# Statistics Lecture 7.4: Confidence Interval for the Sample Mean, Population Std Dev — Unknown

okay well today we're going to talk about how to estimate the population mean when we do not know Sigma which is our population standard deviation this is actually a more realistic case I told you that the last couple of minutes of class last time I said that really are we gonna know the population standard deviation most of the time not in practical experience you're gonna you might approximate it every once in a while but you're really not going to know it because you have to know this to even find that you got to know that so maybe based on past experience you do know what this is but it's it's rare that's not gonna happen all the time so perhaps a more real-life scenario is how will the world that we're going to estimate the population mean when we know absolutely nothing about it and that's what the sections about so this is perhaps a little bit more realistic one thing the problem is the one thing that we we have to struggle on a little bit is if you don't know Sigma you can't use AC school the requirements they're not met so if you go back and you check those or prime mrs. from the last section it said random sample great and the next one was you know Sigma do we know Sigma no so we can't use e school we can't use AC critical value that's a problem for us instead what we're going to use we use a T school yeah what what was that again a t-score you've never heard of a tea school before instead we use a t-score long time ago there was this guy he actually worked through brewing factory and he didn't know do you want to do these experiments on beer to find out different characteristics about them but firstly he didn't have large sample sizes didn't have more than 30 different vats to sample from but he knew that the the distribution would be normal so he saw this stuff we'll talk about in a minute but finally he didn't know the population standard deviation because of course you can't have the population of all all beer all the time so he had to come up with something new so this guy he wrote into a journal under the the name of student team student tea he won't he use a pseudonym because you want himself associated with this to do the beer crossover because it was using their their information so that's where we get the T distribution from so the student tea this one guy wrote into a journal someday there's a long time ago and that's what this is all coming from are you with me someday did the work and actually calculated a new table for us here's what we need for a t-score first thing just like always we absolutely have to have a random sample number two besides the fact that we need a random sample we've got to know that our sample is from a normal distribution from populations normativity or and has to be greater than 30 is that sound familiar to you that's the same thing that we've just talked about so and is greater than 30 for sure or it ends not greater than 30 you absolutely must have the sample coming from a population that's normally distributed so or the sample is from a normally distributed population so that's got to be somewhere either your sample size is more than 30 or the population on the industry in the past we used a z-score to represent this we had what's x-bar stand for everybody should know that what's X bar same company great we would do sample mean minus population mean all over Sigma over the square root of n does that look familiar to you I hope that was for those groups right for the average of us of a sample that was it the problem is we can't use a z-score in these contexts because of one thing we don't have that we don't have that piece of information so instead we're now going to translate that to a t-score I think they use a lowercase letter T but I can't write lowercase letters so I use a capital T so anyway it's a t-score regardless it's letter to letter T well we're still gonna be able to find a sample mean that's fine and that even that's okay that's our sample mean we're comparing that to something but we're actually gonna be estimating this but here right here we can't use the population standard edition cuz we don't Miller and we're not going to assume it so what we're gonna do is we're gonna need an S over the square root of n what's S stand for the sample sample what what was this what's that same thing what is it sure this one's from a distance from a family that's from a long time ago we able to use that later that's not a long time but that S stands for a sample standard deviation so are the formulas pretty much identical yeah only this is from a sample instead of the population that's really the only difference of a z-score and a t-score into finding the actual value here now the group of values of course are going to be very similar so critical values instead of Z we're gonna have T's but it's still going to be that alpha remember that alpha what's alpha for a 90% confidence level how much is alpha for a 90 how much is alpha you look it up you should no no nothing not the critical value I'm saying alpha 1 1 sure if your confidence is 90% you're out this point 1 if your confidence is 95% once your alpha point zero 5 point 5 no no that's good you were saying how about your alpha for 0.99 which alpha was your 1 sure those things have to add up to 1 right they're complementary so we split that because we get those two tails for our confidence intervals that's the same stuff that we've been talking so critical values given by T's and set of Z that's basically all I'm saying there there's one more definition I need to give you it's a definition called degrees of freedom here's how you find degrees of freedom it means the degrees of choice or org or choice in your sample here is degrees of freedom we're gonna signify with a D April 3 V 8 it is an n stand for again sure sample size minus 1 very easy formula use this one really ever get n minus 1 sample size minus 1 so basically just your sample size minus 1 so let's have a sample of 90 people degrees of freedom would be 89 okay we just subtract 1 it is not a whole thing it's not a good question sample size of 50 would give you degrees of freedom of everybody excellent awesome well I'll give it now yeah you subtract one from the number and that gives you your theories are free of why why is it working you get a simple example if I am picking let's imagine this alright what we're talking about averages here you're with me we're talking about averages that's sample sample averages seven means true or false if I take ten numbers I can make them have an average of 100 true can you pick ten numbers to have an average of 100 sir 10 10 10 10 10 10 10 I'm sorry that's ten numbers that have an average of 100 right you can always do that you can pick five 100 ones and 599's that would have an average of one right those are very simple examples but here's the idea if I give you nine numbers so set in stone I give you nine numbers set in stone and I asked you for the last one and I say the average of these things you don't write this down just just watch okay if I say you're putting ten numbers on that have to have an average of 100 I say these first night I don't care what they are do you believe me that it does not matter what these 9 numbers are that last number has to be one particular number to give you an average of 100 does that make sense so for instance I'm gonna pick a random number shoot negative 3 5 13 negative 110 4378 21 negative 304 and positive 2 that's one two three four five six is there a way to make these average out to 100 absolutely but would that never be set in stone yes absolutely so these numbers whatever I give you I have nine choices for picking out whatever ten numbers are gonna average up to 100 have nine choices I can make them whatever I want for the last one will be set in stone the lasso can be nothing else besides one number if I change any of these ones that one would have to change this one is dependent on the other nine choices so if you choose nine things the last one seven stone that means your degrees of choice you have nine choices but the last one will be guaranteed to be a certain value does that make sense to you so that's where we give this degrees of freedom from it says yeah okay you've got almost the whole sample size with the choice but if you're trying to get a certain average that last one is going to be set in stone it's not going to move anywhere now people understood the idea it's a kind of a weird idea right kind of a weird idea but that's where we're getting that from okay you ready ready for some some real stuff how to learn how to use that t-values good thing in here today right so this is new this is new stuff finally let me give you a quick example this is going to illustrate how to find a T critical value let's say you have a sample of 23 from a normally distributed population I lays that statement meaningful to you if I didn't have this statement wouldn't be able to do anything why not so this 23 that's our that's our end this means that I can use this stuff departments or meta now because I have it from a built in student population at some point what I want us to do is find the critical value that's t el tovar to for a confidence level 90% find a T critical value for compliment with 95% okay let's do this thing firstly can't you tell me what my it is how much is my end okay how much is my alpha can you tell me what my output is please how you finding point zero five sure great because I was thinking two things are complementary they have to add up to one this is point zero PI that's great you know what be on the alphas pointer but okay can you tell me how much is my degrees of freedom what's my degrees of freedom how you get 22 yeah you just subtract one from that so degrees of freedom is the sample size minus one or 22 now we're gonna find T up over – that's our critical value and I say critical value that's that's what that means we're gonna go ahead and we're gonna find that I need you to take out your tables you know that pull out sheet that you'll have looks like this or it's in the back your book it's Table eight three so find your eight three for me you see that we hope whoa there's burner redness and my redness smells crazy retinas are your eyeballs right anatomy okay so we have this t-distribution oh it is a lowercase letter T see I told you I just can't write lowercase anyway so we had a t-distribution it says critical t-value so that's exactly what we're looking for let's look at this charges huh look at it for a second what's on the left-hand side hey do you not find that now so you're not looking up the sample size you're actually looking up a degrees of freedom t distributions are often very often used for small samples I'll talk about why in just a bit okay but most of the time your samples are under 30 you just have to know they're coming from an old normally distributed population are you with you on that so most of the time they're under 30 no problem now if you look up at the top it says something weird says area in one tail areas in two tails I need you to understand that on our graph we have a normally distributed curve right we have this bell-shaped curve and we have how many tails 2 tails if we're dealing with a 95 as you write this down before reasons for do with a 95% confidence level the alpha is 5 percent right point 0 5 that's the area that's combined in both tails you within this if the area in both tails is point zero 5 how much is in one tail quite zero to five you figured out how we get point zero to five look what this that's if you're talking about the area of two tails these things mean the same thing it's just to buy it before you look at that area in two tails our tails combined is point zero five you with me or if you want to think about as one tail how much area is in one tail point zero two five these calls mean the same thing Ariane's one tail will be point zero two five or the area in both tails combined would be point zero five either way we're in this calm Ridge and if you understand that good okay so for our confidence intervals you can think about alpha as being in both tails so this right here is your alpha this is your alpha and this is your alpha over two on your table if you want you go ahead and write alpha alpha over two go ahead and write that right there if you'd like to that's for your main right confidence intervals above that little column so right here I'd be putting finer paper I'd be putting a confidence interval to see nine right here and put alpha over two no I had a package of story here put up up the cuff over to here put CI here that's for confidence intervals okay yo with me votes so let's go ahead let's do our T critical value can y'all tell me what was our degrees of freedom that we should be looking at here 22 okay so we're gonna go down to 22 that's right here we're gonna go over to our either our alpha which is this column or alpha over two which is this call me it's say that we're going to go all the way down till they meet up 22 that should be two point zero seven forged y'all find two point zero seven four you should have your table as well be doing the same exact thing two point zero seven four and you with me now stop think back to z-scores they came to Z scores if I said ignore this table don't look at this table if I said to you hey find me the critical value for a 95% confidence level you would be telling me 1.96 you'd be telling me 1.96 you're with me for a 95% comfortable what's a t-distribution do it's a bigger smaller bigger that means for the same level of confidence you're gonna have a wider spread which means that your t-distribution are as accurate why not because you do not know the population standard deviation does it make sense you're estimating with a simple that's why that's why it's gonna be a little bit wider now I need you to look what happens okay check this out look what happens for a very very small sample this would be a sample of two you're not gonna have a sample of one okay that would be irrelevant you'd have all information right there but if you have a sample of two look at that that's your critical value that's way bigger than 1.96 right that's gonna be a huge huge spread my final screen though a wide that spreads gonna be that's crazy but look what happens as soon as you start getting bigger bigger samples what's happening to this critical value okay now I have a critical thinking type question for you or Dingman critical value so critical thinking that's right into this as my sample size goes up what do you think this number is going to approach what do you think no I'm not zero zero would mean you'd have no spread whatsoever not to let me ask you another question what's the critical value for a z-score when you know the population standard deviation for a 95% comfort zone 1.96 it's for a z-score when you know the population center view you would be now if you take samples big enough it's gonna be so large that that sample standard deviations can be pretty close to the population standard deviation right pretty darn close as you go higher and higher in sample size look what happens look at the very bottom of your table what's it safe for Oh look at this we're still in this column right this isn't that this was a 95% comes over you with me on that this column right here so we go to to to over the ones so obviously it doesn't max out a 2 right that's just for a sample size of 61 we come down here oh my what's it getting close to why to say enlarge that means as you're getting really big big samples why does that happen because as you take large samples the sample standard deviation is really closely approximating the population standard deviation therefore 80 score will automatically become a z-score value wise as we get big enough does that make sense be sure you followed that so our T scores the same as the Z scores no clearly not they change for every sample that you take that's what's weird about it right you take a sample of 35 it's different than 36 it's different than 42 it's different than 25 that didn't happen with z-scores so for every different size of sample you're gonna have a different critical value are you gonna really need to know how to read this chart yeah yeah absolutely you're going to need to know how to do that because for every different sound that you've got a different value but as you get big enough as your sample approaches well it says above 2,000 but as it's approaching infinity really is when we have our scoring exactly matching our t-score that's how this thing works for big enough it works just fine look at this one how about 4 mm familiar that's 4 90 that's our 90 column that one familiar it's pretty close to point 5 7 this is 7 6 but we should have two point five seven five person you scored rights okay would you please raise your hand if you understand how to find 18 T critical value good that's the only new thing I got teaching you know why because after we do a T critical value everything else is almost exactly the same as what we did this was two point zero seven four if I remember right but after you find that after you do your T critical value it's the hardest part you're going to do the same exact thing with your II the same exact thing with your confidence intervals they'll be exact like your homework that you just turned in was it two point zero seven four yeah okay so we'll talk about our e that's our margin of error the EES gonna look awfully similar to the e for our last section that we were estimating the population mean knowing the standard deviation for population we're still going to have a critical value we're still going to have a standard deviation and we're still going to have a sample size but the only thing that changed that doesn't change we use the letter it is the sample size and as n no matter what you're talking about so when we're doing this sure we're gonna have a critical value here and we're gonna be multiplying by something over the square root of n hey tell me thinking back to the last section what normally would go here for a good and what symbol that we use to represent that for a population it was Sigma are you gonna know Sigma in this section no so we have to use a different standard deviation the whole reason why we even need a t-score because we're using not the population turvy ation but the and what lever represents that all is great so the only difference here is that we're using s now the critical value can I use a T critical value or a Z critical value which one t why and over to Sigma that's why do we have that's there right if we knew Sigma we never see the Z and Sigma go together T and s go together that's not my team does the e book familiar to you it's just a number of critical value that's a standard deviation over the square root of our sample size and that's all it is and if that's the case then our confidence intervals are exactly the same as last time we're just going to take our X bar minus E and X bar plus me and surround our population mean with that because E is still the maximum difference between our point estimate number our point estimate and our population parameter it's still the maximum difference so if I subtract it from the point estimate and add it to the point estimate it's giving me that range of numbers to which I'm a certain level of confidence that it's gonna the actual population parameter is going to fall in that range if said they're like 50 times now but that's always the same interpretation do okay so far now I normally list out the steps but honestly the steps are exactly the same as the last one number one you're gonna check to see if your requirements are met same as last section you check that right for this you community to end well I've obviously random samples so I understand well the N has to be bigger than 30 or if it's not has to come from a normally distributed population second thing or a third thing I guess if you consider that random sample is system one you have to not know Sigma because if you knew Sigma there's a better way to do it there's a more accurate way to do it you would get that you know Sigma you're using z-score because why hey it's more accurate you're gonna get a better range of numbers we just found out that T's are wider than Z's what would you rather have a narrow range or wide range narrower you wanna be more accurate so if you've had Sigma why are you so T you're not gonna do that if you don't know Sigma you're left to only having it this is your option that's it just T so we're gonna check those requirements Sigma is not going to be knowing the next thing you do is find your teeth you're the reason freedom because you're going to use that to find your T critical value so one requirements to you're going to look up for your degrees of freedom three that's going to let you find your T and once you find your T you're going free you got me you got your you're comfortable what you guys like to do an example you don't look so excited about the example oh you're hurting my heart this should be fun we're almost done with our class we should be excited about every last example there's not many left [Applause] we're going to construct a 95% confidence interval for the age for the average age of people denied a promotion you see these people they were there in this business and this company would would always promote young people but not old people because they figured all they're gonna go away anyway so they're just promoting young people and so these these people who file a lawsuit and said hey you're being ageist here you know we have just some amount of qualifications why you're not nobody else so this is this is where the comes from so we're gonna construct a 95% common set interval to find out the average age of people denied a promotion from this this company okay and here was the information they got in a random sample of 23 people the average age was 47.0 with a standard deviation of 7.2 instructor 95% comes in with average age of people command motion in the sample 23 people average age was 47 with the sand deviation of 7.2 assume this sample comes from population is normally distributed let's go ahead let's see what we can do what we can use personally can you tell me what my different color here what my is 23 23 yes 23 absolutely from my end can you tell me my degrees of freedom member n does not equal degrees of freedom so and it's different than degrees of freedom it's is actually different by exactly 1 all the time what is my degrees of freedom n 22 ok now can you tell me do I have an X bar do I have an X bar or do I have a mu where is it coming from is this average age from a sample or the population am I going to tell you the population for a senator for a common suitable no that's what we're trying to estimate I mean why would I tell you hey the mean age is 47 for the population you would even have to do this right now I'm gonna do that let me waste of time so the sample mean is 47 here's the big one okay this is going to tell you what to do I need to be ultra clear on this you're gonna look right here because it's always going to give you a standard deviation always let you do the reports remember we're in the second half of this first half as proportions you only do one thing what he use for proportions Z or T Z no matter what that's proportions you just use e over here it means we have two scenarios we have Z's and we have T's it all is based around this statement where everything else can be wording these at the same but this statement right here is going to tell you whether using a Z or whether using the T that's it so in this statement is this a population standard deviation or a sample standard gauge in other words is this a sigma or is this a nest and that's what you got to know if it's a Sigma you're gonna be using which one is zero T Z if it's an S if it's a sample segregation you're gonna be using a that's it if you can manage that these comfortables are pretty easy right where the math is not hard on them if you can manage that then you can manage doing the test just fine because you're going to get a whole lot of confidence intervals on your test at least for at least for common signals so if you can tell the difference between Sigma and s population deviation and sample by reading the problem you'll be just fine if you can't you're gonna mess up you're gonna do a Z when you should have a T or a T when you should have a Z nians get crazy on them right so let's read real careful it says in ace a random Animus vendor sample for 23 people that average age was 47 that's where we get our X bar with a with a what's the width a mean it says in relation to that sample that I'm taking right there in relation to the sample it was 7.2 is that a sigma or s definitely nest a Sigma would have said this it would have said assume the population standard deviation is used here the differences there it'll say population standard it will specify it for you it's not going to leave you in the dark okay if it's population starvation it will tell you that if you don't have that it's sample evolved okay that's that's the whole idea here so here we have not a Sigma Sigma you don't know no signal there is no population aggregation even said anywhere in this problem we have and that's that's a 7.2 by the way can we even use a t-score are the requirements met we should be checking that firstly do I have a random sample boom got it random sample the second thing was is my sample size greater than 30 Oh No so can I use it what comes from a population so there's treatment that's great that's exactly what need to see lastly I would check to see if my Sigma is known or unknown if my Sigma is known I would be using AZ distribution for a critical value is my Sigma note no my X's no my sample segregation that means I'm using routine so we're going to go ahead and continue this thing we've already found the degrees of freedom we've already done this nice stuff we're gonna go ahead and continue to find our alpha and our T critical value what is our what's our output here folks are out though cool we now should have enough between these two things to find our critical value we've already done this on the board but take some time go through it again look at your table right now look up your degrees of freedom which should be 22 look over to the appropriate column in this case you have the point zero five for your outlets why had you write that on a table right so you just follow that what is your T alpha over two things yeah we just did that example in cooking class when we give the same information right here after you find that mat you're set I mean you've got the T critical value you've got your s you've got your end now it's time to find your eat so I'd like you to do that now Oh one thing one thing please watch this very carefully you'll be closed if you mess this up because it won't be exact I'll be looking for it when you're doing the end should I use 22 or 23 3 and it's 23 degrees of freedom is only 2 you use the degrees of freedom to find this one they go back to your hand did you find it did you find it how much did you get 3.1 141 let's get three point one one four yes is it okay that that's greater than one we're gonna do with horses anymore so there's no matter what say it again let's just stay three point one one okay that's okay because look at our mean is intense Liz leaving we can actually do intense that you want to party that's fine because we're not doing fortunes we don't need to go to like the third or fourth decimal place that would be important because if I what percentage is there here we're talking about years three point one years is gonna be just fine enough for us we don't need three point one one four years that we'd be like almost an hour no care we want three point one years give it to me say mister averages given you follow okay cool what do I do with that II put now intervals make up my interval go ahead and do that now okay so we should be taking x-bar minus e wrapping that thing around mu that's our population mean that we're trying to estimate here x-bar plus e so we have our exponent should be given to you already that's gonna be explicitly stated somewhere you're not gonna have to find it we could have forty seven point zero minus three point one that's me less than mu which is less than forty seven point zero plus three point one so this is getting this arranged in years right here you should get what what is that forty three point nine and fifty point one now comes for the interpretation hopefully you've had this thing interpreted in your mind that I use the same interpretation now for at least three weeks right so interpretation right here is I don't know exactly what the population mean of ages of people being denied promotion is I don't know what the population is for this company because my samples only twenty three right I don't know the population mean is but I am 95% sure that it's gonna fall between these ages so here's what this means for this company you're 95% sure that the people who are being denied a promotion are between 43 and 50 44 44 and 50 right that's is that older than normal I don't know that's that's for someone else to decide but you can provide them that information okay so it says is this significantly different than like people being denied a promotion at 30 yeah absolutely most of these people 95 percent blur mean you 95 percent sure that most of these people are going to have an average age of summer between 44 and 50 that's that's older than if that's older than all the other people get promotions and now they probably in big trouble right there but at least now you can find the interpretation and give information someone else to make those decisions so again that the interpretation is I don't know exactly what the population averages of people being denied emotion but I'm 95% sure it's gonna fall within this room that's the idea how many people understood this example very very similar to before right only difference is that T that T's crucial though you've got to know when to use it if you have the Sigma perfect you got Z if you don't have the Sigma you get s you got T computations exact the same finding the numbers are a little bit different notice that if you use them if you did use a z-score you're gonna be pretty close right you're gonna be pretty close to this but it's gonna be off right here by just a little bit just a little bit and just a little bit that's not that's not good enough for us we need to be pretty pretty precise on this stuff if you used a Z will be 1.96 that's not all that different that's only different in the timeflow about ten to eleven hundred difference but it will make difference okay last thing we can do here before we do one final example just like before when we had a confidence interval you should be able to break that up and find the the x-bar like we found the P hat to find the e like we found the e last time and really it was all just about whether you can take to find X bar the two bounds add them together and divide them by two basically averaging them you remember doing that with a B now do you hopefully you do to find the the X bar X bar is right in between these two numbers so if I add them up and divide by two by average then it's gonna give me X bar so you take the upper to lower divided by two to find e you take the upper I'm sorry plus alone hi Sarah you upper plus lower I think I said plus I broke minus a minus the lower I divided by two so in our case here we take twenty seven point two one eight plus twenty four point zero six five and divided by two that's going to give me X bar whatever that is to find the e the maximum difference between those that margin of error you take the twenty seven point two one eight you subtract the twenty four point zero six five and you divide by two and that's gonna give your deep it's that clear enough for you you go ahead and do do math on your own that simple mathematics but that's about that all we can do we'll do one more example on this next time so we're doing one last example about confidence intervals this is this is it folks after this we getting the hypothesis testing and we are rolling good stuff after this so in a random sample of babies born to cocaine using mothers the average weight was found to be 2700 grams with a standard deviation of 645 grams construct a 99% comes to them for the population average birth weight the first thing we got to do is determine what section of confidence intervals were in are we dealing with proportions here is there anything to do with success versus failure anything like that no we're not calling you know cooking baby success it's a horrible world thing so that's that's not good so we don't have successes or failures we don't have any sort of proportion we have averages going on you with me so what my averages so instead of dealing portions we're over here in this section averages and in averages we have one of two categories we have we know the population segregation or we don't know the population standard deviation if we read through this let's try to identify some things here firstly the most important thing is do you have well do you have a standard deviation to say that yes you're going to have a standard deviation in these problems it's just you need to determine whether it's the population or the sample if it's the population standard deviation it will say specifically the population standard deviation is assumed to be or assume the population standard asian somewhere it's going to say population standard deviation yeah does that say that you're so do we know the population standard deviation definitely not definitely not what is the standard deviation then so is that Sigma or s that is s exactly right so we know a few things right now firstly do you know let's see you know what I didn't tell you what the sample size was sample of let's say a hundred 1990 babies babies do you know n what is n stand for sample size great so n is 190 do you know the Mew do you know the mute light on the way don't we know the mute hey if you do them you you wouldn't even be doing this problem but because we know the Mew right now what you're trying to do is estimate the Mew are you clear on that why would you be estimated of you if you knew them you that'd just be silly that let be done we won't do that so of course you don't know the view that's what we're trying to estimate so we we don't know that what do we know good how much is next bar and of course that came from the sample then average was 2,700 okay and you said we knew Sigma or we knew FS which one depends okay sure the 645 that is our RS now is the time when you determine whether you're going to be using a z-score or a t-score and it's all based right here it's not may some of this it's not based on this it's based on what that letter is if that's a Sigma what do you use if that was a Sigma the candidate what you use a Z is that signal and that's what do you use now T score that's the determination that I could walk you through that last part of class right as you follow that chart down so this is this is s that means we're gonna be using a a T critical value but there's one more thing that we need that we don't need with a Z that we do need with a T what is it so that's right hundred here the recent freedom what is it okay that's our first step identify all these letters I think we have all the list so the next thing we got to do find your critical value so this was like step one check to make sure your requirements are met step two is let's see if we can find our T in this case critical value by the way what is your zero even finish are you good today no top of the on top of the game and point zero one is your alpha that's the complement of your confidence level so we have point zero one that's stemming from a 99% confidence and will be with me on that it's very good now take out your tables look at this together perfect now you look over that table that these numbers are kind of kind of small I'll zoom in in just a moment your degrees of freedom was listed on the left you should use this column a whole lot less homework right that you just used by the way did they throw in any z-scores or were they all T scores just two for you to determine between right okay yeah but then all the problems should have had to do with TSP if you use z-score on your homework company you better take that homework back today and change that so over here on the left-hand side we start seeing these these degrees of freedom but it's but down here past 40 notice past 40 no it goes up by five and then by ten and then ultimately by a hundred you see that that's because as you're getting closer and closer to these large sample sizes your T scores really starting to approximate your z-score right and and so it's getting really really close and so they can afford to make these jumps I mean look at that between 90 and 100 is only occupied for thousands that's not much between a thousand and two thousand is only off by mm that's not much at all okay that's a little a little bit so that the difference between sample sizes as you get really large doesn't make that much of a difference so whether we have a degrees of freedom of 189 or two what goes our degrees for you well it's been 189 if we had two degrees of freedom of 189 verses of your ease and freedom of 200 it's really gonna make that much of a difference not not really I mean those things are very close sample size wise so when you have a degrees of freedom that's not listed specifically on your chart small sample size yes it matters a whole bunch one being off by one when you lay up there it makes really different T values T critical values but down here it doesn't so when you're dealing with like 189 what's 189 closer to 100 200 take the 200 okay take the 200 so by the way looking back to the top can you read which column were in are we in this column no that would be an 80% confidence interval that's area 2 tails this one to be a 90% yes this would be a 95% true this would be a 90 98 percent that's a 90% confidence interval right there this is your 99 why do you say that there's there's your alpha right there did I put up to the side that this was alpha and this is alpha over 2 for you so if this is our alpha we have point zero one that's why I had you around the board you're gonna go all the way down do this on your table look up you're out alpha and correspondence with the appropriate degrees of freedom which are ones closest to and then give me my T critical value do that on your own don't sell out everyone do it and what is it hello sorry now just to refresh your memory if we deal with a 99% confidence interval with a z-score this is two point five seven five that number should be like in your head right so this is gonna be a little bit wider than a 99% z-score that would be even we had known the population standard deviation so here it's a little bit wider because we don't know that we don't know the spread of the population what now larger they're great so fight me or eat that's that one really from here on out it's just putting in numbers honestly the key part of doing this is knowing whether you use tears that's it have you found your eat have you found your e6 45 divided by the square root of 190 then you multiply that by two point six zero one in this particular case and you get how much now that seems kind of large but when you think about it that I mean the average weight was twenty-seven hundred grams right I said that's large with the sanitization 645 so your margin of error when you do it means can be almost anything there just depends on what your your context is what do you do now do you leave it oh yeah we could find the interval so the way we find our interval is we take the point estimate remember where the point estimate is comes from a sample minus e which is the maximum difference between your point estimate and your population parameter so that gives you a range that gives you the furthest bounds that you could be away from your population parameter which is in this case well it goes in the middle of this yeah why not P we're not deal with forces anymore we're deal with averages and the interpretation once you find this what you should be working on right now go ahead and do that you know the interpretation is you don't know exactly what the population mean for cooking babies is you know you don't know but you're 99% sure it's gonna fall within these two intervals these two to this range why is that important well if you're producing something that's that's geared towards helping these these children these these unfortunate children well then you're going to want to make sure that it holds up to most of the kids right but for most of the kids you want to make some that only helps 20% of these cocaine babies maybe it's something that weighs them accurately or something I don't know some sort of drug that is is based on their weight you want to produce that drug that says okay give this dude those to them right when they're born it'll help out a lot it'll help 99% of these kids okay the one person we need to find something else for them but 99% were pretty worse we're 98% sure it's gonna help these babies from a baby does that make sense to you okay so we got our got our mean which is our and we'll subtract the 121 0.71 and we'll add it what are you get on the left interval let sorry left boundary like this I mean like this okay I think I can so given this information you're not positive you're not positive that the actual population mean is going to fall within this range but you're 99% sure it is that's enough to make a good decision right that's enough to say I'm going to produce this one item that I'm 99% sure it's going to work for these babies because you're pretty sure the averages is long yeah you can really work we have too many decimal places here you could easily do a 0.3 and 2.7 you don't want to get too much offer from that so if you're using the rounding rule then yes that's perfectly appropriate last step it really wouldn't matter since you're adding it and subtracting it that doesn't make that much of difference if your one decimal place pass with your mean what your information is giving you then you'll always be ok the problem is is that if this was like 2,700 points something something and you've rounded to the tenth here that would be a problem how do you understand our example here now this ends the part that's gonna be on your test on Monday right now we're gonna get into the last chapter that we really look at in depth we might go back and do a couple sections if you notice we've skipped a couple of them we might go look back and do that last week of school for right now we're gonna talk about chapter beta but is what you try to get to in statistics chapter 8 is it it is called hypothesis testing it's what we worked all this semester to get to it's the exciting part it's the part that you're actually going to see oh this is actually useful are you starting to see more and more use in this stuff as we go through a semester and again you're like why would I want to put data in a table I don't care Carol is freaking but now we're actually able to make decisions and determine whether something is a good choice or a bad choice based on actual factual data that's important

is there a way to do t score on your calculator like you did z score on your calculator

Thanks professor Leonard, you just saved my one year in university!!!

Hi, Professor. I'm a dropout from bsc computer science who now learns statistics from your playlist. I love learning mathematics, and now at the near of completing the playlist. I'm not sure what to do next ?

Whether to go deep in statistics [ If yes, is there any way to pursue statistics in deep by remote ]

Or,

learn other branches of mathematics like calculus, linear algebra as such ? [ As i have noticed, there is no courses in youtube which offer deep math ]

I thought you can help me with this…

Thank you SO much for your playlists by the way. Your teaching style was AWESOME! 🙂

Sir please complete chapter 9 to 12. Thanks

dunno if this is a stupid question but when do you always use 2-tailed vs 1-tailed distributions. I did not see an example using just one tailed yet

Professor Leonard I must send my gratitude and thanks to you for putting these videos online. I am a 42 yr old returning student and these videos are helping me greatly. The funny thing is I really enjoy math but I am really not very good at it. Math does get confusing and your videos have explained the things I am having issues with. Thank you again Sir. Wish I would have had a professor like you when I originally was going to school.

As always, thank you Professor Leonard!!!

You are so good.

The explanation on D.F is golden.

You have been amazing HELP! Thank you. I have literally rewritten all my notes! Thank u

Prof Leonard, you are the best. I struggled to understanding statistics for long, but with your videos all is understandable now. I thank you for continuing to help the mankind all over the world.

how to calculate t-score from calculator

13:45 why do we get alpha = 0.05 for a confidence level of

95%?was distracted how hot you are. haha

One thing I don't understand is how can you compute T or Z in those formulas at 6:00 when you don't have mu (the population mean)? There are two unknowns in the equation. You don't have sigma or mu. I understand that you use S (sample standard deviation) to represent sigma, but then you also have to use the sample mean to represent the population mean, in which case shouldn't the formula be x – x bar on the top?

The Z-score and T-score formulas he writes on the board are never used to solve any of the problems in this section. Not since the material on the Normal Distribution was introduced did he use the Z-score formula to convert between non-standard units and standard units. My question is: why do we need those two formulas?

The formula used is just sigma (or S if you don't have it) divided by the square root of n (sample size) multiplied by the desired Z or T value from the table.

It would also be nice to know how the values in the table are calculated. Thanks. 🙂

Than you!! I passed my statistic class with an A thank to you.

Suppose that a lottery pays out $1 million with probability one millionth, and pays back $.10 half of the time. If you sample 40 tickets to find a confidence interval for the value of a ticket, then your method for computing a confidence interval will fail to produce a confidence interval containing the true mean value almost all if not all of the time. Did you leave out an assumption?

for T Test what should be sample Size? u said n>30 where as in example u used n =23

for t test n should be less than 30 but here on board it is mentiones as greater than 30

who is watching in 2k17….. from india

Your a boss. would've been staring at my textbook for four hours without this. THANKS

u ARE VERY MUCH SKILLED AND ACCURATE IN WHAT U TELL pROFFESOR lEONARD

so grateful for these videos! <3 REALLY helpful for my university stats class!

– Ally (ON, Canada)

what a great teacher

this is still considered 7.3