in order for us to find the p value we're going to find the p value by saying 1 minus the value we find on the table and we need to multiply that value by 2 because there are two sides because of this two tail test it means there are two sides so we did go and find the value on the table which was 1 minus our value on the table was 0.9938 right because of it's positive in the positive value we need to subtract it from 1 why we need to subtract from 1 if you look at this it says we are getting the values from the greater side whereas we should be getting the value from the smaller portion so we're looking for the smaller portion and the smaller portion is the opposite on this one so it will be that value there which is what we are finding which is equals to 2 times 0 comma 0062 if it was only a one tail test this would have been correct but now because we're doing a two tail test we have to multiply by 2 so therefore we take the answer and multiply that by 2 and the answer of the p value should be 0 comma 0 1 2 so which makes number five the incorrect statement yay and we are almost at the end what we didn't cover was number 20 because there is no other exercise after this except the ones that I am giving it to you guys to do on your own so if the session was two hours we would cover also the rest of the other exercises okay so we do have other exercises that have included in the handout as well so you can go through them and if you need any help you can chat with me on whatsapp and then we can I can help you to answer some of these questions some of these questions so here is the first question and then the second question is more about the sign rank sum as well and the third one is about the sum ranks wilkinson's rank sum test and then the sum sign rank sum test with the pairs matched pairs the before and the after okay so with the last few minutes let's recap on what we have just gone through we have learned three ways of calculating wilkinson's or non-parametric test right with the wilkinson rank sum test you are able to calculate it for two ways or in two ways so when the sample sizes are small then we we're going to find the ranks and then we take the test statistic will be the test statistic of the rank sum with the sample size with the smaller sample size and then we need to go and find the critical value and then use the critical values where we will find them on the you will find the upper and the lower limit in order for you to determine where your original rejections are that is the first one the second one in terms of the wilkinson rank sum test is when you have a bigger or a larger sample size and there we are using approximation and therefore we use the Z test and then you need to know how to calculate the mean and the standard deviation and as well as the Z test statistic and then we also follow the same process as we have done with the previous one the test statistic or the sample test statistic will be your or your sample mean will be the sum rank of the smaller sample size which will be your t1 and then you substitute into your Z test to calculate because you would have calculated your mean and your standard deviation using those formulas and then you can calculate your test statistic also you need to go and find the critical value but the critical value here you find them on the Z table and you can find it in two ways either by using table eight or by going and using the standardized standardized cumulative standardized normal distribution table which has the positive and the negative side you just find the alpha value divided by two or alpha value inside the table and go find the critical values outside which are your Z values and that is wilkinson some wilkinson rank sum test in terms of the signed rank sum test you need to find the difference you need to assign the ranks by discarding the negative and the positive by just assigning the rank and then putting back the negative and the positive and calculating the the sum ranks however while you're doing that anywhere where there are differences of zero you need to omit those ones and then only just calculate the sum ranks and then you need you can also make decision based on two things one is based on the test statistic or the t1 that do the farm which is the sum rank sum of the positive values and you go to table seven table seven and go find the critical values on table seven where you just need your alpha value and the value of your n and then you can make your decision based on that otherwise you can use the approximation to normal distribution by using the Z test which will be your test your sum rank positives minus your expected values divided by the standard deviation and therefore it means you need to know how to calculate those formulas of their expected value and the standard deviation in order for you to calculate the Z value and then to make a decision you need to go and find the critical values on the z table table eight please pay attention when you do your critical values because if it's a two-tail test you divide your alpha by two if it's a one-tail test you do not divide your alpha by two and with that said that concludes today's session if there are any questions or answers on comments the platform is yours thank you and enjoy the rest of your evening i will see you next week unless if they cancel the session i won't know please make sure that you complete the register thank you thank you all right


of these cells, dendrites coming out. And both of these cells are projecting into the brain. They're pulling part of the optic nerve project to the brain. And so, he recorded from these two cells in their fire room. Now, this mouse at this time has over 100,000 of these cells. What are we recording from these two? What about all the other ones? So, to get a picture of what the other ones do, we do, instead of recording from cells, we do an imaging of the cells of the entire retina, or a good chunk of the retina, to see what happens. And what you find is something quite amazing. And I think it's a movie. Maybe it won't work, but let's see. Let me just try this again. But what I was about to show... Excuse me? You want to try it? Maybe you should come out of power trinus. I don't know. Maybe it's nothing, but it's actually quite an amazing thing. So, what you see is this. If you're imaging the entire retina, what you see is, what I just showed is called an activity. But what you see is a wave of activity, like a wave, it's called a retina wave, sweeps across the retina, in one direction. What the heck was that? Now you wait for... Can you go backwards? Yeah. Do you have a question? Yes, go ahead. There's no relation between the frequency of the spikes with the temperature or background radioactivity, for instance? I'm sorry. It's temperature and what? Background radioactivity. Background radioactivity. I don't know about radioactivity, because we try to avoid radioactivity. So we don't look at radioactivity. With temperature, yes. Okay, so now let me just get to this. So here's an area of the retina. It's about two by two. And what you'll see is, this works, is a cloud going across this. So it's just background activity. See that cloud? That's thousands of cells firing. Keep your eye on that. Now, this is five times real time. You're looking at this thing, nothing, nothing, nothing. Five times real time, speed it up. Now watch this in a second. See that? You see that dark thing going across that way, that dark cloud? That's thousands and thousands of cells firing the action potentials. This is imaging with a calcium imaging dye. Now, see that little? It's random across the retina. These waves spread. See that? Do you see that? It's an amazing thing because about every five minutes it happens, so it varies, okay? This is just background activity, nothing, nothing, nothing. There's no light here, you know? See, there's no light, there's no photoreceptors. There's another one like that. And these waves occur during a specific period of development when connections are being formed in the brain. So now I'm going to get to your question. Radiativity, I don't know, because there's no reason why we test that. But with temperature, yes. So what happens is these are done in body temperature, in normal body temperature. If you lower the temperature, the waves become much slower, much slower in frequency. But a lot is known about these waves. They must be a thousand. They must be now a thousand. They must have written about them. And we've looked at these in mouse and ferret and fetal monkey. That happened when you were in utero when I was in utero. They just spontaneously stopped like that. So my first point is do no harm, because if you do harm, you can impact a number of things, not the least of which is intellectual activity. That's important for refining connections to the brain. And then the connections are not what they normally should be. So the baby is born at a disadvantage. The second thing that is absolutely certain from the data we have, including the work of two prize winners in my field, which I'll tell you about in a minute, is, and this has had a major impact on how children are treated in American hospitals, and I assume here as well, shortly after birth, correct sensory impairments at an early possible time. So here is about 3% of the babies in the world are born with this condition. It's called ambiopia. You see this eye? It's deviated, like so. That's a pretty serious deviation. Sometimes it's slight, which is hard to catch. Here's what happens. That eye will not be properly stimulated during the early development. And then when a child goes to school and they can't read well, they try to correct that, they find they cannot correct it with anything. And the reason is this, that because this eye is not getting proper activity during development, the connections of this eye to the brain are completely disconnected. Completely disconnected. Not just functional, and atomically they shrink to about 1-100 of what they should be depending on how severe this is. This was fired to discover working on cats and monkeys by David Hubel and Thorsten Wiesel who were not by price, but his work at Harvard Medical School. Hubel died about a year and a half ago and Thorsten Wiesel is still alive. He's in his late 80s now. And so as a result, now when a child is carefully examined and has any kind of deviation that they corrected as soon as possible, because if you wait, it's too late. That brain is functionally and structurally disconnected from the visual centers of the brain. Nothing you can do. And as a result, a lot of effort has been made into testing children for vision. So here's one. A woman called DeVita Teller at the University of Washington, Seattle developed this very simple. You know, we talked about computers, high tech, this and that. You know, everybody's into that. They're saying that simple costs $5 to make. It's been cited thousands of times in literature. So here's what Dr. Teller did. She wanted to test what the kids see when they're first born. They don't see the way you and I see it. So she gets a mother. That's the mom. That's the baby. She has the baby look at this. And here's what the baby sees. That's the baby looking. Either stripes like this on one side, here's where the baby's first pointing on this thing, or nothing. What happens? Babies like people like to look at something as opposed to nothing. So when this comes on, sometimes comes on here, sometimes comes on here, the baby's eyes, looks through the stripes, okay? Because there's nothing to look at here. And so what she does and what she actually does is changes the frequency of these stripes, more stripes, higher spatial frequency, and thereby determining the purity of the baby, okay? At some point, the frequency of these stripes is such that the baby can't tell the difference between this and this. In which case it's looking as random. So they're able to do exactly what optometrists does, because spatial frequency is a bit. So you and I, under optimal conditions, if your vision is perfectly corrected, can see 40 stripes per visual angle, 40. Light, dark, light, dark, light, dark, like that. A baby can see much less and so here's the kind of thing she was able to get. So this is the spatial frequency vision of babies. From one of age, they can barely see the purity of the green, barely half a cycle per degree. So what that means is when they're looking at you, they just see, like a blur, a blur. Eyes like that. And rapidly, look, first year, spatial purity gets better and better and better and better and better like that. Up to four years of age when it's really kind of reaches almost adult limit. So the first four years of life, the spatial purity of the baby is improving, improving, improving. And with this kind of graph, a mother can take her baby to the clinic and say, where does that baby fall in this thing? Here or here or where? So you notice a normal, a normal, how do we fix it? By the way, these studies have also, an visual system, have also been used. So every baby born in a teaching hospital in the United States is immediately tested for hearing. And they provide different earphones, they provide different frequencies and they look at that and they can immediately tell you 10 minutes, usually 8 minutes if the baby has normal hearing or not. If the baby doesn't have normal hearing, they do what I told you. Fix it as soon as you can. They implant electrodes into the cochlea. That baby's hearing will be perfect. Language will be perfect. They get it very early. So about 1% of the babies in America are born with impaired hearing.  They can now do that. Not just can you hear or not, but do you have a hearing impairment by giving different frequencies and recording brain waves? So that all came out of these studies on the visual system by Hubel and Liesl. Now this is something, before these studies were done, a picture from the New York Times many, many years ago, before these studies were done so that we knew that activity in the two eyes was absolutely essential to have the normal connections. So this is a picture that they had of a boy, this little boy who fell and bruised his eye all around here. So they patched the eye. So now he doesn't get normal activity through his eye. They're actually making this eye disconnected from the brain by patching his eye. And by the way, two or three days is enough to make a change during a certain period of time. So he patched his eye. So this kid, of course, doesn't like it. So what he does is, he rips the patch off. So what they do is, they put these arm guards on him. So not only are they disconnecting his eye from his brain, the probation has arms as well. And this is like a cute picture saying, oh, look how Tommy tries to smart the doctors, but the doctors are smarter than him.  They never do that anymore for the reasons that I just showed you. So the third thing is, and this is something most parents know, is optimal periods for learning do exist and should be taken into account in design learning programs. So there are these people, critical periods. The idea of critical periods actually came from work done by animal behaviorist Conrad Lorenz in Germany. And this is him. He won another prize for this. And what he found by accident really is that if you have ducklings follow you a certain part after hatching, they treat you like the mother and they follow you everywhere. They ignore other types of ducks. And it's just a certain period of time. And so here he is with these ducks going out. Now they're adults.


Although he may have played the Wolf of Wall Street role better than anyone else could've in reality you can call him the Wolf of Los Angeles real estate. Leonardo DiCaprio has used his fortune to buy and sell plenty of million dollar mansions over the years. At the moment he owns a few properties in LA and New York and rents some of them out. I've been getting a few requests for a Leonardo DiCaprio house tour so today we're going to take a look at the place he currently calls home, some of his other properties and more. Leonardo DiCaprio is one of my favorite actors and I know a lot of you guys would agree. Aside from his A-list acting, he's also a producer and an environmentalist which can explain why he's planning on opening an eco lodge. But I'll get into that a little later. At the time of this recording he's 45 years old and he's been in the business since he was just a kid and was working even before his memorable role in What's Eating Gilbert Great. Leo played so many iconic roles that got him nominated for Oscars including the likes of The Aviator, Blood Diamond, The Wolf of Wall Street and many more but it seemingly took him forever to get recognized. Finally with the Revenant he got one. Leo is also the founder of Appian Way Productions which has produced some of his films as well as the Leonardo DiCaprio Foundation which is a non-profit organization devoted to promoting environmental awareness. He has used a lot of his fame to bring awareness to causes he's passionate about, has produced several documentaries on the environment as well as regularly supports these causes. Among his many accomplishments Leo's films have earned over 7 billion worldwide so far and he's been placed time and time again on the list of the world's highest paid actors. Taking all this into consideration his current net worth is at an estimated $260 million. Hey guys it's Kara and today we're doing another house tour here on Famous Entertainment. You guys requested this one and I was super into it so today we're seeing where Leonardo DiCaprio lives including his LA house, some of his other multi-million dollar homes he rents out and more. If you liked these videos make sure you're subscribed and ring that bell to be notified when we post you can always be up to date. We've also done house tours and the likes of Ben Affleck and Brad Pitt and we'll have links to some at the end. Follow me on Instagram because I love connecting with you guys. I've also been reading all your comments and I'm going to be responding to some at the end. I need you to let me know who to do next in the comments down below and whose home you'd like to see. Now let's get into this video. We'll start with Leo's Malibu real estate. I found this part kind of confusing because he's bought and sold a few properties here in recent years. Back in 1998 Leo bought this house on the fancy stretch of Carbon Beach also known as Billionaires Beach for 1.6 million. Malibu prices these days are extremely expensive and you'll need multi-millions to buy anything here. It's also a celebrity favorite along the beach. Leo's Malibu cottage was 1675 square feet of living space with three beds and two baths laid out in a bungalow style. Although the size is humble it's been well preserved and updated through the years. Leo had put this place up for rent at one point asking between 23k to 50k per month. He put it on the market he first asked over 10 million but then lowered it to a price of around 9 million. Either way that's quite the profit. His previous Malibu cottage had a gorgeous courtyard leading to a beach and private hot tub on a big oceanfront deck. Inside the decor was bright and clean with an indoor and outdoor vibe and plenty of ocean views all around. Although this home was far from mansion size Leo made sure there was complete privacy and a gated entryway. Leo also sold another Malibu house but this one was much larger. He was able to pocket over 17 million on the sale and he only paid 6 million for it back in 2002. He rented it out too while it was on the market for a price of 175k per month. Luxurious estate had three separate buildings including a four bedroom main house, a two bedroom guest house and a loft with an extra bedroom, gym and office or media room. They all sat on one large lot spanning 2,633 square feet complete with a gorgeous garden and lawn. Leo also had a beach facing deck, hot tub, spa and fire pit. Some other features of this slightly more upscale Malibu home included private parking, maid quarters and private beach entry. Although Leo decided to unload these properties it seemed he wasn't done with Malibu quite yet. In 2017 he decided to invest in a piece of property on Malibu's Paradise Cove. Despite the fact that the small house spent a mere 610 square feet and is the size of a modest New York apartment it sits on 1.76 acres of land and costs Leo a massive 23 million dollars. The cliffside bungalow has only one bedroom but he's in an exclusive neighborhood full of other celebs neighbors like Courtney Cox. Moving on the most recent purchase reported by Leo in Los Angeles was a house in the Los Feliz neighborhood. Back in 2018 he bought a fully renovated mansion that used to belong to Moby. Los Feliz is a relaxed yet trendy area in LA that draws in a lot of up and comers and creative types. Not to mention it's close to Griffith Park in the observatory offering a ton of amazing hikes and scenery. Leo bought this traditional style home for just under 5 million and it sits on just under an acre of land in a hillside neighborhood. Although the house was built back in 1926 it underwent a two-year renovation and was reimagined with modern amenities yet kept the original crossmanship. It spans 4,644 square feet and has five beds and eight baths with two full master suites. His Los Feliz mansion is full of grand common rooms, high ceilings and hardwood floors. Upon walking in the home you're greeted by the large entryway and then to the dramatic living room complete with a fireplace. There's a state-of-the-art kitchen and pantry which connects you to the outdoor living area. Upstairs the two master bedrooms come with fancy spacious bathrooms and walk-in closets. One even has its own study or yoga area but from listing photos it looks like a mini yoga room. On the lower level of Leo's home there's an assistant's office and guest suite that has a secret entryway connecting you to another full bathroom and the steam room. Other features of the house include the gorgeous swimming pool and spa back that's surrounded by gardens and landscaping. There are plenty of tall trees surrounding the property which is perfect for Leo considering he likes to stay far away from the paparazzi. These spots aren't all that Leo owns though. He loves his real estate and he's kept other properties and rents them out. First there's his Hollywood Hills compound because every A-lister needs a place in the Hollywood Hills. At one time in the early 90s Leo bought this home from none other than Madonna for somewhere between 2 and 4 mil. This mega mansion is the result of two neighboring properties merged together and Leo carried out a full renovation in order to create the ultimate palace. His compound spans 12,530 square feet with five beds and 8.5 bathrooms. Some features of the Hollywood mansion include a home movie theater, wine cellar, spa and other fancy add-ons. Not to mention outside there's a giant infinity pool, hot tub, full-size basketball court and subterranean garage. Leo still owns his compound to this day and although he doesn't rent it out to the public, he lets some of his actor friends use it as a crash pad from time to time. Even his buddy Toby Maguire lived here. Leo also maintains a large property in the Vacation Area of Palm Springs which is a city and southern California desert best known for its hot springs, fancy hotels, spas and of course Coachella Valley. It's a lab site Kris Jenner also owns homes here. Leo bought the property in 2014 for just over 5 million but staying at his ranch style estate will cost you 4,500 per night these days. The house spans 7,100 square feet of living space and sits on 1.34 acres of land, offering six beds and 7.5 baths. It's described as a mid-century modern gem. The house was built for actress Dina Shore back in 64 and has clearly been renovated since. Leo's Palm Springs home has light-filled interiors and expansive rooms ideal for entertaining despite the fact it's a one-story house. There's a sunken living room with a fireplace, grand piano and cocktail bar as well as a formal dining room, a kitchen with breakfast bar and an office. The master suite has its own sitting room and the bathroom has its own spa like bath with two saunas. The main house has multiple terraces for dining and relaxing and there's an additional guest house on the property with a kitchenette. Outside there's a pool looking out into the mountains and a tennis court within closed pavilion. Leo's Desert Oasis is situated in a private enclave so you can expect complete security. On top of these places, Leo owns some real estate away from California and in New York City. He owns a condo with the eco-friendly building, the River House and Battery Park City that cost him almost 4 million and a few years later purchased the neighboring unit for 8 million. This place overlooks the Hudson River and the building has a 50-foot pool, yoga studio, 24-hour concierge and much more. Leo's place at the River House is not to get confused with the previous New York apartment he used to own at the Delos and Greenwich Village. He rented this out for a few years as a landlord and his building was also green. There are features like vitamin C in few showers, purified air and water, heat reflexology flooring and a circulated aromatherapy air supply. By all those luxury accommodations, he ended up selling this one. So now we've gotten the rundown of Leonardo DiCaprio's properties including the ones he rents out to guests and residents. I'm not sure which he spends the most time in but I'm gonna guess the place he got an LA in 2018 since it's the most recent. I mean he could be out in Malibu but that house is tiny. If you guys know be sure to tell me. What place did you guys like best about Leo's? I mean he's a bit real estate obsessed and with all those places it's hard to choose but I'd have to say the vacation home in Palm Springs. Not to mention this environmentalist also owns his own island off of Belize on Black Adore K. He bought in 2005 just for 1.75 mil and he's hoping to turn it into an eco resort. Although they were aiming to open in 2018 it's been postponed due to permits and stuff like that. Leo's eco lodge would be the first if it's kind of fully powered by renewable energy and guests can even participate in restoration projects if they choose. Word on the street is that it may open this year but I guess we'll just have to wait and see. Okay guys now read out some comments from previous videos. On our Mariah Carey house tour Brandon Forrest wrote, thank you so much for doing an Icons house tour. I definitely think her NYC penthouse was amazing. It was fit for a queen and he also requested a house tour for Nicole Scherzinger. Thanks Brandon, I love Mariah's penthouse too. On the same video Justin Johnson commented saying, this is so far my favorite tour. Stay safe Kara and take care of Mike. Thanks Justin, I appreciate it. I totally will and you too. All right guys that's all I could finally in order to caprio's multiple properties. I didn't know the man owned so much real estate but I can't say I'm surprised. What did you like best about Leo's homes? I like the fact that even though he spends millions many of the places he actually lives and remains somewhat humble. Let me know in the comments as well as what other celebrity houses you'd like to see. Follow me on Instagram if you wanna chat more and I'll see you next time. Bye.