This is stock number 10447. We are here at Summit Automotive in Fond du Lac, Wisconsin. Your new and used Jeep and Jeep Wrangler headquarters. Today we are checking out this super clean 2013 Jeep Wrangler Rubicon 2-door with the 10th anniversary package on it. This vehicle has 3.6 liter Pentastar V6 motor and has been fully safety and inspected by our service shop. It has a fresh oil and filter change. All the fluids have been checked and topped off. And this Jeep is 100% ready to go. It has the color match hardtop. We shoot all of our videos in 1080p 60 frames per second so if you have HD capabilities on your computer tablet or smartphone device turn them on right now because it is like you're right here looking at the Jeep with me and definitely your best way to check out the vehicle before seeing it in person comes with the painted and polished aluminum rims. You get the little Jeep red Jeep logo on the rims there. It has Goodyear Wrangler Dura Track tires. These are LT 285 70R 17s with about 50 to 60% of the tread left. It has a Fox shock Jeep suspension on it. Might be a two inch suspension lift on here. Has some fog lights. Has an ARB off-road metal bumper with some IPF. Baja lights has a worn winch on here. This is a 9.5 CTI S winch. That's a nylon rope on there. You get the power dome cowl induction hood. Now this is labeled as a 10th anniversary. No scuffs or scrapes on that rim but it's very very similar to the newer hard rock package. As you go down this side just take note of how clean that body is, how reflective and mirror like that paint is. Hard top is in excellent shape. All the doors are in really nice condition. The fenders are in nice shape. His back rim is in nice shape as well. No scuffs or scrapes. It has the metal bumpers back here which you'll note. If you know anything about Jeeps they made the hard rock package with the metal bumpers as well. Toeing package, four pin and receiver hitch. And this thing right here is if you wanted to put a heavy-duty tire carrier on here. It was already equipped to do so. There's a brand new Dura Track tire so compare that to the tread that's on the tires right now. And this matching rim on the back here is in excellent shape. The back storage area is very clean as well. It does come with the factory subwoofer. And this back seat actually comes out. It'll go down as well.  You do get storage underneath here. Very clean back here. Has the spots for your roof and door bolts.  There's three quick bolts on each side. And then two quick disconnects right here. And then there's bolts on the top as well. Rear gates in pretty nice condition. And that shuts nice and smoothly. And down this side of the Jeep just take note once again how clean the body is, how reflective and nice that paint is. The color match hard top. That is an option. You get the heated power mirrors inside. The 10th anniversary Rubicon package gives you the black leather interior with the Rubicon stitched 10th anniversary into the backrest. There are no rips or tears on the seats. Driver seat height adjuster. And we have weather tech floor mats inside the vehicle. You get the red axle lock and sway bar disconnect buttons on there. Power locks as well. And as we hop inside the Jeep you can see that this one has 43,475 miles outside temperature and compass display. You get the leather wrapped steering wheel, cruise controls on the right, Bluetooth and information center controls on the left, and audio controls on the back. This one has the 430N radio which gives you the AM FM and Sirius XM radio capabilities. It also gives you your factory navigation system.  Five-speed automatic transmission, your 4x4 shifter on the floor. You know it's kind of dark in here. A little bit late in the day. We've been doing videos all day. It has the hard top pieces that are in nice shape. And to control all those lights in the front, here are your controls. So we'll go check those out in just a second. And it does come with this Midland CB transceiver. It's a Midland 75822. You also get the Rubicon 10th anniversary badge on the handlebar there. And it does have a 115 volt, 150 watt plug-in as well. The back seats are just as clean as the front seats. No rips or tears back there. And the floor is very clean as well. We'll take a quick look under the hood. I would personally like to thank you for checking out the video today. And hopefully from this HD video you've been able to tell just how clean this Jeep is all the way around. You can see all those lights are lit up and all working great. Under the hood we have the 3.6-liter Pentastar V6 motor. Engine bay is very clean. Runs very smooth. This truck has been fully safe and inspected by our service shop. Has a fresh oil and filter change. All the fluids have been checked and topped off. The Jeep has been gone through mechanically 100% and is 100% ready to go. Really nice Jeep all the way around. I'd highly recommend it from a quality and condition standpoint. And to see more pictures of this Jeep or one of our other 450 new and used cars, trucks, SUVs, minivans, Wranglers, halftons, three-quarter tons, one tons. The name we got it go to our website www.summitauto.com. Full pictures and descriptions of every single vehicle from two locations all at summitauto.com. If you'd like to check out more HD videos you can go to youtube.com slash summitauto. Remember to like, subscribe, and share on this video and all the videos that you see there. In fact in a second you will see a link to subscribe to our YouTube channel on your left. You'll link to more Wrangler videos like this one on your right if you have not been to our website on the bottom link to this vehicle on our website. Click those, check us out. We're super excited to help you with this super clean 2013 Jeep Wrangler Rubicon 2-door with the color matching hard top. Thanks again.


Hello everybody and welcome back to my channel now I just wanted to do a quick little power test on this and we'll just see what we're getting for our buck but I'm just gonna quickly show you the way I've got this connected up so just at the moment because I don't have an adequate NTC which is basically gives me a bit of a soft start and NTC is a negative temperature coefficient resistor so basically when it's cold it has a high resistance and as it gets warmer as it's being used it has a low resistance that's a pretty much zero and what this is for is to when you first especially on a circuit like this with a big old capacitor bank like this when you first apply the power this power bank is gonna load a power and the problem with it doing that is this thing is only good for two and a half amps and the power supply itself is only good for a few amps so you can really do some harm to your turns on here and cause the turns on there. This thing will suck in quite a few amps at the start and it's only for a fraction of a second but it's that fraction of a second that does the damage and these things are what prevent that from happening just by slowing down the inrush current and making it all just a little bit easier on the on the system there so for now I'm using this because I can slowly wind it up as a soft start and it just means that we don't put our stress on the system so over there as you can see so let's go so and at the minute I think we've got 170 millivolts going in there I'm just gonna put this on to 600 millivolts RMS which as you can see when we do that it takes that signal out of the range of the scope and it's graticures so it's a bit better out of the range of the scope and it makes a horrible mark on the FFT but if I am to just back that down a little tiny bit and make that a bit better we can see that the FFT cleans right the way up and we're all good so what I'm gonna do now is I'm gonna see we get a little bit closer because this is actually applying the power to it and let's see what's going on so as you can see then like I say we've got this transformer coming in now this is 160 VA but we've got to take into account things like power factors on these and the power factor from 0 to 1 on this type of load it's inductive it's not going to be as good as if it was a non-inductive fully resistive load so because this doesn't transfer the energy from this part of the AC the high voltage side to the low voltage side without costing and to be honest with you I don't think these are any more efficient than like 55% something like that maybe somebody can tell me in the comments but they're not very efficient what they are is simple and clean unlike a switch mode power supply which if you get a modern one you might be talking 94 95% efficient so if you get one 10 years ago you're talking sort of high 80s efficiency but you've got a lot of induced noise from those and that's not what you want when you want a nice audio system but you know maybe maybe you're doing a lot of classy stuff is like that anyway so that's a completely different thing so here we got our first of all we've got a an X2 capacitor which goes across the main line coming in the DC early AC sorry and that is in parallel with the upright 35 amp full bridge rectifier and this just makes sure that there's no like the line noise coming through this X2 capacitor and then we're straight into our capacitor bank we've got about 15 millifarads of sider on the left and the right it's like pushing it to every vehicle so I don't know why not they were there and here we've got our circuit I have actually screwed the transistors now to the heatsink which makes it look a lot neater rather than big ass clamps from our output there I've got a connection to the scope I've tried to do it as close as I can to the circuit because that's what I've learned that is the best way of doing it and we've got a dummy load connecting up there I've also got a jigsaw I was just trying to keep an eye on what we're going on on the output but I don't really need that at the moment the reason why I was doing that and just see what's going on over there it pretty much tells you the same thing over here 35 to 32 and all that is going on downstairs look there's the people have been doing a lot of work down there so what we're going to do now is we're just going to turn this up and just see what we can actually get down here as our output and then we can just do a quick little bit of math and we'll work out what powers coming out this before we see a clipping off tops and bottoms so let me just turn this up a little bit so I'm going up to 700 millivolt RMS going in that's 800 that's 900 that's one volts going in RMS so I'm going to just move it down the scale now because we don't want to turn that too much higher because I think we're going to start clipping any second now there we go start seeing that just chopping off on the top you can see how messy the FFTs come so let's just back that down back that down back that down there we go that's cleaned up on the FFT and if we just look across there we can see basically we've got 22.55 volts and over here just zoom into that it says we got 23.05 volts so we'll work with what we see on the screen and let's do some quick calculations and I'm just going to turn that down while we're doing that just because I don't want to be over stressing everything it's going to look a bit messy on the FFT because we're not actually putting enough signal on there that's why you can see the sinusoidal waves are very small and that's just not enough for it to do its calculations we're actually down to about two and a half volts now so rather than playing around with anything too much let's just go over here and let's just do our quick calculations what do we have we have 23 or 22.55 volts let's just do it like that so we're going to put 22.55 we're going to square that and then we're going to divide that by the atome load and we should now have our output power in watts so we've got there it says 63.5 63.56 watts so let's just say 60 watts a little bit over 60 watts which is exactly what this amplifier is supposed to give us at 8 ohms that's quite nice can't really say that's too too bad at all all right well from there then what we're going to do is we're going to do a little bit of a change in setup and what I mean by that is rather than let me just get this thing sorted rather than have the oscilloscope on and waveform generator over there I've got a new bit of kit I want to play with and I want to show you and we'll see what that says I'll be back in a moment I got a new piece of software now I'm using Linux and you can't actually use this new software in Linux I'll put a virtual machine in and I'm using it within the virtual machine so let's just make that a full page and I'm gonna tell you now what this is this is my new bit of kit for the bench this is basically a board for audio it does have a whole bunch of other capabilities but I've bought it to get around some of the issues that we got when we're trying to use the FFT and audio measurements on an 8-bit oscilloscope and that's basically this say an 8-bit oscilloscope you can do it you know you need more bandwidth in you need more bits and this thing here is a 14-bit USB oscilloscope now it comes with or you can get free with it from Digilent some software called waveform and this is the waveforms here but I don't use waveforms I use waveforms in order to calibrate this thing because it's got a bit of a calibration process so you got to go through but I don't use that I'm using a piece of software from a chap and Jake his name is and his this piece software is by the Stuffmade now the Stuffmade is a YouTube channel I'd say if you're into your audio you'll want to look at this sort of software in that go there he's a great fella I had a couple of questions he answered straight away for me which is really brilliant and it's allowing me to progress with this now what we're going to do is we're going to go I've just got on the spectrum analyzer part here and I just want to say straight away that look you can see with a noise for always down a down a hundred and the highest peak that I've got on this is a minus 72 minus 73 point 2 dB now we're going to do a couple of little quick measurements on this the actual amplifier is on it's all all go there with the connector block down and amps on there you see the little green LED slightly illuminated power supply of course we need that and what we're going to do is because this isn't going to look very good so we're going to use the screenshots of this on recording in Linux I'm recording the actual screen of this and I hope it does work out to you know to give me the actual screen and not just inside Linux hey it's all a big learning curve and we're going to find out together so we're going to go for this first of all THD noise versus frequency and what I'm going to do is I'm going to set the level to 1 watt and in actual fact what I'm going to do is I'm going to set the level to 5 watts yeah we're at 8 ohms this is just the range for the top that's a plus 10 percent distortion we don't we we're not going to need that I don't believe but the highest we go is is 1% there so we'll have to put it at 10% and the bottom is 0.1 we can have another order of magnitude and then again but I don't think we're going to actually be needing that particularly here we're going to go for the THD of the distortion side of things and I've given it a band with 30 hertz and I'll tell you what that will be in a minute averages one I'm going to put the average to two and we're going to have 50 steps on this and please forgive the banging it's probably the nicest it's been for the last three days but downstairs is having some work done in the bathroom a new bathroom a new walk-in bathroom very nice so let's this was a measurement I took a little bit earlier when I first just started the amp-up but I've made a few differences like I've switched from times 10 on my probe two times one so let's run this it's going to take a few moments just to get it going especially at the lower frequencies and then we'll see the start coming across the screen so here we are this is at a 26 hertz now we can see the marker and we're at like 0.08 percent which isn't terribly great and 0.06 I do that little dip and this is remember is at a five watts of power going through this and I can see over on there I just show you quick what's going on over here you get to see what's going over what's going on over there as that runs through that is the AC voltage on the output a bit more seeable okay well that's that completed so that's the that's the noise as well as the total harmonic distortion so let's just put it down to be in the total harmonic distortion on this one without the noise and again we're going to run it at five watts I'm just gonna let that start doing its thing at the lower frequencies it does take a little bit longer to to process it okay so we started off down there we're at 30 hertz now at 0.05360 percent 0.04 it's quite nice it's good to 70 hertz I do like the idea on the bottom left you got the spectrum analyzer and you get your scope of course on the right hand side I do like that but the spectrum analyzer at the moment is most things are there sort of like around about 70 minus 70 any anything else you can see in that now there's gonna be things there's gonna be things from this entire setup you're gonna see me you're not gonna get you know this I suppose unless you spend a lot of money a pure type of thing going on there so it's all cut as I find about 17 17 kilohertz there so but that's quite nice you get to look down and we get to go all the way back to the start and have a little look what was going on there at 20 hertz 0.06991% go a bit further in we go here 25 hertz 0.58 and of course we also get to sit down here so we go for that because that's our where are we there that's at 80 80 hertz but even that's minus 66.6 more than 66 is a 666 69 we look at these peaks there that's minus 78 nearly minus 79 there so it's good it is a and it's all below the 0.1% line which is which is pretty good pretty good now we can look at a THD versus power I think that asked you up to 100 watts now so I'm gonna do I've set it for I'm gonna put it onto 1% distortion where I want it to cut off like I don't want to I don't want to push this too high and I'm not even sure if I can without breaking this device it's gonna be the first time I'm gonna push it if I can to 100 watts make sure our load impedance is 8 ohms which it is I'm just gonna do a quick temperature check that's well on the dummy load there's nothing there's some warmth on the heatsink to the amplifier but the dummy load is actually cold so we're gonna give that a go then and we are gonna be doing stopping at 1% bandwidth is gonna be 20 hertz we're gonna be going from steps 50 yeah can we go we start from 1 watts all right just bring it straight in at 1 watts I haven't done this before so this is gonna be well this is gonna be whatever it's gonna be let's give it a go let's just start running that okay as you can see the power is rising we're still under like 1% under 0.1% I should say and we are at so we're at 10 watts now 0.06 on the THD and noise this is noise as well we're at 40 watts whoa not quite 1% 63 watts okay but this is only good for that amount of power because in the measurement I shown you before we did Bob was it 22.5 volts or something was it 23 volt I can't remember I'm sorry but we're good for 63 watts so that tallys up quite nicely with that and then we got into being at 1.064 percent distortion which is it's cut out because it is gone above now let's say we get that under 60 we're still 0.8 and down here we get to be 0.7 so that's at 50 watts but the jump is above above 40 watts we start going up we're not quite on 1% distortion yet but below that we're below up to 40 watts we've got a distortion of 0.055 percent and that's at 40 watts that's pretty good I mean I don't think I'd begin it to do any more work and that anyway to be honest you're in this small room that's fine I am thrilled absolutely thrilled with this software too I'm thrilled with this because it is going to open up new doors you can play around with employers and testing them and yeah I'm I'm thrilled with this particular circuit anyway it's it's still below 1% distortion all the way up till we get past where I know it clips anyway and that's what we're seeing that's what it would have been seen straight away and I've just this thing's got a built-in cut off I set it at 1% distortion THD and that's exactly what it's done it's cut off there beautiful so I can't damage anything I am thrilled so look guys are we're going to do one more test actually we're going to do a frequency response and we're going to do that out round about let's say we're going to do it at a one one volt RMS a frequency response right so we're going to go in from a range of 10 to 50,000 Hertz so I know it's high we need that but we're going to do that across there anyway and we should probably change that to 30 let's change that to 30 and this is just our range top to bottom on DB and we're channel one and let's run it let's run our frequency response to see what it's like at one volt across the frequency okay what we don't want to see is deviation doesn't really matter what the numbers do we just don't want to see it deviating all over the place we want to see it go into a pretty much level area and then stay in there all the way to the end and if we get that I'm going to jump around the room didn't look it is some weird noises coming out the other thing but yeah so well we stayed within a decent parameter there what we got there we got a 0.69 dbr at 27 db which is wow that's pretty much as we can run this thing actually 27 db that would equivalent to what it said it was on the game that's 18 Hertz and we started off down here at minus 0.41 but remember that we were running that at a reasonable rate and not so that's one one watt of power one volt of power all right let me just do this just one more time and this time we're gonna have a little look over here and see what happens on there look at the voltage all right that's just as it got game with the higher with the lower frequency was at the higher voltage let's just run that that's peaking remember 22 volts 22.5 volts is what I could get out of it so this is absolutely peaking at that one watt VRMS oh it wasn't happy there was it and you can actually hear that I think all right so we're not gonna push that any harder on there and the result is pretty much the same anyway we've still stayed within that area which is less than the db of change so that's all good and our spectrum analyzer if we just put that on for a level this is at just under a volt we should probably go to something that's a lot kinder to it let me just feel the temperature temperature the heat out the dummy load that's very very hot I tell you what I'm gonna call that quits for now we can look back at this and do some other in-depth stuff if I think there's anything really worth pointing out but just for now I'm gonna say let's just do a quick single shot and we get to have a little look this is a 70 dB down here and we are measuring over there minus 73.5 dB at 2 Hertz this is our fundamental that we're putting at a point to VRMS at one kilo Hertz okay well guys have you got this far I hope this has been interesting to you and this is hopefully just a start for me playing with this audio stuff and thank you very much to Jake out there and the stuff made that's a big thank you because without this software this wouldn't have been possible and yeah I will catch us in the next one bye for now


So this is where we work with the wireless charging. So you put it on, it snaps on, it opens the app all in one go. Yes. So we're here with a CNA. So hi, so who are you? My name is Park, just assisting the CNA. And who are you? My name is Dongsutang, and CEO. CEO. And so this one has a motors in the arms, and it grabs it, and it also launches the app. It's for Android or iOS? iOS and Android, or it's also OK. And it goes directly into the Google Play? And it holds it nicely. Is this for sale soon, or? Sorry? How soon is for sale? When is it ready for shipping? Actually, we are planning to find the customers, yeah. And nobody else do something like this? Actually, we are selling this product, and making this project with Dimitri Japan. How does this work? This one is also the table, so you can use for the, the function is the same as this one. You can set up the application that you want to pop up. So it just sends a trigger, open this app. It just works every time? Yeah. But it needs to be unlocked? Yeah, it needs to be unlocked. So when you put it on, it's unlocked? Yeah. And then the other one has a different app, maybe? So this one is for the commercial use, for the restaurant or cafe. Wi-Fi. I also can call it to the Wi-Fi without putting any password. So this is for the commercial use, like the cafe. The customer can place an order. Nice, you can choose the coffee? Yeah. Cool. Work with Android IOS? Yeah, both are OK. Nice. This is the web-based, so no need to install any applications. Is this the standard to be able to do this? Is it like NFC or what? Yeah, NFC. So your phone need to connect with NFC? Yes, yes. All right. So the manager can see the order list, and also they can notify the customer to pick up their food. So when they pick up, they will be on. Oh, cool. The sound of the... Already selling this? Yeah, we are making a project with the Japan. What is that? Japan. Japan, yeah. All right, in Japan, they love to have a lot of automatic ordering your food kind of things. Every restaurant needs this. Actually, in the hotel industry, yeah. What is this? This one is just a video. Video going on here. All right. And is there a price for this? This one, the retail price, $30. $30? And this is going to be? $50. $50. So you already sell many? A big company or just a small startup? Yeah, just a startup. Startup. All right. So all these things coming out, that was a loud speaker.


Okay, continuing with our example involving 3 factors in the 2 power 3 factorial design first we will calculate the error sum of squares. We have to calculate the total sum of squares first and then we subtract from the total sum of squares the sum of squares of all the effects. So the total sum of squares is 30.018 with the grand mean equals 3.5 i is equal to 1 to a the index i running from the levels of a here we are having only 2 levels and j is equal to 1 to b is representing the 2 levels for b and k equals 1 to c will represent the levels of c obviously a is equal to b is equal to c is equal to 2 and n represents the number of repeats. So that is given by the index l, l running from 1 to n. So you have y ijkl i for factor a index j for factor b index k for factor c index and n corresponding to the repeat index this is the overall mean and when we carry out these computations we get 30.018 and this represents the total sum of squares and the sum of squares due to the effects would be 29.890 from a b, a b interaction, a c interaction, b c interaction how did we calculate the effects sum of squares we have this formulae which I had shown you earlier. So using these formulae we can calculate the respect to sum of squares for the different effects and with this we get the error sum of squares which is rather small when you subtract 29.89 from 30.018 we get 0.128 and for carrying out the analysis of variance we first of all calculate the critical f value which is f.05 where 0.05 represents alpha which is the level of significance 1 numerator degree of freedom because each effect has 1 degree of freedom and 16 represents the degrees of freedom in the denominator it also represents the degrees of freedom for the error and the degrees of freedom for the error would be a b c into n-1 a b c would be 2 x 2 x 2 which is 8 n-1 would be 3-1 which is 2 so 8 x 2 would be 16 so that is why you have 16 degrees of freedom in the denominator. And hence we can divide the ANOVA table into source of variation degrees of freedom sum of squares mean square f not as before and you can see that the degrees of freedom are the same for all the effects and the interactions error I just told you as 16 degrees of freedom the sum of squares due to a would be 14.727 b is 1.233 c is 10.881 error is 0.128 and the mean square is obtained by dividing the sum of squares by the degrees of freedom. Here it is quite simple 14.727 divided by 1 is the same value 1.233 divided by 1 is 10.233 10.881 divided by 1 is 10.881 again but here it is 0.128 divided by 16 which comes to a value of 0.008 and this total is 26.969 and once you have the mean square you can divide it by the mean square error to get the f value. The f values are pretty large at 1841, 154 and 1360 the f values I repeat are obtained by dividing the mean square of that particular effect or the interaction with the mean square error. So we have the numerator degrees of freedom corresponding to degrees of freedom for the source of variation which we are considering and the denominator degrees of freedom corresponding to the degrees of freedom of for the error and similarly you can do for the binary interactions and the ternary interaction and we get these values. And interestingly the critical f value if you recollect was 4.49 these values considerably exceed this 4.49 and hence they lie in the rejection region. The critical f value was 4.49 the actual f value is much higher than the critical f value. So these lie in the rejection region and hence these indicate that a, b, c are important factors and hence the null hypothesis which says that those factors are insignificant should be rejected. Similarly you have a, b interaction to be quite significant it is much higher than 4.49. However interestingly b, c interaction is negligible because it is lying in the acceptance region the f value is lying in the acceptance region the critical value was 4.49 but this f value is only 3.34 and hence you can reject the other hypothesis null hypothesis. But for this case b, c you accept the null hypothesis that the factor b, c or the interaction between factors b and c is negligible. Now when you look at the a, b, c interaction again the f naught value is 0.12 which is considerably lower than 4.49 and hence we also accept the null hypothesis that the a, b, c interaction is not having any effect on the process. So this completes on our table but we can also find the p values, p values are very useful it tells us that by what margin the effect was considered to be significant or it was considered to be insignificant. What I really mean here is that the effect a for example or factor a for example became significant narrowly it just squeaked through or it was a very dominating factor okay. If the p value is very small then that particular factor or that particular interaction is strongly significant it is lying well in the rejection region. If the p value is let us say 10 power-3, 10 power-4 that indicates that the f statistic was lying firmly in the rejection region and you can reject the null hypothesis comfortably. But if the p value is 0.049 or 0.501 it is a marginal case it is hovering between rejection and acceptance okay. So it tells us the p values are a better indicator of how much we were able to reject the null hypothesis or accept the null hypothesis. Usually p values of 0.049 or 0.501 are unusual the p values are usually in the order of 10 power-3, 10 power-4 for cases where you reject the null hypothesis or you get rather high p values like 0.46 or 0.35 or even 0.10 when you accept the null hypothesis okay. But the p values here are very useful okay it tells us whether the effects were rejected by a comfortable margin or they narrowly got rejected. So instead of finding out f alpha effect degree of freedom and error degree of freedom you find fp effect degree of freedom and error degree of freedom okay with the associated mean square error ratio you find the probability associated with that ratio okay. So mean square effect by mean square error gives you the value and you find the probability associated with that particular value okay. So this is inverse of what we did earlier we just had the f value computed the critical value and compared the actual f value the critical value and then said whether we accepted or rejected but here we are given the f value and then you are also given the degrees of freedom you have to find the probability associated with that f value and this probability we compare it with 0.05 if it is much lower than 0.05 we reject the null hypothesis if the computed p value is greater than 0.05 we accept the null hypothesis. So when you look at these p values corresponding to abc you find these values are pretty small and so you can comfortably reject the null hypothesis saying that these effects are useless. So you have to say with lot of confidence that abc effects are significant but when you look at bc the p value is 0.086 which is greater than 0.05 and hence bc is a insignificant interaction in the process whereas abc is having a p value of 0.734 which is quite huge and hence you can reject the null hypothesis sorry you can accept the null hypothesis that abc effect is insignificant. The same conclusion we saw earlier also we did not consider bc and abc to be significant let us go back to that particular table bc 3.34 was smaller than 4.49 abc 0.12 was again smaller than 4.49 and hence we considered bc and abc to be insignificant. Significant factors will have high p values and significant factors or significant interactions will have very low p values please remember this. From the analysis it may be concluded that temperature, sturdier speed and rpm influence the extraction all the binary interactions except those between sturdier speed and particle diameter are significant a ternary interaction between temperature, sturdier speed and particle diameter is not significant. So we have the data the most important analysis after the experiments are carried out when the model has been developed is to analyze the residuals. We have seen the analysis of variance table from which we can detect the important effects. Another way to detect the important effects which I have not indicated so far is the use of the normal probability plots. The different factors and their interactions are plotted in the normal probability plot and we can find out which of the effects are significant. So in this plot we can see that abc and bc are lying close to the solid line whereas c, ac, b, ab and a are lying quite further apart. This means that the effects corresponding to factors c, ac, b, ab and a are significant whereas bc and abc are insignificant. The next important concept in design of experiments is to analyze the residuals. The residuals are defined as the difference between the actual experimental value and the prediction from the model. So you can see that eijkl this is the residual corresponding to a single point. There is a difference between two values. The first value is the response corresponding to the ith setting of factor a, jth setting of factor b and kth setting for factor c because now we are looking at 3 factors and l corresponds to one of the repeats corresponding to the setting of i, j and k and this is the predicted value. The difference between the two would be defined as the residual. Now ideally we would like the residual to be 0 but in a real world there will always be some random error component. We want to see whether the difference between the actual experimental data and the model predicted value corresponds to random behavior. The difference is the residual and we want to see how random the behavior of the residual is. Similarly we calculate the residuals for all other experimental data points. So we may say the residual as the leftover effect after all the model the main effects in the interactions have been accounted for. So the experimentalist wants to see whether the model he has developed is adequate or additional terms are necessary. So we will be looking at a lack of fit concept in regression where more would be said about the adequacy of the proposed model. So what do we do with the residuals? We want to see whether the residuals are normally distributed, whether the residuals are independent and whether the residuals are having constant variance. We know that the basic assumption in the model development, in the linear model development was the error terms where independent and identically distributed in a normal distribution with zero mean and constant variance sigma square. By looking at the residuals trend or how they pan out, we can check these assumptions whether the residuals are normally distributed, whether they have constant variance, whether they are independent from one another. So the residuals may be plotted in a normal plot that is one option. The residuals may also be plotted in the time order in the sequence in which the experiments were carried out. The residuals may be plotted against predicted values and the residuals may be plotted against each of the factor levels to see the spread of the residuals at each factor level. So if the residuals are normally distributed, they will fall on the straight line on the normal probability plot. So as far as this experiment is concerned where we considered the three factorial design given the yield of the medicinal compound, we see that the residuals are behaving reasonably well. They are falling on a straight line more or less. We do not see any gross deviation of the residual from the straight line. The next check would be to inspect the pattern formation of the residuals. The residuals are ideally speaking impartial quantities. They do not really depend upon the sequence of the experimentation. They do not depend upon the fitted value depending on the experimental setting. So when you look at the spread or pattern of the residuals, you should really see no distinct trend. If the residuals are plotted against the fitted value and they show up a definite trend then something is amiss or something is not correct. If the residuals when plotted against the fitted value show a funnel kind of pattern. So they are spread over a smaller distance at low value of the fitted value and if the residuals spread out when the fitted value increases then something is not correct. So this indicates that the residuals increase with the fitted value and the error variance is not constant. That means it is not homogeneous and residuals are more broadly scattered when the fitted value increases. For example when the instrument error is proportional to the reading measured then the residuals are going to increase with the increase in value of the fitted value. Hence the residuals may not be normally distributed and they come from skewed distributions. In such cases you may have to go in for a stabilizing transformation for the variance and you may instead of using y directly may want to consider log y or root y. So the residual analysis is subject on its own and a fascinating one at that and because of lack of time we cannot go or dwell deeper into this subject. On the other hand there are some excellent books on residual analysis. One is the book by Draper and Smith and the next one is the one by Montgomery. They have been cited in the references. So when you look at the residuals as the against the fitted value you can see that they are pretty much scattered uniformly. You do not see any kind of funneling arrangement. For example you do not get a kind of diverging residuals. The next check is to see whether the residuals are independent of one another. So the residuals are plotted against run order and we have to see whether there is a systematic trend of positive residuals and negative residuals. What I am trying to say here is whether the residuals are over a certain time period increasing continuously and over the another time period they are decreasing continuously. This also means that the randomization is not properly done and the experimental response was affected by other unaccounted factors. For example if the temperature was not controlled properly or there was a sudden shoot up in the temperature then the sequence of residuals may show a continuously increasing trend or there may be some other unaccounted factor. So over a sequence of experimental runs the residual values may be continuously declining. So this shows that the randomization was not proper and there were unaccounted factors which affected the experiment in a certain systematic manner. When you look at this particular plot you can see that you do not find any sort of residuals showing a continuously declining trend over a significantly long time interval or along the order of observations. You can see that it is declining here but then it is increasing and here also it is increasing decreasing. So you do not really get 4 or 5 consecutive positive values or 4 or 5 consecutive negative values. So you do not see a systematic trend of sequences of positive or negative values as given here. And another important thing is the errors are having constant variance sigma squared. So are the residuals showing more or less the same variance is about the mean in this case 0. The residuals are plotted for different settings of a given factor and if the spreads are not unusually different even though some inequalities may exist then we may say that the assumption of constant variance is not bad. So here we have plotted the residuals as function of the 2 different settings of the temperature in the coded format and you can see that the spread is reasonably uniform around the origin. So the variance does not really change drastically from one setting to another setting. You have another case here, well you can see that this variance is smaller than this variance but in real data we cannot exactly get a uniform spread and so on. So unless there is a gross deviation we can assume the residuals are behaving in a reasonable fashion. For example if the residuals are clumped over a very narrow region here and the residuals are occurring over a broad region here then the assumption of constant variance may not be true but here the spread is comparable I would not say exactly identical but it is comparable and hence we can live with the assumption of constant variance. And another important thing is the presence of outliers, outliers are unusual data points. They may be thought of by the experimenter as rogue data point or rebellious data point, something which is not following the general trend, this lying way out of the general trend. The reasons for outliers may be pretty simple, measurement error or a calibration error of the instrument, some careless mistake or the settings were not noted properly. Usually the experimenter gets into a routine and he does not make these kind of elementary mistakes even though they are possible and if you find any outliers it is very important that you do not brush them under the carpet but pay closer attention to the reasons for the occurrence of such outliers or outlying data. They may tell you something different, something unusual is going on which is not accounted for in your approach to the experimentation. For example if there is a condition at which the data point shows an unusually high yield obviously that is going to be profitable to the company if it is true and the experimenter may like to inspect the data point more closely. Now coming to example 3, the problem statement goes like this, greenhouse gas is removed in a packed absorber, the variables studied are gas flow rate, solvent flow rate, solvent type and packing type. The experiments are repeated and the results are shown in the table below. Looking at the background for this example we know that carbon dioxide is a greenhouse gas and its emissions are very harmful to the environment leading to global warming and so on. So one way to reduce the emissions is to absorb the carbon dioxide in using a certain solvent. The solvent may be monoethanol amine or diethanol amine and the equipment in which the gases are removed by absorbing or putting it crudely, dissolving them in the liquid solvent is called as the absorber and if you put packings in them it becomes a packed bed absorber and we want to make the process quite effective and we want to see what variables are significant. So the variables studied are the gas flow rate, the solvent flow rate, solvent type and packing type, the experiments are repeated and the results are given below. This company apparently running these tests wants to keep the details confidential so the levels are expressed in the coded format. Since there are 4 variables you have 2 power 4 that is 16 runs and if you want to do at least 2 repeats per setting that would mean 16 into 232 runs. Pilot scale studies are quite expensive and hence the management may feel those 32 runs are quite a lot in terms of investment time and power and so on. So it may even consider telling to you what would happen if you do a fraction of those runs and see what results we get and then we decide to move on from there. Factorial designs of experiments are so structured that it is indeed possible to construct a fraction design and implement it. So the main thing is to how will it go about carrying out these tests and how may the results be interpreted and reported. So a full set comprises of 16 experiments so we can start with one half fraction of the full 2 power 4 design and we have 2 half fractions. Since there are totally 16 experiments one half fraction would involve 8 runs which would correspond to a 2 power 3 design but here you may ask wait a minute you are having 4 variables you are having a 2 power 3 design by making a 2 power 3 design I will be able to handle only 3 variables what about the 4th variable it does not seem to be there. So the answer to the question is even the 2 power 3 design we are considering all the 4 variables the only thing is we are not doing the full set of possible experiments as envisaged in the 2 power 4 design we are only having a 2 power 3 design and we are investigating all the 4 variables with the truncated design. So this means that while we are gaining on the effort in experimentation we are losing out on some information whether the losing out of the information is serious or not the results will tell us. So we are losing on information when trying to save on experimental effort. So how do you go about doing the design you are having 4 factors you are considering all the 4 factors so now let us look at the highest possible interaction among the 4 factors it is very simple it is A, B, C, D. Now when you look at the design matrix of an experimental design you will find that it has a set of columns each column containing some pluses and some minuses the number of pluses would equal to the number of minuses. And so in a 2 power 4 full design you will have 8 pluses and 8 minuses in every column the column may be A, B, C, D, A, B, A, C, B, C, B, D and so on. So it will have 8 pluses and 8 minuses. So look at the column corresponding to A, B, C, D all the pluses will constitute one set all the minuses will constitute another set please consider the experimental settings corresponding to the pluses that will be the first fraction. So the table of contrast comprises of 8 pluses 1 and 8 minuses 1 we use the set of 8 pluses 1 in the A, B, C, D column to define the first fraction and the remaining set of 8 minuses 1 in the A, B, C, D column into the second fraction. So all the pluses will be corresponding to one fraction all the minuses will be corresponding to the next fraction. So we define a design generator I is equal to A, B, C, D and use this to set up the 2 fractions. Once we define a design generator I is equal to A, B, C, D and use this for setting up the 2 fractions. So you can see that the design matrix is set up you see some of them are blue and some of them are red what is the reason for this it is quite simple. We take the A, B, C, D column whatever is 1 we label it as color coded as blue and whatever is minus 1 we color coded as minus 1. The experimental data were coded into plus 1 and minus 1 for the sake of convenience now I am color coding the plus ones with blue and the minus ones with red. So we collect all the plus ones and you can see that the experimental settings are also color coded according to whether A, B, C, D is 1 or minus 1. So this one represents a case where all the factors are at their low levels A, B, C and D are their low levels A is the setting corresponding to A at a higher level, B is the setting corresponding to B at a higher level. In such cases all other factors would be at their low levels. If you have A, B then both A and B are their high levels and all other remaining factors C and D would be at their low levels. Anyway this we have already seen I am just bringing it to your attention in case you have a doubt you just go back and then see this table things will become clear A is plus 1 here and A is minus 1 but B is plus 1. So that is quite simple but we do not look at these we look at A, B, C, D and we look at the color codes and we collect all the ones together then we collect all the minus ones for the second fraction and once we do that we will get the 2 fractions and we can now look at the design generator I is equal to A, B, C, D to find the aliases. You can see that A is equal to B, C, D here B is equal to A, C, D, C is equal to ABD and D is equal to ABC. So main factors are aliased with only 3 factor interactions. So even though the A and B, C, D are coming together B, C, D interactions are usually negligible. So the full effect of A is felt in even in the partial factorial design that is because the higher order interactions like third order interactions are usually not important that is very good. But sometimes in addition to the main factors the interactions are also quite significant or quite important but unfortunately in this design the 2 factor interactions are aliased with one another as you can see here AB is aliased with CD, AC is aliased with BD and AD is aliased with BC. That means when the information is presented we are unable to uniquely determine the interaction due to AB and interaction due to CD both of them are felt together. So A, C and BD are felt together and AD and BC are also felt together but whether that causes serious issues depends upon the responses and depends upon the particular experiment we are considering and let us look at the results and see what really happened. So the first fraction you have one AD, AB, AC, AD okay you can go in the sequence AB, AC, AD, BC, BD and then CD, AB, CD so that is what you have as the first fraction and if you look at it it is one AB, AC, AD, BC, BD, CD and AB, CD so we are having the first fraction and since we are done 2 repeats you have the same settings repeated twice and you have the experimental observations presented up to 2 decimals under the column percentage extraction. So I have given you the design table once again for the purpose of calculating the effects.  However even though you are doing it under A it is also having the aliasing with the BCD so will BCD have the same entries in its column when compared to that of A the answer is yes because A is aliased with BCD you cannot really distinguish between A and BCD because their column entries are identical. So let us verify this if you look at this BCD it also has minus 1 plus AB plus AC minus BC so let me write it down so that there is no ambiguity.  So when you look at the entries for A and also for BCD they are identical and when you are calculating the effect of A you are also finding the effect of BCD in addition. So the same idea will also apply for factor B which is aliased with ACD and for factor C which is aliased with ABD and factor D which is aliased with ABC. If you want you may take any main factor and it is ternary alias and see whether the column entries are matching it is a good idea to do this to verify so that we have set up the design matrix correctly. Similarly the 2 factor interactions are also aliased with one another so you can easily find that for example AB would have the same column entries like with CD okay and BC would have the same column entries as AD. So when you are finding the effect of A even though you write the linear contrast for A as LA it is actually having both effects of A and BCD in it and you have plus 4 entries and minus 4 entries so you divided by 1 by 4 for averaging purposes. So just as I showed on the board the effect of factor BCD is identical to that of factor A. Since there are 4 positive and 4 negative entries we take the average with respect to 4 that means we divide by 4 and you are also having 2 repeats. So we effectively divide the contrast by 8, 4 for the averaging and N for the number of repeats. So using this formula we can calculate numerically the effect of A and we find that A value is 5.25 so the entries are given here. So this is minus 1 and that the value is 19.22 plus 17.52 it is minus of 1. So 1 value is 19.22 and 17.52 so minus 1 would correspond to minus of 19.22 and minus of 17.52. Similarly you can find the values corresponding to the other settings please note that we are doing repeats and so for each setting there will be 2 values or 2 responses. So effect of B is effectively given by 6.56 that includes both B as well as ACD.  And let us just check it just 1 38.28 and 39.59 AB 38.28 and 39.59. So we are on the correct track and so others can also be written down in the similar fashion and we calculate the effect of B as 52.48 divided by 8 as 6.56. Similarly you can find the other effects so take care to divide the linear contrast by 8 to account for both the averaging and also for the repeating. The next important step is to calculate the sum of squares. Once you have the contrast calculating the sum of squares is also a piece of cake and the sum of squares is given by 1 by n into 2 power k into contrast squared and let us see how to calculate the sum of squares. The effect was given by contrast by n into 2 power k minus 1. Here k is equal to 3 because we are only looking at a 2 power 3 design 1 half fraction of a 2 power 4 design. So we put k is equal to 3 so 3 minus 1 is 2 2 power 2 is 4 2 repeats 4 into 2 8 and that is what we did when we divided the effects we divided the contrast by 8. So that is an order and hence we can calculate the contrast as effect into n power 2 power k minus 1 or effect into 8 will give you the contrast. Once you get the contrast you can square the contrast and divide by a suitable number to get the sum of squares. So contrast is 8 times the effect for contrast a you first find the effect of factor a and effect of a and then multiply by 8 and then you get back 42. This 42 corresponded to the 42 you had calculated earlier when finding the effect of a. And once you have the contrast then you divided by n into 2 power k. So contrast square is 8 times the effect and that becomes 64 times the effect squared divided by n into 2 power k which is n into or 2 into 2 power 3. So you get 64 effect squared by 16 or 4 effect squared. Sum of squares of a is 1 by 16 into 42 squared n is equal to 2 into 8 is 16 contrast we just now saw for a as 42. So you get 42 squared by 16 which is 110.25. Likewise we can find the contrast for b based on the effect of b and then square the contrast divided by 16 n into 2 power k and get the value for b. Similarly you can get for c, d, a, b, a, c and a, d. So using all these information we can construct the ANOVA table. The error sum of squares for doing so first we have to calculate the total sum of squares and the total sum of squares is given by this huge relationship y ijkl-y bar triple dot. So we are finding the total sum of squares now we are subtracting each experimental data observation with the grand total not the grand total the grand average add up all the numbers in the matrix divided by the total number of entries you will get the grand average and that grand average is subtracted from each and every individual observation and that will give you the difference which is squared and when combined or added completely gives you the total sum of squares. To make sure that you have understood this correctly please do the calculation on your own and see whether you get the correct number because more than the correct calculation it is the understanding of the procedure which is more important you have to implement the procedure in the correct manner. We have found the sum of squares due to the various effects by using the contrast squared and dividing it by the n into 2 power k okay I seem to forget it all the time. So that is 16 in our case so we calculated the sum of squares for all these effects and once that is done you also have the total sum of squares with you the sum of squares from all the effects are calculated and so you can find out the error sum of squares. So total sum of squares is equal to sum of squares of effects plus error sum of squares which comes to 583.2067 equals 580.2674 which is sum of squares of all the effects plus sum of square of error the sum of squares of error comes to only 2.9397. The next step would be to construct the mean squares and find the f values and see whether the f values are lying in the rejection region or not based on the critical f value. So we will continue with this example in the next lecture. Please revise the portions discussed in this slide discussed in this lecture rather and make sure that you have understood the important concepts make sure that you have understood how to form the different fractions because the understanding here will help you when you are actually carrying out the experiments. Please pay particular attention to the calculations of the errors how to account for errors in your experimental data and analysis okay the residual analysis is also very important. We will continue with this example in the coming lecture. Thank you.