Practical Session: Case Studies and Probabilities

okay welcome to today&#39;s practical session everyone um so today we are going to focus on uh practical three we are going to look at case studies one two and three and then next week we are going to look at binomial distribution so we haven&#39;t done binomial distribution in class yet um we will get to that next week and we will integrate next week&#39;s um practical sessions with the lectures to get you guys um well acquainted with what&#39;s what&#39;s going on sorry not next week the week after um so the week after the the taste we will work on the binomial day so um i always forget there&#39;s a taste week in between that we&#39;re not doing a prac in that week so the rest of this week we are going to do q a sessions and you will notice here on my slide i did not say q and a specifically for prac three obviously it is for break three but if you have any remaining questions on prac one and two you are also welcome to come to the q and a sessions and we can refresh on things like that so we had some questions before we started about the group mean and the group standard deviation we can cover those in q and a sessions this week as well for the practicals um so let&#39;s jump straight in and i&#39;m going to start sharing now two practical three so we only have six practicals in this course so this one marks the halfway mark and hopefully you guys are getting the hang of it it looked quite positive in the chat earlier uh so let&#39;s hope that it stays like that um so let&#39;s look at what is in brack three so i&#39;m just going to go and give you guys a quick overview again as always read this there&#39;s some extra information added into this bit here you&#39;ll notice most of it is the same with saying that it&#39;s sample data and all of that but i&#39;ve given you a note here saying that for two of the case studies we are actually going to prove that this is population data and the giveaway there is the word census remembrances deals with the entire population now objectives you can go read through that you can go see all of the things that you need to see here you can see here that the submission link will be available midnight um after your your submission this friday um why did i make it only then oh so you don&#39;t have a submission this friday i don&#39;t know why i said that sorry but i&#39;m i&#39;m going completely crazy clearly i need a holiday as well so submission link is going to be available on saturday morning i could maybe even make it a little bit earlier on friday if there&#39;s a need for that it will be ready for that so you guys can just let us know if you prefer to be available a little bit earlier i&#39;m happy to do that um three attempts allowed only the best one will count and i&#39;ve given you guys 50 minutes for this one so it&#39;s a little shorter than the others but it&#39;s also com using some of the skills you&#39;ve gained previously so that should um help you as well now the functions that you may need to use are given here this binom dist will only be used in case study four so we are only going to cover that in the practitioners after the semester test it&#39;s therefore not included in the scope of your semester test but any of the other things we can include in the scope of your semester test but all you know we&#39;ve used sum product you&#39;re going to use it again count we&#39;ve used you could even use the count f function for some questions if you wanted to and then the square root function is important as well now just to give you an overview of what we&#39;re going to try and achieve with this practical probabilities using cross tabulations you&#39;ll see this follows on from practical ones case study so it&#39;s important that you&#39;ve mastered practical one so that you can do this question we&#39;re going to do this one second we&#39;re actually going to start today&#39;s session with empirical probability distributions which we haven&#39;t covered in detail in class yet so you guys will um still see this kind of thing in class um but don&#39;t don&#39;t stress about it it sounds much worse than it is like i said before if you see the word empirical just say change it to observed so it&#39;s an observed probability distribution and you&#39;ve actually done the basics of this um in class already and we&#39;ll get back to that expected value and variance you haven&#39;t done in class yet but we&#39;ll get to that um and i&#39;ll this will give you guys a head start on what&#39;s coming in the lectures and then counting techniques we&#39;ll do today as well and then the binomial probability question so when we get to the binomial probabilities like i said linda and i are going to integrate those quite nicely with each other um so that the classes the lectures and the practitioners link up and you guys can grasp the concepts quite easily then so let&#39;s get back to this first case study that we&#39;re interested in which is case study two so here we&#39;re telling you to assume for the purposes of this exercise that the day trips data set contains population data and you&#39;ll notice in the formulas that we&#39;re using here that we actually refer to the population mean so it&#39;s just the technical thing the reason why we do uh say assuming its population data is because expected values and variances are based on information for the population so we are just going to pretend here for this concept this is population data now what we want to do here before we can worry about expected value and variance and we&#39;ll we&#39;ll talk about that a bit later we first need a probability distribution now you&#39;ll remember in last week&#39;s class linda did mention the methods that we can use to get probabilities one of them is the classical method and the classical method of getting calculating probabilities assumes that each outcome is equally likely so each possible outcome you can have will have the same probability attached to it now the next method is the relative frequency method and that&#39;s what&#39;s applicable in this case now um we&#39;re going to work with specifically that method in the practicals then the last method is the subjective method which is pretty much taking an educated guess what the probability would be so i could say the probability of you getting a distinction in the semester test is 0.3 or 0.2 or 0.4 whatever it is i can base that maybe on what i&#39;ve seen in the past but it&#39;s just a rough get so that&#39;s the subjective method we&#39;re not gonna really work on that um now we are going to like i say focus on the probabilities um using the relative frequency uh this method and for that we first need frequencies and that is something you guys have actually done before so we&#39;re going to work with the number of trips that people have taken and we first need to get a list of frequencies for these now let&#39;s move over to excel so what is the easiest way to get frequency a frequency distribution in excel i want you guys one or two people to tell me so what what method can we use to just figure out what values yes pivot table okay so we&#39;re going to click on cell a1 and insert our pivot table and you guys might remember last week i told post on the discussion board where someone ran into some weird issues with groupings and things just to to remind you when we do a pivot table do not tick this last box so this add this data to the data model box should not be ticked that&#39;s why that person had issues i went and i tried it and that i ran into the exact same issues as she had so all we want is we want to have the um range selected and i&#39;m going to show that to you guys hopefully nicely now we&#39;ll just wait for the zooming in thingy to activate the slide there we go okay so we want to go and select the table or range and you want to make sure that all of your data is included so yes you can see goes from a1 up to cr17518 so all of my data has been selected and then the next thing we need to do is just say go put this in the new worksheet we never want to put a pivot table in the same worksheet and that box there needs to be unticked then you click on ok and let&#39;s just take that thing away again so now we need to go and set up this table so we want to use the number of people or the number of trips that we have taken so we&#39;re just going to start typing number and then we find number of trips and i&#39;m going to drag and drop that into the rows box then i&#39;m going to drag and drop the same thing into the values box and let&#39;s go compare this i&#39;m just gonna do that and let&#39;s just find the right file there we go um let&#39;s zoom in okay so when i look at this i can see that i have a lot more values here than i have in my um instruction document we&#39;ll talk about that now and here the first value matches but then things are going really really wrong and when i was preparing for this this morning i had a very quick look at what what i&#39;d said up and i saw this and i thought but i&#39;m using the right variable what&#39;s wrong any idea what&#39;s wrong what&#39;s causing this issue anyone exactly it says sum up big so we don&#39;t want sum we want to change this over to count okay so what it was doing is it was summing all of those values um so it summed all of the ones we had summed all of the twos we had summed all of the threes we had and so on and that&#39;s what we&#39;re interested in we are interested in knowing how many of each we have so if we change that to count that solves our problem there now if we go back here you&#39;ll notice that we said in the instructions that we don&#39;t want to work with all of the data um we are just for simplicity going to take a look at people who took 10 or less trips so that is in what we are interested in so we&#39;re pretending here that you can only take up to 10 trips we are not going to worry about people who took 90 trips those are people that probably live far away from home more than 40 oh sorry far away from work more than 40 kilometers from work so pretty much every day they go to work as a day trip so we&#39;re not interested in those people so we&#39;re just going to filter this out and change it so that we only have everything up to and including sen in our data set and there we go so now this matches okay so then this means we now have a frequency distribution for the number of trips that people took what we are actually interested in is not the frequencies we want to know the probability so i want to be able to say what is the probability that a person has taken four day trips in the last three months or what is the probability that they&#39;ve taken nine day trips in the last three months and how do we calculate a probability probability is just this frequency in each of the categories divided by the total number of observations in my sample so we&#39;re going to work with this total here now there&#39;s two ways of doing this first way i&#39;ve shown you guys in this links up again with case study one so you&#39;ll see a lot of what we&#39;re doing now here in case study two actually links up with case study one and you can also use it just this is a big hint you can use it for case study three as well so i&#39;m going to first just change this and now my mouse has decided to die let me just get the mouse working again my mouse is still taking a weekend so we&#39;re going to change this to the percentage of grand total okay and now we don&#39;t want percentages we want probabilities so all i&#39;m going to do is i highlight all of these i go here to the home ribbon and then i&#39;m going to change it here from percentage i&#39;m just going to take it to number and once it&#39;s a number i can go and increase my number of decimals so if you go back to we had here you&#39;ll see that these values match and you&#39;ll also notice here that this now has a grand total of one which means if i sum everything i&#39;m going to end up with one so and that has happened there except it&#39;s still a percentage so because i didn&#39;t highlight it so i want to make that a number as well okay in andrea i see you are a little bit confused which part is confusing you where did i lose you okay how do i get to the percentage so if i right click here let&#39;s say this would have been what it looked like originally so to get from a frequency to a percentage and you&#39;ll remember you did this in one of your previous practicals as well and i&#39;m going to show you a different method to do this as well now you can just go and change this to percentage of grand total and once it&#39;s a percentage we can then make it a count now another way that we could have done this so i&#39;m just gonna undo so we go back to the frequencies i&#39;m just gonna copy this pivot table and remember what i told you guys if i paste it because i want to do calculations with this data i want to paste it as values so let me just add that thing again so you guys can see the button properly um so when i do my paste you&#39;ll see that there&#39;s a few different options so we have here normal paste we have with no borders we have all kinds of things so that&#39;s the with formatting and things then there&#39;s some other paste options at the bottom but the middle bit which is paste values we want to use this bit that says values you can also do it so that it has number formatting applied or source formatting applied we&#39;re not worried about that we are just worried about the values so i&#39;m going to paste this as values and then we&#39;re going to work with that in a moment so first thing i&#39;m just going to go through this again to show you how we are getting from using the pivot table tool to get probabilities and then we&#39;ll do it in a different way so again right click on one of the frequencies and change this to show values as percentage of grand total we highlight all of this we change it from percentage to um number and then we can go and display our decimals as well okay now remember we are recording this so you can always go back and just watch this again slowly now the other way of doing this would have been to work with these and do this from scratch for all of the values so what i want to do first is i&#39;m going to change my headings here so these are my x values i&#39;m just going to make this center so we can see it nicely so i have my x values here so remember x represents my outcome so i&#39;m for now just going to write this here as a reminder these are my possible outcomes so i could see that someone has taken one two three four up to 10 trips then the counts are actually my frequencies so i&#39;m going to just call this fi and like i represent which um one we&#39;re talking about so the first f1 will be the first frequency f2 is the second one f3 is the third one and so and it&#39;s just for the the math notation to make sense that i&#39;m calling it that you could just call it f as well now we know from experience and from being in class that the formula for a probability if we&#39;re doing a relative um frequency approach we&#39;re going to take f i over n or just f over n so that is the frequency in for that related to that outcome divided by the total number of observations so that is our formula and we can then go and apply this so if i want to calculate my probabilities which is a little confusing with your textbook some textbooks call this px your textbook calls it fx so fx is just my probability let&#39;s actually write that out of x taking on a specific value equal to and i&#39;m gonna write this mathematically correctly your textbook doesn&#39;t distinguish between capital letters and lowercase letters but i&#39;ll talk about that just now again so don&#39;t worry too much about this now we&#39;ll we&#39;ll talk about it just now now that probability is just going to be my first frequency divided by my total number of observations as always i do an f4 just to fix onto this and i don&#39;t multiply by 100 because i don&#39;t want a percentage so i want it frequency and you can see all i do then is drag this down and i get all my probabilities that i got up there using my pivot table approach as well so you could do it either way now if all of these and i&#39;m going to show you guys a shortcut for something you could type the function or if you click in this cell there and you go back to your home red bin and you click on autosum it will automatically go and do the sum of everything that&#39;s above that cell so that&#39;s a short way of doing it i tapped only a4 not control f4 okay so it&#39;s just a 4 and that is covered in section 1.7 of your practice now these are my probabilities now what this probability represents this is actually just the probability that x is equal to one this is the probability that x is equal to two and so on so we can carry on like this and get all of these probabilities over here so you&#39;ll see that each number&#39;s probability is just related to that outcome over there so this is just the placeholder at the top so if i had written uh for instance f1 then that is just the probability that x is equal to one to that it&#39;s just that x is just the placeholder for whatever outcome i&#39;m interested in and if i fill that in it actually just means if one is probability that x is equal to one which is just the probability that a person has taken one day trip okay so um i&#39;m seeing a question what does the equation be for the third column this is just the frequencies divided by the of the total so we&#39;re just converting from frequencies going over into probabilities okay now i hope that makes sense to everyone um i see a bit of chat there the fn is just there&#39;s an fn button on some computers so mine is next to my control button bottom left of my keyboard and not all computers have an f in button sometimes it&#39;s called something different okay now this notation again just to give you a little bit of clarity if i wrote for instance if nine what i&#39;m actually asking you to calculate is the probability that x is equal to nine and in this case we can just go find it over there so that is my probability that x is equal to nine okay now that is the first part of this um question that&#39;s the easiest part i think it&#39;s just setting up your probability distribution and knowing what it means and you&#39;ll notice that here i&#39;ve also indicated that we&#39;re our probabilities with this effect sorry fx notation because that is what they use in your textbook some textbooks will use a px so if you are looking at a different website or a different resource somewhere and they&#39;re using px same thing not a problem now um the next thing we want to do is we want to then go and calculate expected value and variance now expected value of x just means what do we expect x to be equal to and if you looked at a data section i said to you well what&#39;s the expected value what what do you expect to see here what is the the expected number of trips that a person will take the obvious thing to go to is the average so an expected value the expected value of x is actually just the average value of x so i hope that makes sense you&#39;re going to get to the stall in the lectures don&#39;t stress about it now on average so you can see here expected value of x in this case we&#39;re going to calculate it now um and that&#39;s very closely linked to what we did with the group data so we&#39;re going to get back to that again this is just the average number of trips we expect people to take during the last three months so you can see here it&#39;s a value of 2.1874 blah blah blah so just over two trips in the last three months on average now what i want you guys to pay attention to here as well is that this value obviously has to fall between one and ten so often we&#39;ll see in a test situation that students calculate their expected value and it&#39;s way outside the range because they&#39;ve made a mistake somewhere it&#39;s not in this range of possible outcomes which doesn&#39;t make any sense because if you calculate an average that average has to fall between your minimum and maximum possible values it can&#39;t be outside that range because then how on earth did you calculate your average so just pay attention to that your variance variance can take on quite a lot of different values standard deviation again you need to go and think does it make sense remember variance is a squared value so don&#39;t put too much thought into what that specific value is but when you get a standard deviation always also check that it makes sense so if i calculate the standard deviation here and i get my standard deviation is 120 that means that on average my values are 120 units away from the mean and how can you be 120 units away if your range is only 9. it doesn&#39;t make any sense so i want you guys to start thinking about those things when you do a calculation always check is my answer matching like does it make logical sense because that&#39;s a whole point of status to make sense of the data okay so now we are going to get back into the expected value and variance calculations now you&#39;ll see here expected value of x is just the sum of x values the possible outcomes multiplied by their associated probabilities so we&#39;ll look at that just now and then you&#39;ll see that variance is calculated as again x minus mu squared times fx now i&#39;m going to quickly open here just last week&#39;s practical because i want to show you guys something here so this is the practical that you have already submitted now if you look at the formulas here when we come to the group data you should notice that here we had midpoint minus x bar squared times the frequency and we had to then divide by n minus one and there was stuff happening here as well but you&#39;ll notice here we have x minus mu so again something minus the mean or a mean square it and we multiplied by something else so everything you learned in practical two for that case study is again applicable yes so those skills are completely transferable so let&#39;s actually start with the expected value of x now in lost fixed practical you had to say frequency times midpoint and sum all of that and divide by n here we are going to say each outcome times each the probability that goes with it and we just sum all of that so basically what if we look at this i want to say one times eight thousand sorry not eight thousand not the frequency the probability so one times this probability plus two times the probability that goes with two plus three times the probability that goes with three and so on so that is a concept behind expected value so i&#39;m going to write my formula down here i&#39;m just going to delete the stuff here because it&#39;s going to get in the way and let&#39;s actually delete that bit as well now our expected value of x i&#39;m just going to write that up here the expected value of x which is actually just the population mean is going to look at the sum of all of the outcomes times their probabilities so if i wanted to write this out in a little bit more detail just so that we think about the formula first what i&#39;m doing is i&#39;m going to take the first outcome which is one and i multiply it by the probability of observing a one then i&#39;m going to take the next outcome which is two so i want to put a one there and i&#39;m going to multiply this by the probability of observing a 2 plus 3 times the probability of observing a 3 and we&#39;re going to do this until we reach the last outcome so that is the formula we want to use and now because you&#39;ve already done the group data you know that there&#39;s a quick way of doing this because we want to sum a bunch of products so we are going to use the sum product function so i&#39;m going to just do that over here my expected value of x which is equal to mu and this is how you type mu greek letter is just going to be the sum product of and let me actually just maybe zoom in a little bit for you guys so you can see exactly what i&#39;m typing so it&#39;s the sum product of my x values which are my outcomes not my frequencies my x values and my probabilities so you can see that there i&#39;m using sum product of the x values the outcomes and the associated probabilities and i press enter and there we go okay um so i see someone&#39;s asking how do you get the bottom table again not allowing you to copy and paste so that&#39;s a little odd uh let me just quickly go back there so all i did was i highlighted the whole table and went to copy now scrolled a bit down and then you should be able to pay just make sure that it&#39;s completely clear that there&#39;s nothing written in that area because sometimes that can block you from pasting but then i went to paste values and i pasted that there so you can also see and i&#39;m actually glad you asked this question um if you chose to calculate these probabilities using your pivot table when you want to do calculations with this it&#39;s also a good idea to just copy this and paste it as values and work with that um straight away so that you don&#39;t end up um with uh maybe this column sometimes people accidentally use this column in their calculations and you don&#39;t want to do that and also when you do the variance you need to do calculations based on these values and we know pivot tables don&#39;t like that okay now okay so let&#39;s look at this this is now our mean this is our population mean let&#39;s actually just write this this is our population mean okay up there now population of population okay so now we have our value of mu so we said the next thing we want to calculate is our population variance so our population variance we always denote with sigma squared and that is the sum so let&#39;s just write it like this as well this is the variance of x and again this variance is interpreted exactly the same as the variance you did in the previous weeks of this course it&#39;s just in a slightly different context that we&#39;re calculating it but it still means exactly the same thing so this is going to be the sum of our possible outcomes minus the population mean and square it and we multiply that with our probabilities so with the midpoint or the group data that we worked with last week we took midpoint minus sample mean because we were working with sample data and we squared that and then we could use the sum product function and again same thing here we need a column for these values and we need a column for the probabilities which we already have so all we still need to do is calculate a column of the x minus mu squared values so i&#39;m going to write that here i need a column for x minus mu and we are going to square it once we have those values we can use our sum product function and we can finish the rest of the calculation so in this case again remember your brackets we are going to take our x values again outcomes so this is why it&#39;s important to label these columns so that you when you look at your formula you know which one you&#39;re referring to we are going to subtract the population mean we&#39;ve just calculated so i&#39;m just going to press f4 there to fix onto it and then i need to go and square it so i&#39;ve done that and i can drag this down now and copy my formula so same as what we did last time and again i do a spot check and i take that i&#39;ve done my calculation correctly we can see here yes i&#39;ve taken an x value i&#39;ve subtracted the population mean and i&#39;ve squared it so everything is working as expected now we want to calculate the population mean which is just sigma squared which is just the variance of x and again it&#39;s just the population variance nothing odd about it so there we go and let&#39;s just actually make it so that we can see it after i type okay so we have our sigma squared which we want to calculate using this formula and again it&#39;s just going to be our sum product because it is the sum of these products the sum product now what i need here is firstly my x minus mu squared column so i&#39;m going to highlight that and then the next thing i need is my effect so my probabilities and i&#39;m going to highlight that and again the common mistake that we see students making with this work is highlighting the wrong f values so some students would go and highlight the counts instead of um the probabilities probably because it both of them have an f in in the heading so just make sure that you know f x represents the probability that x is equal to some value and when we say if just f like that then it is the um account so just make sure you understand the difference between the two so there we go now that is my variance and now if i want to calculate my standard deviation and again notice this is my standard deviation for the population then let&#39;s just call this sd of x let&#39;s make that alignment like that then all i need to do is i need to go calculate the square root of the variance okay and this one like i said earlier make sure that it makes sense so this value 1.64861 remember standard deviation tells us on average how far away our observations are from the from the mean and you can&#39;t expect the standard deviation to be massive because we only have a range of nine so we would expect our standard deviation should be less than nine because our values can&#39;t be further away than the maximum or the minimum value so i hope that makes sense um to you guys so um again just to recap for this question we have said here we weren&#39;t using all of our data because obviously there were quite a lot of rows for this variable and the more rows there are the more confusing the calculations can look so i&#39;ve just chosen to just show you guys this example with people who took up to 10 trips so our population we could make the argument logically saying that we are only interested in people who took up to 10 trips and that makes them our population so remember our population is who we are interested in so even though there are people who took more than 10 trips we are only interested in people who took up to 10 trips so we can say that this is our population data we&#39;re pretending this that we&#39;ve observed everyone um in in this data set otherwise we can&#39;t calculate an expected value and a variance so it&#39;s just a technical issue we&#39;ll come back to that again so i see a question just to show you how i did mu again remember mu is just outcomes times their associated probabilities um sum all of that so we could have done it and i&#39;ll quickly maybe show you we could have also said here let go do a column of x times the f x values so that would have been manually if we didn&#39;t have the sum product function we could have said this value times the probability that goes with it and get all of that and why when we then go and sum it so i&#39;ve just done that calculation for all of them you can see we get to exactly the same value or we could save ourselves some time and use the sum product function and just say let&#39;s use the counts we&#39;re going to use the probabilities and it needs to take the corresponding um of not the outcomes and the corresponding probability multiply them together and then we sum all of those answers okay so i think this is probably um one of the more challenging case studies it might also be because you guys haven&#39;t done this in class yet but now when we look at this in class you will look at it a little bit differently and i&#39;ll very quickly show you the variance one again again same as with group data all we did was we went and calculated the column of x minus mu squared values there it was midpoint minus x bar squared values same thing um same same concept behind it and then we use the sum product function so i want you guys to go work through this on your own and then come back and ask any questions that you have so you should be able to replicate this now so what you should be able to do for this question is you should be able to take one of the variables in your day trip data set set up the empirical probability distribution for it which is what we&#39;ve done here and then use that to calculate expected value variance and standard deviation so that is what you should be able to do so i want to move on to the next case study um i&#39;m going to ask the moderators to just keep track if there&#39;s any other questions that if there&#39;s time we can come back to that again we&#39;ll do that at the end of the session but i just want to go through the other two case studies so that you get uh um start on those as well so this one should be really really easy for you guys now this is our case study from prac one so let&#39;s quickly go set up that probability distribution again but this probability distribution differs from the one we just did because here we have two variables so there we had only one variable that we were working with here we&#39;re going to work with a cross tabulation with two variables in it so we need the type of dwelling and whether or not a person has a homophone so if we just go back to excel we need to find the right tab i&#39;m gonna again insert a pivot table so this i&#39;m gonna go through fast because we&#39;ve done this one a few times already so we are going to take the type of dwelling and put that in the rows i&#39;m going to quickly throw it into the values box as well and then we are going to take home phone and put it in there and again like we did last time filter out the blanks we can sort it so it&#39;s yes no and then here i&#39;m also just going to filter this out so i&#39;m doing this one first because you guys know how to do this one already now again here we have currently all of these as frequencies and if we want to change it to probabilities nice way of doing it quick and easy same as the previous case study we show values as a percentage of the grand total so there we have them as percentages now and if we want to move over and make its probabilities all we do is we go back and we say change this to a percentage of not to a percentage to a number and we can of course then make the um values or give it more decimal places so if you go back to the case study here you&#39;ll notice that in this case study i&#39;ve taken you step by step through this showing you what the percentages look like how to change this in an alternative way from what i i just showed you guys the shortcut this is the longer way of doing it you can do it like this as well and what your table should look like in the end now what i want you to notice here is this is just a joint probability table where we have our joint probabilities here in the middle it&#39;s easy to remember because the joint probability deals with two values or two variables jointly here on the side we have our marginal probabilities they&#39;re in the margins and same thing there those are also marginal probabilities so this if i looked at this i can tell you the probability that a person has a home phone is 0.1125 and i get that directly from reading this off now these are our joints and marginal probabilities you could also go and calculate conditional probabilities by changing instead of percentage of grand total you just change it to percentage of row total or percentage of column total so that is um that&#39;s basically what we showed you in track one as well so i did show you how to do that here now um for decimal places i saw that i saw tristan and people have answered thanks very much but i&#39;m gonna quickly answer as well there number of decimal places always pay attention to what we tell you on the system so and follow that so if you see all of my answers are to four decimal places you give your answers to 40 small places if i give them to six and you have to complete a table you also want to give them two six so you want to stick with what we&#39;ve given you there as well and then if you um your numbers is quite strict if you give less decimal places then it asks you for it&#39;s going to give you zero because obviously you could have had a few different numbers that round to the same value so you always want to make sure you follow those instructions exactly to the letter um there was something else i wanted to also say about that um so with these questions i want you guys to also for this question go back to practical one and go review those questions that you had below your cross tabulation so there was a post on the discussion board where someone asked us how do you know which values to use what to divide by and that same question is going to come up with these questions so we are again going to ask you similar questions so i could ask you what is the probability that a person who lives in a caravan or tent has a home phone so you need to know that that is a conditional probability that i&#39;ve just asked you for um so i&#39;ll get to your question now chase um so yeah you need to be able to go from a question that&#39;s asking words and identify is it a question related to a joint probability a marginal probability or a conditional probability and once you have that you will know which type of table to use and you&#39;ll see that in your practical what&#39;s going to happen is we are going to give you some variables that you need to use to construct a cross tabulation and then you need to change it to percentages of the grand total um rotators and column totals and then we are going to ask you joint marginal and conditional probabilities but we&#39;re not going to tell you which one of the three tables to use to answer each one so that is the challenge of the question and seeing whether you can put theory and practice together so that&#39;s very important now to swap those around the yes and the no tables uh columns i actually used a little bit of a shortcut here i right clicked on them and you can go say that you want to move the one that you clicked on up or down but since i only have two categories here yes and no if they&#39;re in alphabetical order it&#39;s going to be no first then yes so i just said instead of doing it manually i went here to sort and i said sort from z to a so i just reverse sorted it for me and that&#39;s how i did it so fast um so just to go back again to the probabilities the probabilities here on the inside of the table are your joint probabilities because it&#39;s a joint event there&#39;s two things happening here so you have the dwelling and whether or not the person has a home phone to consider so it&#39;s a probability that joins up two concepts now on the sides these are in the margins so those are known as marginal probabilities they&#39;re in the margins and each of them relate to only one event so if i look at this um 0.887 whatever here it actually is just the probability of a person not having a homophobe so that&#39;s in the margins it only deals with one variable if we did a conditional probability then obviously it&#39;s conditional so the one thing is given the other thing is the unknown so you just have to identify what&#39;s the given what&#39;s the unknown so now let&#39;s go back to more practical there&#39;s one more case study to look at and we only have four minutes but we can do this um so this case study you&#39;ll notice that we&#39;ve said here work through the examples in section four of your back eye and last year i saw students were struggling a little bit with this question but if you think about the question it&#39;s actually very straightforward so this question doesn&#39;t have a specific case study now let&#39;s quickly go back to your pack guide if we look at the prac guide and we go look at counting rules in excel i don&#39;t know if you guys noticed but you can click on a um let&#39;s just actually go up i don&#39;t think you guys saw that but in your index here we&#39;ve made it so that if you go to counting rules and you click on it it will take you straight there so you don&#39;t even have to page through the document you can go straight to those so this one just goes through some counting rules and it has a few different things that you can calculate so you need to be able to do factorials you do that with the fact function combinations are done with a combination function permutations done with the permutation function and now you can see the person who set this up had their excel set up differently from mine because if you look at the separator here between the different arguments she had a semicolon where i usually have a comma so you need to pay attention to how your computer is set up so that you know whether you need to use semicolons or commas now um in this example it just takes you through how to calculate these and different software packages actually calculated differently but it&#39;s literally one page that you need to go and read and you&#39;ll see here for combinations and permutations we first tell you how many values we are trying to select from and then what our sample size is so when it comes to your questions that you&#39;re going to get on click up for this we are going to give you a scenario and say you are interested in this group of people so you just then need to go and figure out how many people fall into that group and we will also then tell you on click up you want to sample so many people from this group so then you need to in that given scenario figure out is a factorial the right thing to calculate is it a permutation or is it a combination and then you use the skills you have that you guys illustrated very well at the beginning of the session you know exactly how to go calculate how many values are in a group you said it yourselves just go to a pivot table it&#39;s a quick way of figuring out how many values are in a specific category that is in your population size we&#39;ve explained that here and then you just need to know is it a combination a factorial or a permutation that you need to calculate and that is pretty much it so this question actually should be really easy marks you just need to combine the skills that you&#39;ve had used in the past practicals and this one to go and do the calculations so i think that&#39;s everything i wanted to show you guys the binomial probabilities like i said we will get to those the week after your semester test so if you want to you can if you&#39;ve worked with binomial before you&#39;re welcome to go try it but we&#39;re not going to cover it this week um so the remainder of this week you can bring your questions for the first three case studies here as well as previous practicals okay so i hope you guys have an awesome day and we will see you in the lecture and like i said please bring your questions to the q and a sessions the rest of the week

okay welcome to today&amp;#39;s practical session everyone um so today we are going to focus on uh practical three we are going to look at case studies one two and three and then next week we are going to look at binomial distribution so we haven&amp;#39;t done binomial distribution in class yet um we will get to that next week and we will integrate next week&amp;#39;s um practical sessions with the lectures to get you guys um well acquainted with what&amp;#39;s what&amp;#39;s going on sorry not next week the week after um so the week after the the taste we will work on the binomial day so um i always forget there&amp;#39;s a taste week in between that we&amp;#39;re not doing a prac in that week so the rest of this week we are going to do q a sessions and you will notice here on my slide i did not say q and a specifically for prac three obviously it is for break three but if you have any remaining questions on prac one and two you are also welcome to come to the q and a sessions and we can refresh on things like that so we had some questions before we started about the group mean and the group standard deviation we can cover those in q and a sessions this week as well for the practicals um so let&amp;#39;s jump straight in and i&amp;#39;m going to start sharing now two practical three so we only have six practicals in this course so this one marks the halfway mark and hopefully you guys are getting the hang of it it looked quite positive in the chat earlier uh so let&amp;#39;s hope that it stays like that um so let&amp;#39;s look at what is in brack three so i&amp;#39;m just going to go and give you guys a quick overview again as always read this there&amp;#39;s some extra information added into this bit here you&amp;#39;ll notice most of it is the same with saying that it&amp;#39;s sample data and all of that but i&amp;#39;ve given you a note here saying that for two of the case studies we are actually going to prove that this is population data and the giveaway there is the word census remembrances deals with the entire population now objectives you can go read through that you can go see all of the things that you need to see here you can see here that the submission link will be available midnight um after your your submission this friday um why did i make it only then oh so you don&amp;#39;t have a submission this friday i don&amp;#39;t know why i said that sorry but i&amp;#39;m i&amp;#39;m going completely crazy clearly i need a holiday as well so submission link is going to be available on saturday morning i could maybe even make it a little bit earlier on friday if there&amp;#39;s a need for that it will be ready for that so you guys can just let us know if you prefer to be available a little bit earlier i&amp;#39;m happy to do that um three attempts allowed only the best one will count and i&amp;#39;ve given you guys 50 minutes for this one so it&amp;#39;s a little shorter than the others but it&amp;#39;s also com using some of the skills you&amp;#39;ve gained previously so that should um help you as well now the functions that you may need to use are given here this binom dist will only be used in case study four so we are only going to cover that in the practitioners after the semester test it&amp;#39;s therefore not included in the scope of your semester test but any of the other things we can include in the scope of your semester test but all you know we&amp;#39;ve used sum product you&amp;#39;re going to use it again count we&amp;#39;ve used you could even use the count f function for some questions if you wanted to and then the square root function is important as well now just to give you an overview of what we&amp;#39;re going to try and achieve with this practical probabilities using cross tabulations you&amp;#39;ll see this follows on from practical ones case study so it&amp;#39;s important that you&amp;#39;ve mastered practical one so that you can do this question we&amp;#39;re going to do this one second we&amp;#39;re actually going to start today&amp;#39;s session with empirical probability distributions which we haven&amp;#39;t covered in detail in class yet so you guys will um still see this kind of thing in class um but don&amp;#39;t don&amp;#39;t stress about it it sounds much worse than it is like i said before if you see the word empirical just say change it to observed so it&amp;#39;s an observed probability distribution and you&amp;#39;ve actually done the basics of this um in class already and we&amp;#39;ll get back to that expected value and variance you haven&amp;#39;t done in class yet but we&amp;#39;ll get to that um and i&amp;#39;ll this will give you guys a head start on what&amp;#39;s coming in the lectures and then counting techniques we&amp;#39;ll do today as well and then the binomial probability question so when we get to the binomial probabilities like i said linda and i are going to integrate those quite nicely with each other um so that the classes the lectures and the practitioners link up and you guys can grasp the concepts quite easily then so let&amp;#39;s get back to this first case study that we&amp;#39;re interested in which is case study two so here we&amp;#39;re telling you to assume for the purposes of this exercise that the day trips data set contains population data and you&amp;#39;ll notice in the formulas that we&amp;#39;re using here that we actually refer to the population mean so it&amp;#39;s just the technical thing the reason why we do uh say assuming its population data is because expected values and variances are based on information for the population so we are just going to pretend here for this concept this is population data now what we want to do here before we can worry about expected value and variance and we&amp;#39;ll we&amp;#39;ll talk about that a bit later we first need a probability distribution now you&amp;#39;ll remember in last week&amp;#39;s class linda did mention the methods that we can use to get probabilities one of them is the classical method and the classical method of getting calculating probabilities assumes that each outcome is equally likely so each possible outcome you can have will have the same probability attached to it now the next method is the relative frequency method and that&amp;#39;s what&amp;#39;s applicable in this case now um we&amp;#39;re going to work with specifically that method in the practicals then the last method is the subjective method which is pretty much taking an educated guess what the probability would be so i could say the probability of you getting a distinction in the semester test is 0.3 or 0.2 or 0.4 whatever it is i can base that maybe on what i&amp;#39;ve seen in the past but it&amp;#39;s just a rough get so that&amp;#39;s the subjective method we&amp;#39;re not gonna really work on that um now we are going to like i say focus on the probabilities um using the relative frequency uh this method and for that we first need frequencies and that is something you guys have actually done before so we&amp;#39;re going to work with the number of trips that people have taken and we first need to get a list of frequencies for these now let&amp;#39;s move over to excel so what is the easiest way to get frequency a frequency distribution in excel i want you guys one or two people to tell me so what what method can we use to just figure out what values yes pivot table okay so we&amp;#39;re going to click on cell a1 and insert our pivot table and you guys might remember last week i told post on the discussion board where someone ran into some weird issues with groupings and things just to to remind you when we do a pivot table do not tick this last box so this add this data to the data model box should not be ticked that&amp;#39;s why that person had issues i went and i tried it and that i ran into the exact same issues as she had so all we want is we want to have the um range selected and i&amp;#39;m going to show that to you guys hopefully nicely now we&amp;#39;ll just wait for the zooming in thingy to activate the slide there we go okay so we want to go and select the table or range and you want to make sure that all of your data is included so yes you can see goes from a1 up to cr17518 so all of my data has been selected and then the next thing we need to do is just say go put this in the new worksheet we never want to put a pivot table in the same worksheet and that box there needs to be unticked then you click on ok and let&amp;#39;s just take that thing away again so now we need to go and set up this table so we want to use the number of people or the number of trips that we have taken so we&amp;#39;re just going to start typing number and then we find number of trips and i&amp;#39;m going to drag and drop that into the rows box then i&amp;#39;m going to drag and drop the same thing into the values box and let&amp;#39;s go compare this i&amp;#39;m just gonna do that and let&amp;#39;s just find the right file there we go um let&amp;#39;s zoom in okay so when i look at this i can see that i have a lot more values here than i have in my um instruction document we&amp;#39;ll talk about that now and here the first value matches but then things are going really really wrong and when i was preparing for this this morning i had a very quick look at what what i&amp;#39;d said up and i saw this and i thought but i&amp;#39;m using the right variable what&amp;#39;s wrong any idea what&amp;#39;s wrong what&amp;#39;s causing this issue anyone exactly it says sum up big so we don&amp;#39;t want sum we want to change this over to count okay so what it was doing is it was summing all of those values um so it summed all of the ones we had summed all of the twos we had summed all of the threes we had and so on and that&amp;#39;s what we&amp;#39;re interested in we are interested in knowing how many of each we have so if we change that to count that solves our problem there now if we go back here you&amp;#39;ll notice that we said in the instructions that we don&amp;#39;t want to work with all of the data um we are just for simplicity going to take a look at people who took 10 or less trips so that is in what we are interested in so we&amp;#39;re pretending here that you can only take up to 10 trips we are not going to worry about people who took 90 trips those are people that probably live far away from home more than 40 oh sorry far away from work more than 40 kilometers from work so pretty much every day they go to work as a day trip so we&amp;#39;re not interested in those people so we&amp;#39;re just going to filter this out and change it so that we only have everything up to and including sen in our data set and there we go so now this matches okay so then this means we now have a frequency distribution for the number of trips that people took what we are actually interested in is not the frequencies we want to know the probability so i want to be able to say what is the probability that a person has taken four day trips in the last three months or what is the probability that they&amp;#39;ve taken nine day trips in the last three months and how do we calculate a probability probability is just this frequency in each of the categories divided by the total number of observations in my sample so we&amp;#39;re going to work with this total here now there&amp;#39;s two ways of doing this first way i&amp;#39;ve shown you guys in this links up again with case study one so you&amp;#39;ll see a lot of what we&amp;#39;re doing now here in case study two actually links up with case study one and you can also use it just this is a big hint you can use it for case study three as well so i&amp;#39;m going to first just change this and now my mouse has decided to die let me just get the mouse working again my mouse is still taking a weekend so we&amp;#39;re going to change this to the percentage of grand total okay and now we don&amp;#39;t want percentages we want probabilities so all i&amp;#39;m going to do is i highlight all of these i go here to the home ribbon and then i&amp;#39;m going to change it here from percentage i&amp;#39;m just going to take it to number and once it&amp;#39;s a number i can go and increase my number of decimals so if you go back to we had here you&amp;#39;ll see that these values match and you&amp;#39;ll also notice here that this now has a grand total of one which means if i sum everything i&amp;#39;m going to end up with one so and that has happened there except it&amp;#39;s still a percentage so because i didn&amp;#39;t highlight it so i want to make that a number as well okay in andrea i see you are a little bit confused which part is confusing you where did i lose you okay how do i get to the percentage so if i right click here let&amp;#39;s say this would have been what it looked like originally so to get from a frequency to a percentage and you&amp;#39;ll remember you did this in one of your previous practicals as well and i&amp;#39;m going to show you a different method to do this as well now you can just go and change this to percentage of grand total and once it&amp;#39;s a percentage we can then make it a count now another way that we could have done this so i&amp;#39;m just gonna undo so we go back to the frequencies i&amp;#39;m just gonna copy this pivot table and remember what i told you guys if i paste it because i want to do calculations with this data i want to paste it as values so let me just add that thing again so you guys can see the button properly um so when i do my paste you&amp;#39;ll see that there&amp;#39;s a few different options so we have here normal paste we have with no borders we have all kinds of things so that&amp;#39;s the with formatting and things then there&amp;#39;s some other paste options at the bottom but the middle bit which is paste values we want to use this bit that says values you can also do it so that it has number formatting applied or source formatting applied we&amp;#39;re not worried about that we are just worried about the values so i&amp;#39;m going to paste this as values and then we&amp;#39;re going to work with that in a moment so first thing i&amp;#39;m just going to go through this again to show you how we are getting from using the pivot table tool to get probabilities and then we&amp;#39;ll do it in a different way so again right click on one of the frequencies and change this to show values as percentage of grand total we highlight all of this we change it from percentage to um number and then we can go and display our decimals as well okay now remember we are recording this so you can always go back and just watch this again slowly now the other way of doing this would have been to work with these and do this from scratch for all of the values so what i want to do first is i&amp;#39;m going to change my headings here so these are my x values i&amp;#39;m just going to make this center so we can see it nicely so i have my x values here so remember x represents my outcome so i&amp;#39;m for now just going to write this here as a reminder these are my possible outcomes so i could see that someone has taken one two three four up to 10 trips then the counts are actually my frequencies so i&amp;#39;m going to just call this fi and like i represent which um one we&amp;#39;re talking about so the first f1 will be the first frequency f2 is the second one f3 is the third one and so and it&amp;#39;s just for the the math notation to make sense that i&amp;#39;m calling it that you could just call it f as well now we know from experience and from being in class that the formula for a probability if we&amp;#39;re doing a relative um frequency approach we&amp;#39;re going to take f i over n or just f over n so that is the frequency in for that related to that outcome divided by the total number of observations so that is our formula and we can then go and apply this so if i want to calculate my probabilities which is a little confusing with your textbook some textbooks call this px your textbook calls it fx so fx is just my probability let&amp;#39;s actually write that out of x taking on a specific value equal to and i&amp;#39;m gonna write this mathematically correctly your textbook doesn&amp;#39;t distinguish between capital letters and lowercase letters but i&amp;#39;ll talk about that just now again so don&amp;#39;t worry too much about this now we&amp;#39;ll we&amp;#39;ll talk about it just now now that probability is just going to be my first frequency divided by my total number of observations as always i do an f4 just to fix onto this and i don&amp;#39;t multiply by 100 because i don&amp;#39;t want a percentage so i want it frequency and you can see all i do then is drag this down and i get all my probabilities that i got up there using my pivot table approach as well so you could do it either way now if all of these and i&amp;#39;m going to show you guys a shortcut for something you could type the function or if you click in this cell there and you go back to your home red bin and you click on autosum it will automatically go and do the sum of everything that&amp;#39;s above that cell so that&amp;#39;s a short way of doing it i tapped only a4 not control f4 okay so it&amp;#39;s just a 4 and that is covered in section 1.7 of your practice now these are my probabilities now what this probability represents this is actually just the probability that x is equal to one this is the probability that x is equal to two and so on so we can carry on like this and get all of these probabilities over here so you&amp;#39;ll see that each number&amp;#39;s probability is just related to that outcome over there so this is just the placeholder at the top so if i had written uh for instance f1 then that is just the probability that x is equal to one to that it&amp;#39;s just that x is just the placeholder for whatever outcome i&amp;#39;m interested in and if i fill that in it actually just means if one is probability that x is equal to one which is just the probability that a person has taken one day trip okay so um i&amp;#39;m seeing a question what does the equation be for the third column this is just the frequencies divided by the of the total so we&amp;#39;re just converting from frequencies going over into probabilities okay now i hope that makes sense to everyone um i see a bit of chat there the fn is just there&amp;#39;s an fn button on some computers so mine is next to my control button bottom left of my keyboard and not all computers have an f in button sometimes it&amp;#39;s called something different okay now this notation again just to give you a little bit of clarity if i wrote for instance if nine what i&amp;#39;m actually asking you to calculate is the probability that x is equal to nine and in this case we can just go find it over there so that is my probability that x is equal to nine okay now that is the first part of this um question that&amp;#39;s the easiest part i think it&amp;#39;s just setting up your probability distribution and knowing what it means and you&amp;#39;ll notice that here i&amp;#39;ve also indicated that we&amp;#39;re our probabilities with this effect sorry fx notation because that is what they use in your textbook some textbooks will use a px so if you are looking at a different website or a different resource somewhere and they&amp;#39;re using px same thing not a problem now um the next thing we want to do is we want to then go and calculate expected value and variance now expected value of x just means what do we expect x to be equal to and if you looked at a data section i said to you well what&amp;#39;s the expected value what what do you expect to see here what is the the expected number of trips that a person will take the obvious thing to go to is the average so an expected value the expected value of x is actually just the average value of x so i hope that makes sense you&amp;#39;re going to get to the stall in the lectures don&amp;#39;t stress about it now on average so you can see here expected value of x in this case we&amp;#39;re going to calculate it now um and that&amp;#39;s very closely linked to what we did with the group data so we&amp;#39;re going to get back to that again this is just the average number of trips we expect people to take during the last three months so you can see here it&amp;#39;s a value of 2.1874 blah blah blah so just over two trips in the last three months on average now what i want you guys to pay attention to here as well is that this value obviously has to fall between one and ten so often we&amp;#39;ll see in a test situation that students calculate their expected value and it&amp;#39;s way outside the range because they&amp;#39;ve made a mistake somewhere it&amp;#39;s not in this range of possible outcomes which doesn&amp;#39;t make any sense because if you calculate an average that average has to fall between your minimum and maximum possible values it can&amp;#39;t be outside that range because then how on earth did you calculate your average so just pay attention to that your variance variance can take on quite a lot of different values standard deviation again you need to go and think does it make sense remember variance is a squared value so don&amp;#39;t put too much thought into what that specific value is but when you get a standard deviation always also check that it makes sense so if i calculate the standard deviation here and i get my standard deviation is 120 that means that on average my values are 120 units away from the mean and how can you be 120 units away if your range is only 9. it doesn&amp;#39;t make any sense so i want you guys to start thinking about those things when you do a calculation always check is my answer matching like does it make logical sense because that&amp;#39;s a whole point of status to make sense of the data okay so now we are going to get back into the expected value and variance calculations now you&amp;#39;ll see here expected value of x is just the sum of x values the possible outcomes multiplied by their associated probabilities so we&amp;#39;ll look at that just now and then you&amp;#39;ll see that variance is calculated as again x minus mu squared times fx now i&amp;#39;m going to quickly open here just last week&amp;#39;s practical because i want to show you guys something here so this is the practical that you have already submitted now if you look at the formulas here when we come to the group data you should notice that here we had midpoint minus x bar squared times the frequency and we had to then divide by n minus one and there was stuff happening here as well but you&amp;#39;ll notice here we have x minus mu so again something minus the mean or a mean square it and we multiplied by something else so everything you learned in practical two for that case study is again applicable yes so those skills are completely transferable so let&amp;#39;s actually start with the expected value of x now in lost fixed practical you had to say frequency times midpoint and sum all of that and divide by n here we are going to say each outcome times each the probability that goes with it and we just sum all of that so basically what if we look at this i want to say one times eight thousand sorry not eight thousand not the frequency the probability so one times this probability plus two times the probability that goes with two plus three times the probability that goes with three and so on so that is a concept behind expected value so i&amp;#39;m going to write my formula down here i&amp;#39;m just going to delete the stuff here because it&amp;#39;s going to get in the way and let&amp;#39;s actually delete that bit as well now our expected value of x i&amp;#39;m just going to write that up here the expected value of x which is actually just the population mean is going to look at the sum of all of the outcomes times their probabilities so if i wanted to write this out in a little bit more detail just so that we think about the formula first what i&amp;#39;m doing is i&amp;#39;m going to take the first outcome which is one and i multiply it by the probability of observing a one then i&amp;#39;m going to take the next outcome which is two so i want to put a one there and i&amp;#39;m going to multiply this by the probability of observing a 2 plus 3 times the probability of observing a 3 and we&amp;#39;re going to do this until we reach the last outcome so that is the formula we want to use and now because you&amp;#39;ve already done the group data you know that there&amp;#39;s a quick way of doing this because we want to sum a bunch of products so we are going to use the sum product function so i&amp;#39;m going to just do that over here my expected value of x which is equal to mu and this is how you type mu greek letter is just going to be the sum product of and let me actually just maybe zoom in a little bit for you guys so you can see exactly what i&amp;#39;m typing so it&amp;#39;s the sum product of my x values which are my outcomes not my frequencies my x values and my probabilities so you can see that there i&amp;#39;m using sum product of the x values the outcomes and the associated probabilities and i press enter and there we go okay um so i see someone&amp;#39;s asking how do you get the bottom table again not allowing you to copy and paste so that&amp;#39;s a little odd uh let me just quickly go back there so all i did was i highlighted the whole table and went to copy now scrolled a bit down and then you should be able to pay just make sure that it&amp;#39;s completely clear that there&amp;#39;s nothing written in that area because sometimes that can block you from pasting but then i went to paste values and i pasted that there so you can also see and i&amp;#39;m actually glad you asked this question um if you chose to calculate these probabilities using your pivot table when you want to do calculations with this it&amp;#39;s also a good idea to just copy this and paste it as values and work with that um straight away so that you don&amp;#39;t end up um with uh maybe this column sometimes people accidentally use this column in their calculations and you don&amp;#39;t want to do that and also when you do the variance you need to do calculations based on these values and we know pivot tables don&amp;#39;t like that okay now okay so let&amp;#39;s look at this this is now our mean this is our population mean let&amp;#39;s actually just write this this is our population mean okay up there now population of population okay so now we have our value of mu so we said the next thing we want to calculate is our population variance so our population variance we always denote with sigma squared and that is the sum so let&amp;#39;s just write it like this as well this is the variance of x and again this variance is interpreted exactly the same as the variance you did in the previous weeks of this course it&amp;#39;s just in a slightly different context that we&amp;#39;re calculating it but it still means exactly the same thing so this is going to be the sum of our possible outcomes minus the population mean and square it and we multiply that with our probabilities so with the midpoint or the group data that we worked with last week we took midpoint minus sample mean because we were working with sample data and we squared that and then we could use the sum product function and again same thing here we need a column for these values and we need a column for the probabilities which we already have so all we still need to do is calculate a column of the x minus mu squared values so i&amp;#39;m going to write that here i need a column for x minus mu and we are going to square it once we have those values we can use our sum product function and we can finish the rest of the calculation so in this case again remember your brackets we are going to take our x values again outcomes so this is why it&amp;#39;s important to label these columns so that you when you look at your formula you know which one you&amp;#39;re referring to we are going to subtract the population mean we&amp;#39;ve just calculated so i&amp;#39;m just going to press f4 there to fix onto it and then i need to go and square it so i&amp;#39;ve done that and i can drag this down now and copy my formula so same as what we did last time and again i do a spot check and i take that i&amp;#39;ve done my calculation correctly we can see here yes i&amp;#39;ve taken an x value i&amp;#39;ve subtracted the population mean and i&amp;#39;ve squared it so everything is working as expected now we want to calculate the population mean which is just sigma squared which is just the variance of x and again it&amp;#39;s just the population variance nothing odd about it so there we go and let&amp;#39;s just actually make it so that we can see it after i type okay so we have our sigma squared which we want to calculate using this formula and again it&amp;#39;s just going to be our sum product because it is the sum of these products the sum product now what i need here is firstly my x minus mu squared column so i&amp;#39;m going to highlight that and then the next thing i need is my effect so my probabilities and i&amp;#39;m going to highlight that and again the common mistake that we see students making with this work is highlighting the wrong f values so some students would go and highlight the counts instead of um the probabilities probably because it both of them have an f in in the heading so just make sure that you know f x represents the probability that x is equal to some value and when we say if just f like that then it is the um account so just make sure you understand the difference between the two so there we go now that is my variance and now if i want to calculate my standard deviation and again notice this is my standard deviation for the population then let&amp;#39;s just call this sd of x let&amp;#39;s make that alignment like that then all i need to do is i need to go calculate the square root of the variance okay and this one like i said earlier make sure that it makes sense so this value 1.64861 remember standard deviation tells us on average how far away our observations are from the from the mean and you can&amp;#39;t expect the standard deviation to be massive because we only have a range of nine so we would expect our standard deviation should be less than nine because our values can&amp;#39;t be further away than the maximum or the minimum value so i hope that makes sense um to you guys so um again just to recap for this question we have said here we weren&amp;#39;t using all of our data because obviously there were quite a lot of rows for this variable and the more rows there are the more confusing the calculations can look so i&amp;#39;ve just chosen to just show you guys this example with people who took up to 10 trips so our population we could make the argument logically saying that we are only interested in people who took up to 10 trips and that makes them our population so remember our population is who we are interested in so even though there are people who took more than 10 trips we are only interested in people who took up to 10 trips so we can say that this is our population data we&amp;#39;re pretending this that we&amp;#39;ve observed everyone um in in this data set otherwise we can&amp;#39;t calculate an expected value and a variance so it&amp;#39;s just a technical issue we&amp;#39;ll come back to that again so i see a question just to show you how i did mu again remember mu is just outcomes times their associated probabilities um sum all of that so we could have done it and i&amp;#39;ll quickly maybe show you we could have also said here let go do a column of x times the f x values so that would have been manually if we didn&amp;#39;t have the sum product function we could have said this value times the probability that goes with it and get all of that and why when we then go and sum it so i&amp;#39;ve just done that calculation for all of them you can see we get to exactly the same value or we could save ourselves some time and use the sum product function and just say let&amp;#39;s use the counts we&amp;#39;re going to use the probabilities and it needs to take the corresponding um of not the outcomes and the corresponding probability multiply them together and then we sum all of those answers okay so i think this is probably um one of the more challenging case studies it might also be because you guys haven&amp;#39;t done this in class yet but now when we look at this in class you will look at it a little bit differently and i&amp;#39;ll very quickly show you the variance one again again same as with group data all we did was we went and calculated the column of x minus mu squared values there it was midpoint minus x bar squared values same thing um same same concept behind it and then we use the sum product function so i want you guys to go work through this on your own and then come back and ask any questions that you have so you should be able to replicate this now so what you should be able to do for this question is you should be able to take one of the variables in your day trip data set set up the empirical probability distribution for it which is what we&amp;#39;ve done here and then use that to calculate expected value variance and standard deviation so that is what you should be able to do so i want to move on to the next case study um i&amp;#39;m going to ask the moderators to just keep track if there&amp;#39;s any other questions that if there&amp;#39;s time we can come back to that again we&amp;#39;ll do that at the end of the session but i just want to go through the other two case studies so that you get uh um start on those as well so this one should be really really easy for you guys now this is our case study from prac one so let&amp;#39;s quickly go set up that probability distribution again but this probability distribution differs from the one we just did because here we have two variables so there we had only one variable that we were working with here we&amp;#39;re going to work with a cross tabulation with two variables in it so we need the type of dwelling and whether or not a person has a homophone so if we just go back to excel we need to find the right tab i&amp;#39;m gonna again insert a pivot table so this i&amp;#39;m gonna go through fast because we&amp;#39;ve done this one a few times already so we are going to take the type of dwelling and put that in the rows i&amp;#39;m going to quickly throw it into the values box as well and then we are going to take home phone and put it in there and again like we did last time filter out the blanks we can sort it so it&amp;#39;s yes no and then here i&amp;#39;m also just going to filter this out so i&amp;#39;m doing this one first because you guys know how to do this one already now again here we have currently all of these as frequencies and if we want to change it to probabilities nice way of doing it quick and easy same as the previous case study we show values as a percentage of the grand total so there we have them as percentages now and if we want to move over and make its probabilities all we do is we go back and we say change this to a percentage of not to a percentage to a number and we can of course then make the um values or give it more decimal places so if you go back to the case study here you&amp;#39;ll notice that in this case study i&amp;#39;ve taken you step by step through this showing you what the percentages look like how to change this in an alternative way from what i i just showed you guys the shortcut this is the longer way of doing it you can do it like this as well and what your table should look like in the end now what i want you to notice here is this is just a joint probability table where we have our joint probabilities here in the middle it&amp;#39;s easy to remember because the joint probability deals with two values or two variables jointly here on the side we have our marginal probabilities they&amp;#39;re in the margins and same thing there those are also marginal probabilities so this if i looked at this i can tell you the probability that a person has a home phone is 0.1125 and i get that directly from reading this off now these are our joints and marginal probabilities you could also go and calculate conditional probabilities by changing instead of percentage of grand total you just change it to percentage of row total or percentage of column total so that is um that&amp;#39;s basically what we showed you in track one as well so i did show you how to do that here now um for decimal places i saw that i saw tristan and people have answered thanks very much but i&amp;#39;m gonna quickly answer as well there number of decimal places always pay attention to what we tell you on the system so and follow that so if you see all of my answers are to four decimal places you give your answers to 40 small places if i give them to six and you have to complete a table you also want to give them two six so you want to stick with what we&amp;#39;ve given you there as well and then if you um your numbers is quite strict if you give less decimal places then it asks you for it&amp;#39;s going to give you zero because obviously you could have had a few different numbers that round to the same value so you always want to make sure you follow those instructions exactly to the letter um there was something else i wanted to also say about that um so with these questions i want you guys to also for this question go back to practical one and go review those questions that you had below your cross tabulation so there was a post on the discussion board where someone asked us how do you know which values to use what to divide by and that same question is going to come up with these questions so we are again going to ask you similar questions so i could ask you what is the probability that a person who lives in a caravan or tent has a home phone so you need to know that that is a conditional probability that i&amp;#39;ve just asked you for um so i&amp;#39;ll get to your question now chase um so yeah you need to be able to go from a question that&amp;#39;s asking words and identify is it a question related to a joint probability a marginal probability or a conditional probability and once you have that you will know which type of table to use and you&amp;#39;ll see that in your practical what&amp;#39;s going to happen is we are going to give you some variables that you need to use to construct a cross tabulation and then you need to change it to percentages of the grand total um rotators and column totals and then we are going to ask you joint marginal and conditional probabilities but we&amp;#39;re not going to tell you which one of the three tables to use to answer each one so that is the challenge of the question and seeing whether you can put theory and practice together so that&amp;#39;s very important now to swap those around the yes and the no tables uh columns i actually used a little bit of a shortcut here i right clicked on them and you can go say that you want to move the one that you clicked on up or down but since i only have two categories here yes and no if they&amp;#39;re in alphabetical order it&amp;#39;s going to be no first then yes so i just said instead of doing it manually i went here to sort and i said sort from z to a so i just reverse sorted it for me and that&amp;#39;s how i did it so fast um so just to go back again to the probabilities the probabilities here on the inside of the table are your joint probabilities because it&amp;#39;s a joint event there&amp;#39;s two things happening here so you have the dwelling and whether or not the person has a home phone to consider so it&amp;#39;s a probability that joins up two concepts now on the sides these are in the margins so those are known as marginal probabilities they&amp;#39;re in the margins and each of them relate to only one event so if i look at this um 0.887 whatever here it actually is just the probability of a person not having a homophobe so that&amp;#39;s in the margins it only deals with one variable if we did a conditional probability then obviously it&amp;#39;s conditional so the one thing is given the other thing is the unknown so you just have to identify what&amp;#39;s the given what&amp;#39;s the unknown so now let&amp;#39;s go back to more practical there&amp;#39;s one more case study to look at and we only have four minutes but we can do this um so this case study you&amp;#39;ll notice that we&amp;#39;ve said here work through the examples in section four of your back eye and last year i saw students were struggling a little bit with this question but if you think about the question it&amp;#39;s actually very straightforward so this question doesn&amp;#39;t have a specific case study now let&amp;#39;s quickly go back to your pack guide if we look at the prac guide and we go look at counting rules in excel i don&amp;#39;t know if you guys noticed but you can click on a um let&amp;#39;s just actually go up i don&amp;#39;t think you guys saw that but in your index here we&amp;#39;ve made it so that if you go to counting rules and you click on it it will take you straight there so you don&amp;#39;t even have to page through the document you can go straight to those so this one just goes through some counting rules and it has a few different things that you can calculate so you need to be able to do factorials you do that with the fact function combinations are done with a combination function permutations done with the permutation function and now you can see the person who set this up had their excel set up differently from mine because if you look at the separator here between the different arguments she had a semicolon where i usually have a comma so you need to pay attention to how your computer is set up so that you know whether you need to use semicolons or commas now um in this example it just takes you through how to calculate these and different software packages actually calculated differently but it&amp;#39;s literally one page that you need to go and read and you&amp;#39;ll see here for combinations and permutations we first tell you how many values we are trying to select from and then what our sample size is so when it comes to your questions that you&amp;#39;re going to get on click up for this we are going to give you a scenario and say you are interested in this group of people so you just then need to go and figure out how many people fall into that group and we will also then tell you on click up you want to sample so many people from this group so then you need to in that given scenario figure out is a factorial the right thing to calculate is it a permutation or is it a combination and then you use the skills you have that you guys illustrated very well at the beginning of the session you know exactly how to go calculate how many values are in a group you said it yourselves just go to a pivot table it&amp;#39;s a quick way of figuring out how many values are in a specific category that is in your population size we&amp;#39;ve explained that here and then you just need to know is it a combination a factorial or a permutation that you need to calculate and that is pretty much it so this question actually should be really easy marks you just need to combine the skills that you&amp;#39;ve had used in the past practicals and this one to go and do the calculations so i think that&amp;#39;s everything i wanted to show you guys the binomial probabilities like i said we will get to those the week after your semester test so if you want to you can if you&amp;#39;ve worked with binomial before you&amp;#39;re welcome to go try it but we&amp;#39;re not going to cover it this week um so the remainder of this week you can bring your questions for the first three case studies here as well as previous practicals okay so i hope you guys have an awesome day and we will see you in the lecture and like i said please bring your questions to the q and a sessions the rest of the week

Transcript for:Practical Session: Case Studies and Probabilities

Transcript for:
Practical Session: Case Studies and Probabilities