 it might be a very short one because we are done with content related fields. But however, I thought with this last session, we should just go through and we do at a high level revision just to refresh our minds in terms of what we have learned up to so far. And then I can answer any question that you have relating to your module if there are still certain things that you want me to respond to. Because you all have different exams and your exam papers are different. I've sat by different lecturers so we cannot do any, we cannot look at an exam paper because then it means I'm going to look at one exam paper and ignore the other modules maybe here because we have STA 1610, STA 1510 and 1501. So for that papers I will see if we can have an exam preparation session for individual modules so that then we can look at your exam papers, your specific exam paper and then look at how you tackle questions from that exam paper. But that does not take away the fact that for every exercise, every activity, most of them that we use, they are similar to what you will receive in your exam whether you're doing 1610, 1510 or 1501, it will be the same. If you're writing a multiple-trance questions, the questions the same that we have been using so far, eight months or so or seven months or so. Okay so we can start with today's session. Please remember to complete the registration, just post it in the chat. And remember if you have any questions relating to the content, if you need any assistance, please send me an email, my email address is eboy at jimesa.ac.az.za and copy city and tact. For those who have been participating in the tutorial session in week out, you do have in the next couple of weeks, you do have the opportunity to request for a one-on-one consultation with me. If you have not been completing your register, it might be very difficult for me to prioritize your one-on-one. So you need to make sure that you always have been completing your register because this is something that I'm going to offer to those who have been part of the sessions to get an opportunity to sit with me one-on-one online and we go through your concerns. You can book a session with me. I will send the link to how you book a session to your emails. I will ask Unisa to send you an email directly with the link to on how you join the session or how you book a session because there is a register and a calendar that you can use. So today we were supposed to look at the techniques of how to effectively answer questions and we were supposed to unpack the exam question papers. And like I said already, because you come from multiple modules, it's going to be impossible to do it in the same session like this. So what I will do is we will go into revision straight away and then we will go into question and answer. But before we start the today's session, are there any questions that you want to ask? Anything that you want me to clarify? You can go ahead. Anything? Anyone? Are you good? Are you happy? It doesn't help for me to be the only person talking in this session as well because I also want to know which area was the most difficult for you guys so that we can spend even more time. Yes, just this. How are you? I'm fine. I am driving. I will be in my destination maybe in 10 minutes time. But there was something which you touched when you were dealing with this presentation with regard to those the graphs or is it graphs something like that whereby on the other side it is going to the left and it is going to the right. I didn't have to go by that time. I'm not sure what is the graphs. If you can touch on them it will assist me. Are you talking about probabilities? Yes. Okay. I will reach my destination I think in 15 to 10 minutes. Okay. Yes. Are there any questions, comments? Things that you want us to spend more time on? Good morning everyone. Morning. On myself I would like for us to spend some time on the probabilities. Okay. And I also wanted to find out when I have a p-value of 5.66 I mean is that value sorry is that value of 5.66 what could the value of the p-value be? Okay. So it also depends on where are you are you talking about hypothesis testing right? Here we're not talking about we're talking about the hypothesis testing. So therefore it means you need to tell me whether are you having a one-tail test or a two-tail test because it depends on that as well. How you will find your p-value or determine your p-value as well as if it's a one-tail test is it in the upper side or is it in the lower side as well as what was the the z-value negative or positive? All those things you need to consider them especially for the p-value but we will get to that just now. Okay. Thank you. Any other question? So I don't have a particular question but I think throughout this entire module though the one or the few things that were but difficult were definitely the probabilities for me I didn't cross but as well as what I did with the other chapters that we were doing so I would want to spend more time on that as well. Okay. So when you say the probabilities are you referring to the basic probabilities or are you referring to the normal probabilities or discrete probabilities because we start talking about probabilities from study unit or basic probabilities. It was always on and it was by nominal. Okay. Really talking about the discrete probabilities. Makes it easy to understand which one you're referring to as well. You can go ahead. Apologies. I also wanted to find and with a normal distribution I know we did this part but I think I'm still struggling with that as well. When the z-value is greater than a negative number do we still go and say one minus the value from the small side from the negative side? Manis. Yes ma'am. I just want to find out which module are you doing? 15.01. Okay. Okay. Your table for the probabilities should be the same especially for the basic probability. For the normal probability should be the same with the negative and positive. I think what I'm struggling with is that I know when it's a positive value I would just go to the big side and say one minus the value from the big side. So now I get stuck when it's a negative value that do I still go to the small table? No ma'am. No. Remember also when we get to the normal distribution I will discuss this further. Remember it depends on what side was that. Did it say you need to find the probability of a greater than or did you say you need to find the probability of a less than? Irregardless of whether it's negative or positive if you got the probability of a less than the value you see on the negative side table or the value you see on the positive side it is that value. If you're going to find the probability of a greater than regardless of whether you find your z value it's negative or positive the value you find on the table you have to subtract it from one. Those are the things that you always need to remember. It doesn't mean that if it's greater than and it's positive you only need to subtract from one if it's negative you don't you still have to do the same but we will get to that and we will discuss. Okay so event you the only person who has answered anything is there anything that you're struggling with or are you happy? Okay so with that one you need to be able to know how to use your calculator properly right because it's just about substituting the values after you have calculated your expected value and then just calculating and remember it's your observed minus your expected squared your square the top part and divide by the expected and then add the next one and it's just like that it repeats itself so it should be easy and straightforward but we will look at that as well so that we can calculate it manually because I remember that we didn't spend more time on the manual calculation or on using a formula for the calculation we spent more time on using the template because my assumptions were by now you should know how to use a formula and do some calculations but yeah we can look at that as well so let's get to it I have prepared a couple of slides and then I realized but it's no use for me to go through the slides we can have a discussion so but we will just go through some of those slides and then we can go and dig deep into the concept that we all raised because I think that's where the majority of you have the challenges with okay so remember for statistics the questions in your exam they will be you will you are required to know the content right which means you need to know how to describe things how to how things are built what are the definition of certain things and what do they mean how to interpret that you still need to know those things on top of it you also need to know how to do calculations and it means therefore you need to always know which formula to use and when to use it and how to calculate it and those are the things you also are expected to know how to use your tables since there are statistical tables that we have introduced since from study unit six or study unit five when we started with discrete probabilities up until you your last study unit which includes chi-square where you need to be able to use your tables so you just need to know how to use them how to find certain things from those tables as well and how to read them you are expected to have a calculator to calculate because it calculations you cannot do mentally you need to be able to calculate them so starting from the beginning remember we start from the introduction of statistics most of you would have forgotten about all that because we did that right at the beginning of the year so you still need to remember that what the statistics what are the branches of statistics that we have which is the descriptive statistics and the inferential statistics which you still need to know how to describe them that descriptive statistics is about describing and summarizing and visualization of the data whereas the inferential statistics is about inferring or making decisions about the population right you also need to be able to know what is a population what is a sample and what are the measures that comes from a population which are called the parameters and what are the measures that comes from a sample which are called the statistics so that you should be able to know how to classify those that is some a sample is a subset of your population on top of this you need to be able to know and remember that from the population of the sample there are things that we collect like your variables which there are two types of variables numerical variables or categorical variables numerical variables also are called quantitative variables and you also have categorical variables which are also called qualitative variables on top of all these variables you need to be able to know that in qualitative variable there are two data or variable or data that we have we in qualitative variable and what type of variables are those those are variables that you can put into categories and for quantitative variables there are two which is discrete and continuous and these are variables that you can either measure or count so if something you can measure it means it is continuous if you count it it is discrete there are also levels within the variables of the data that we have in categorical variables there are two levels nominal ordinal you need to remember that nominal it's data that does not have any order ordinal there is an order or rank you also need to know that in quantitative variables there are also two levels of measurement or scales of measurement which is our interval and ratio one does not have a true zero and one has a true zero for example an interval like temperature it's an interval value because zero from a temperature is another temperature which is it it's cold but it can also go into the negative side a ratio for example ratio we deal with distance zero it has a true meaning meaning is if I moved zero distance it means I haven't moved at all right you also need to know from chapter two or the next level how to visualize the data that we just spoke about you need to know that for categorical data what are the types of variable the types of visualization you can do we can use tables pie chart and bar chart but you also need to know the properties of each one of them how are they built when we talk about numerical data you need to be able to know how to visualize the numerical data by means of ordered array it means sometimes you will need to order your data from lowest to highest and you need to visualize the data by means of frequency distribution histogram a histogram a polygon and an ogif and when you have an ordered array you can use what we call a stem and leaf and remember there is one stem many leaves and the stem and leaf plot can take many for a hundred stem and leaf tens stem and leaf hundred and sorry a thousand stem and leaf or a decimal stem and leaf you need to be able to know how to build each one of them and how to deconstruct one of them so that you go back to the original values as well and in terms of frequency distribution you need to know how to build the frequency distribution the rules that are the properties of building a frequency distribution which is a summary table for numerical data you need to be able to know how to interpret your histogram how to build the histogram how to build a frequency polygon or a cumulative frequency polygon which is also called an ogif in study unit three we learned about the measurement in terms of numerical data that how do we summarize the data by means of two ways of three ways the measures of central location measures of variation and the quantal in terms of measures of central location you need to always remember those three measures of central location that we talk about the mean the median and the mode when we talk about the measures of variation and you need to know how to define them and how to calculate them or how to find them the measures of variation you need to know how to calculate or find the range the the variance the standard deviation and the coefficient of variation as well and also going back to the median remember with the median you first need to find the position after you have ordered your data from lowest to highest and then find the position and where it falls the position falls that's what the median is and if the position it's between two values you take the average of the two you need to be able to know how to interpret your standard deviation for some reason not always but how to calculate them and how to calculate the coefficient of variation and remember some of this you can use your calculator if you are given the data and put your calculator on a state mode zero and just capture the data and calculate your mean your variance your standard deviation and you can also do your coefficient of variation when it comes to the quartiles you must also remember that the quartiles breaks down your data after you have sorted the data from lowest to highest into four parts where you will have quartile one, quartile two, quartile three which represent quartile one represent 25 percent of 25 percent of your data falls below that and your quartile two which is the same as your median and your quartile three it tells you that 75 percent of the data falls below that or 25 percent of the data falls above above that but you also not but actually you need to also remember that with a quart with the quartiles you you need to find the position and from the position you need to find the value using the value you can calculate what we call the interquartile range which is your q3 value minus your q1 value and you can also visualize the quartiles by using a box whisker plot box and whisker plot which gives you the five number summary the smallest quartile one quartile two quartile three and the maximum values okay when it comes to the probabilities I just want to come here you need to be able to remember that probabilities are always between zero and one and a complement of a probability is that probability that is not from the original one but it is part of the sample space and you also need to know that the sum of all probabilities should always equals to one and you need to know that to calculate the probability of a simple event we use the x divided by n which is the number satisfying that event divided by how many there are which are your sample space the joint probabilities if they didn't tell you anything about independent and so on you need to always remember that joint events of probabilities are number satisfying that joint event divided by the sample space you also need to know how to calculate the additional rule remember now in the example in anyway they will not tell you calculate the additional rule you need to be able to read the question and identify that this is your additional rule because they will tell you that you need to calculate the probability of a or b or either one event happening or another event happening and that is the additional rule and that is given by the probability if I have event a and b it will be given by the probability of a plus the probability of b minus the joint probability of a and b and remember that if events are mutually exclusive therefore it means they cannot affect one another then the probability of a joint event will be equals to zero and if that is the case then the probability of event a or b happening will be equals to the probability of a plus b for mutually exclusive events you also need to know how to calculate the conditional probability that is the probability of an event given that another event ate already okay and that is how what you need to always remember and that the probability of a given b is given by the joint probability of a and b divided by the probability of a given which would be b and if they tell you that event a and b are independent therefore the probability of a given b will be the same as the probability of a because b is independent from a whatever even whether b has already happened before has no effect on what is going to happen on a hence the probability of a given b will be the same as the probability of a because the event a and event b are independent you also need to know the multiplication rule that if you are given the conditional probability which is the probability that an event a given b had happened and they ask you to find the probability of a and b you must know that now you are dealing with conditional probability but it is a multiplication rule therefore the probability of a and b will be given by the conditional probability of a given b times the probability of a given unless if they tell you that the probability of a and b are independent therefore because we know that if they are independent the given probability will be the same as that probability that we want to find then the probability of a and b will be the same as the probability of a multiplied by the probability of b because the given probability of a given b for event a and b being independent is the same as the probability of a then the probability of a given b it's equal to the probability of a and b will be given by the probability of A times the probability of B. And that is the basic probabilities. So these are the things that are easy to find and calculate that I'm going to ask you. Remember also for the probabilities you can represent your probabilities by means of using a contingency table where inside the table you will have what we call your joint probabilities and outside the table where you calculate your totals you will have what we call the marginal probabilities or your simple probabilities. And this grand total will be equals to your sample space or you also need to always remember that probabilities are given to you in decimals, events are given to you in whole numbers. So you need to be able to identify those things in the question that this is probabilities and this is events so that then you know how to handle the question as well. So moving to the activities that you said you struggled with which you highlighted that sometimes you're struggling with your discrete probabilities. So there are two things in these probabilities and there is three things. Discrete probabilities is made up of three things. As always remember that it is your properties of the discrete probability where you will be given a table with x and its corresponding probabilities. You remember that right? Then you have also what's called a binomial and you have a Poisson. So with binomial and Poisson two ways. In the exam usually they don't ask you to use the formula to calculate but you need to know how to use the formula for Poisson to find the probability of divided by n is given by the combination rule times the probability times 1 minus the probability of n minus x. So that is finding the probability of a binomial. Otherwise you need to be able to use the table and remember the table has the left hand side values and the top probabilities and the bottom probabilities and the right values. Remember that right? That the top side probabilities from 0 to 0.5 you will find the value by using your n and your x from the left hand side. If you find the probability that it is between 0.6 or 0.55 to 0.99 then you need to use the bottom part and read the value of your n and your x from the left hand side or the right hand side. For a Poisson as well the probability of x is equals to your e to the power lambda or minus lambda times your lambda to the power of x. You need to be able to calculate that divided by your x factorial. You need to be able to calculate that probability by using that otherwise then you will need to use your table where your table is divided by the lambda value and the value of your n and x or your x actually not your n because with the Poisson you do not have nx. You need to be able to know how to find and navigate between the two. Now in terms of this at a high level this is straightforward and easy to do but there are certain things that you always also need to be able to calculate should be able to calculate the expected value of a discrete probability. The expected value of a discrete probability is calculated by the sum of your x times the sum the probabilities so it means you need to multiply and add all your x times its cross-pointing probabilities and add them together. You also are expected to know how to calculate the standard deviation of a discrete probability which is all the variance which is the square root of the sum of your x minus your probability your x minus your expected value not your probability your x minus your expected value squared times its cross-pointing probability. Remember whatever is underneath the square root it is your variance. This is your variance which is sigma squared. You need to be able to know how to calculate each and every one of them. Also I expected to know how to calculate the expected value which is your n times the probability for a binomial or the standard deviation which is the square root of your n times the probability times one minus the probability and remember anything underneath the square root is the same as your variance. You are also expected to know how to calculate the expected value of the Poisson which is the same as your lambda and which is the same as your which is the same as your variance sigma squared. Therefore your standard deviation you must know that it is the square root of your lambda. You should be able to calculate all that. On top of everything else remember the sides. How do we calculate the probability which I should have done it when I gave you the formulas? How do we calculate the probability you always need to remember that at least what does that mean? At most what does that mean? Less than what does that mean? Greater than what does that mean? And between what does that mean? And when it's between inclusive and exclusive what does that mean? So you need to be able to know all that in terms with regards to the side the mathematical side that at least it's greater than or equal and if you are going to use a formula therefore it means you will have to use this formula multiple times to find the aqueous because if it says it's greater than and if your n if your n was five and they say x is greater than three therefore it means four and five will be included when sorry if they say it's greater than or equals to three three four and five so it means this formula you will calculate for three you'll calculate for four you will calculate for five so you need to know how to use the equations or the mathematical functions at most it's less than or equal less than greater than between exclusive and inclusive inclusive it means it will have the both equalities if it's inclusive of a value um exclusive it will have a less than less than so it will depend because based on it can either be it's inclusive and exclusive or it can be exclusive and inclusive so depending you need to be able to know and identify what each one means in terms of that um and how do we find uh each one of them as well and still remember bring back the study unit for content that you've learned the probabilities so let's look at uh I just want to open a share my entire screen as well so I can just open I think I can open a it's let's open a table so that we can go to the tables and we can look at one exercise sorry since it's one of the things that you guys are struggling with let's look at that in detail when I share my entire stream instead of okay so yeah I've got 20 20 tutorial letter I just want to go through one of the questions that deals with um boy by Romeo okay so this is one of the questions so yeah we have autism South Africa has found that 50 of the people with autism disorder struggle with social interaction assume we randomly select six people living with ASD and use the information given to answer the following question so always remember when you deal with binomial distribution as well I'm not going to touch on discrete probabilities because this is one of those questions that I'm assuming that you guys by now know how to do that much so I'm going to the table um directly uh to the question so when you get a question that it reads like this where they only give you one percentage and they ask you questions but also in the exam you will be able to identify that now you are in this type of a question you will see your exam questions are in an order of your study units they are not going to be randomized the way you get a study unit 10 in question two and then study unit uh three in question 50 no they follow the structure of your chapter your study unit chapters so the first few questions will come for the first if they are uh two two two question per chapter the first two questions will come from study units one the first second two questions will come from study unit two like that like that like that and that is you need to be able to identify that when I read this question and I see these things I know which question or which study unit this comes from for example like this one so this you can immediately put the that is your probability of success which means it is binomial and we can answer the question and the other thing that they give you here is your end and in the question they will give you your x and immediately when you read the question what is the probability that only three people with ASD will struggle with social interaction you can just write there on your note the probability because they say that only then it is equal which is exactly um and then three therefore you can either use the formula or you can go and calculate using the table which is easier on the table with the formula you will have to calculate and c r and we'll use x and c x and minus and minus x so you will have to say six three times zero point five to the power three times one minus zero point five to the power six minus minus three and go and calculate that calculation otherwise you go to the table so the question there is zero point five so it means I can find it at the top of the table so when you go to the table you go look for binomial probabilities and that is the table and I can just update it and that is my table and there is zero point five at the top and I must just look for n of six and there is my n of six and the probability is x so I must just go and look for three x is three and that line and the answer will be that and that's how straightforward it would be if you have your tables and the answer will just be option three you can also take your calculator and calculate the same and it will also come to the same conclusion um so if it had said that um x is greater than three let's go there yeah is your question what is the probability that at most person with ASD will struggle with social interaction and by your n is six and your pi is zero point five what is at most that's um less than or equal to so therefore it means we need to find the probability that x is less than or equals to what then it means we need to find the probability that x is equals to zero plus the probability that x is equals to what so if I'm going to use the formula you will use six mc r a zero maybe I should keep the formula down instead of substituted values and see x a x one minus pi n minus x plus n c i mc x pi x one minus pi minus x so you will need to do both add them together so if there were three you will do the same have to go and find all three of them and add them together they will fall we will do the same substitute six zero zero point five one minus zero point five six minus zero plus times zero point five to the power one plus one minus zero point five to the power six minus one and you will solve that otherwise you go to the table we're looking for one and two so you go sorry zero and one which are the two two values there zero point zero one five and zero point zero nine three zero point zero five six plus zero point zero nine three seven you add them together what is the answer six plus seven thirteen three plus five is eight plus one is nine nine plus one is ten where one and the answer is the answer is right here on the side the answer is zero point one zero nine three which is option one that's how you will answer the question so looking at Poisson so you should be able to also be able to answer this type of a question your expected value or the same question and the variance remember I've showed you the formulas on how to calculate each one of them remember those formulas you should be able to calculate them remember everything underneath the square root is your sigma squared so you will use all three formulas to answer this question okay let's move to Poisson I'm wondering who answers the question Chantel you the one who said you were still struggling with how to use this table or was it you I kind of remember was that yes yes so are they making am I making it easier for you or is it still confusing right so that question that you just answered now when you use the formula would you still be able to answer that question by looking at the table yes remember this is how you would have found it on the on the formula if in the exam what I I can tell you is in the exam try by all means to stay away from using formulas use the table because for Poisson for that it saves you time as you can see that instead of me taking my calculator writing out this I could have just wrote this step which is the probability of x is equals to zero plus the probability of x is equals to one and go to the table and found those values because that's what I did I went to the table and I found those two values and I came here and wrote them here even though I wrote the formulas like that you are not going to be marked on how did you find the answer right you are not expected to show how you calculated it you're just going to pick if the answer is one two three or four or the answer is a b c o g that's the only thing that matters so use save time by using tables they create shortcuts and they save you time to spend it on other things as well okay so this is by Nomia for Poisson it will be similar so in the exam you will recognize it because there will be only one question from Poisson so you will not have a lot you'll have usually because you're you're writing um out of 25 like the possibilities are you have 25 questions right of those 25 questions you have 11 study units of the 11 study units therefore if they will be two question per study unit then it means you will be having 22 questions you are short of three questions there are some places where there are it you will be forced to have three questions past that unit and one of those is the binomial because with binomial you deal with discrete Poisson sorry it's that is discrete probabilities in discrete probability you deal with discrete binomial and Poisson in chi-square you deal with three things chi-square when the population standard deviation is known when the population standard deviation is unknown and for the proportion there are three in hypothesis testing you deal with three questions three statement the same way when the population is known when it's unknown population standard deviation and the proportion so there are three so binomial you might get oh sorry discrete you might get three questions uh confidence interval three questions hypothesis testing three questions that will make 25 but that does not mean it will always be like that because study unit one two three they always almost are similar in terms of questions they can be mixed up in those three uh six or four questions from that and that might mean that you might get three questions from either oh three question from either study unit um three way we deal with measures because you also have remember not only uh central tendency you have uh the measures of variation and you have your quartiles as well and so on so please pay attention to those kind of things as well but moving on to uh Poisson so in Poisson you will know when you got the binomial the next thing it's Poisson and in Poisson they talk about the mean and they always can also specify that this is a Poisson including also with binomial they might specify that this is a binomial and that will give you an indication of where you are so the minute they talk about the Poisson and they give you the mean you must know that that is your lambda and also you can either use formula or you can use the table to find the probability what is the probability that on any given day a neuropsychologist will consult with only one adult and also they is the that key weight only so if that is the only then it's equal therefore x is equals to one and they're asking you to find the probability that x is equals to one now in the exam they might not or they might you must pay attention to the information given to you so on this one they expect you to also at least know how to use your formula so we can write the formula remember the formula is i'm looking at how the option on here is i'm going to use the option they search it's lambda x times e to the power of minus lambda divided by x Victoria that is the formula otherwise you can go to the table so let's go to the table we're going to find a Poisson there is our Poisson table and we're looking for the 1.5 is our average it will be on the lambda values and we are looking for x is equals to one and x is equals to one we go to one and the answer is 0.3347 coming here you can see that the answer is not there don't select this and say maybe it's a typing error the answer is not there therefore it means the next best thing is let's substitute the values onto the formula so our lambda is 1.5 our x is one our e times minus our lambda of 1.5 divided by our x is one factorial hey as you can see there it is your answer exactly the same way so you need to be able to recognize things that are given make decisions quickly because in the exam you also don't have time to contemplate them and do a lot of things you just need to go in there being prepared for different scenarios that could happen because they might also not give you questions like this to calculate the probability they might ask you to explain Poisson or they might ask you to calculate the mean of Poisson or there's the variance or something like that so you should be able to know how to so let's look at the next one so we know that our lambda is 1.5 what is at least you need to be able to know what is the probability that on any given day a neuropsychologist will consult with at least seven adults so they is what is at least greater than or equal so it means you need to go the probability that x is greater than or equals to seven immediately before you even start typing there go to the table and look at the table so let's go there the probability of lambda of five where is seven seven is there the rest is eight go there and look at the values so it means I must take all that so it's up to nine remember that with Poisson the challenge with Poisson is for every lambda value I'll be there for every lambda value the table changes right so you can see from 1.1 to 2.0 your x value ends at nine for the next one if the if it was 2.5 the lambda value the x value ends at 12 so you need to pay attention to that so you need seven eight nine so because on the chances are on here because they don't tell you your n value so you won't know whether you need to find the probability that x is equals to seven that's the probability x is equals to eight that's the probability x is equals to nine because you would have went on and on and on and on if you didn't know to calculate so then you will need to go and use the formula if you want to use the formula I'm not going to use the formula on this one the probability that x is equals to a value we know that it is given by the lambda to the power x e to the power minus lambda divided by x factorial do you have any question just this I or someone I yes yes it is it is me can you go back to this table please because I I struggle on the table uh this is the table you say when we start by looking at it you we look at you mark the seven there and uh yeah on the on the vertical line one with zero one two three four five six seven then you went to there and it is under the lambda there yes can you explain how do you mark it because I am struggling okay so we are given we are told what our lambda is right it's 1.1.7 we are also told that we need to find the probability that x is greater than seven so you go to where seven is and we know that it is greater than so it means anything bigger than seven so I went and I highlighted let's remove the ink I think I can remove it this way so I had already highlighted where my lambda is so I went then I go to seven because that is where I need to start and I highlight where the values are and those are the values that I must be interested in just it's seven eight nine and I wrote here the probability of seven the probability of eight and the probability of nine which is zero point zero eight and zero point zero zero one so you just say zero point zero zero plus zero point zero zero zero that's how you will identify your answer and the answer is zero point zero zero five it is as straightforward as that so if they would have said it's less than seven I would have done the same but instead of highlighting all three of them I would have only highlighted the two so because it says it's at least at least means greater than or equal so it includes seven and anything bigger than seven okay this is Leslie thank you okay um oh sorry to interrupt you the full for 1501 our tables they are cumulative right your table is cumulative okay that's the other thing and I don't have your table how possible that I don't have your table your table is cumulative so you probably at this point on your table you are on zero point zero zero nine nine where x is seven and at that point where x is eight you are zero point zero zero nine two nine nine on on my table for eight for seven it's zero point nine nine nine eight you have five decimals no four um point point nine nine nine nine nine eight zero point nine nine eight zero point nine nine eight there won't be a zero are you sure you have four please read them again wait but my debts I might be writing them bro let's do this so on seven you have zero point zero nine no zero just nine zero point nine oh because it's cumulative right yes man wow so it's zero point nine nine nine another nine and eight on under one point five not nine nine nine yes man so on on on eight what do you have then and on eight is one say zero point one and then on six zero point nine and then on six nine nine nine one another nine and one nine and one okay so on your one because you have both of those seven and six and we looking for any value greater than so we need to subtract you need to subtract that value to that so that you can get the value but this is hundred so probably you need to take one minus one minus zero point nine nine nine one you will need to take the one before this value from six so you will say one minus the probability that x is equals to six for you to be able to get the probability of x greater than equals to seven what do you get when you subtract zero point nine nine nine one you should have the same that's what we have yes it's the same so you need to be able to know how to read your table of your cumulative table uh so therefore it means uh for you you need to know that if you are on one one includes the previous value plus that when you are on two it includes all three of them and if you need to only find the probability of two you will need to take your last value on two and subtract the value on one so that you can get the actual value for two and should give you the same so you need to be able to read your table properly and probably when we when we when we have a session on your example we can use your own table to explain because I think the challenge with these sessions I use most of the time the 1610 content because that's what I have at my disposal as an e-twitter for 1610 and I always forget about 150 I must ignore it of that okay we'll do that when we do the example may I ask one more question you can if if if it's x greater than or equals to four would it be correct then to say one minus x less than or equals to three uh since the tables are cumulative if the table yes if if the table are cumulative you will say you will do that so you for example you will take you will do the same thing as as this if x is greater than or equals to four then you will say one minus one minus the value of three that x is equals to three so you will always take the one above but if your question is it's less than or equals at most less than or equals to four therefore for you it will just be the probability of x is equals to four for a less than or equal right so if if your question was the probability that x is less than or equal because your table is already accumulative you just take the value you see on four but if it says it is less than four on your table then it will be different you need to treat it differently so if the question is the probability that x is less than four then on your table it will not be equals to uh four it will be the probability of x equals to three for a less than because it's cumulative when you are at three it includes three two one and zero right oh yes i see that thank you you see that cool cool cool cool cool cool and if they ask you that find the probability that x is equals to four then that's when you can say the probability of x equals to four minus the probability of x equals to three so that you can get the actual value for four so you take the value at the way you want minus the previous value that will give you the difference which will be your actual value that you are looking for so you just need to know how we can look at the scenarios at a later state but you can play around and see if you get it from your side so that is the study unit five so i can't imagine that we are still at study unit five so let's look at the other question you asked was you struggle and i think this comes from you from justice as well because for you you say you don't know whether when you have a less than and with justices it doesn't also know um which side the lower side or the upper side or something like that so how do we then do normal distribution so remember now we are in study unit study unit six which is normal distribution so now with normal distribution there are a couple of things that you always need to remember you have the probability of z less than a value and if the you need to find that you will find the value on the table you also have the probability that z is greater than a value regardless of what value is a negative or positive it doesn't matter then you will have to always say one minus the value you find on the table for the probability of z lies between two values a and b whichever value they are you will always say the table value for b minus the table value for a and that's how you will find the probabilities also remember the z value if they didn't give it to you as z value you will need to find the z value by calculating your x minus the mean divided by the standard deviation and that is what you need to be doing the other one which i'm going to include with the study unit six because these are the the basic things that you need to know it's study unit seven which study unit seven which is um sampling distribution so with that sampling distribution the same concept because we calculate in the z right it will be the same and remember also the equal sign it doesn't really matter that much with the with this two only when we deal with hypothesis testing then the equal sign the at least the greater than the at most those equal sign from those the equal and the less than an equal sign will matter when we do only hypothesis when we deal with normal distribution or sampling distribution they do not matter that much whether you use the less than the greater than you will still get the same concept so with with sampling distribution there are a couple of things that you need to to to to remember and do so here we have two scenarios that can happen you will do for the mean for the mean or for the proportion right but all of them you will be calculating the z so if i'm calculating the z value for the mean it will be your sample mean minus your population mean divided by the standard error divide by the standard error so remember this is your standard error which is called the sampling distribution and standard deviation which is the population standard deviation divided by the square root of a if you calculating it for the proportion then you will have your sample proportion minus your population proportion divided by the standard error which is your population proportion one minus the population proportion divided by n divided by n all of them they use z z z z z z you just need to calculate the z accordingly and then use to find the probability use the table to find the probability so we're going to look at all three of them because in the exam as well you will have normal distribution and immediately after normal distribution the minute they start mentioning the sample size then you know that you're doing the point the sampling distribution so let's look at how the questions look since even this is not the exam but you just need to be able to know how to move from one to the other and usually key words like normal distribution will give you an insight in terms of whether are you now in with the normal distributions okay so this one lucky for us we don't have to calculate any z so this is the question find the probability that z is greater than or equals to negative 2.88 irregardless of whether it was a negative or a positive we do not even worry about that because that tells us which table site we need to go oh the sign is just tell you which site that's it the greater than or equal tells you how how are you going to find it and the how part it is this this is the use which is the how the how right the how is how are you going to find from which side of the table so let's let's look at that which side the negative tells us which side we need to go to the normal distribution table which is called the cumulative standardized normal distribution and I guess you all have the same table it's called cumulative standardized normal distribution if it just says standard normal distribution table don't use that one because it's not the same as this needs to say cumulative okay so now we know that we need to be on the negative side and remember it's negative 2.88 so the two we will find on the side and the eight we will find it at the top right so that's what we know minus 2.8 and there it is and at the top we need to just find the last digit always remember the last digit two decimals last digit at the top where they both meet that's the video that we're looking for that easy yes sorry may I please ask you to to share that this table on the group if you can at this table is part of the notes I've shared it on the notes on the notes okay I'm going to look for it thank you if I if you want me to share it via here let me stop right here and because then I need to go into my I was thinking of printing it for the exams because sometimes the the p-value it's the z value is large and our tables don't go beyond three now in the exam they won't give you values that are not on the table they won't give you things that are outside of that so you just need to look at what you have on your some reason I cannot attach a document on no that is alright if they won't ask anything bigger I know I cannot attach a document on yeah only a link I can only add a link but not a document so yeah but anyway there are on the on the notes section if you go to the schedule thing where you get the recordings you will get the okay the the same table this is lazy as I indicated last week that my phone got lost then I have done a swim swap so okay it is not possible for me to access schedule three it's not possible for you to access the the schedule three because I my phone was lost then last week I had to do swim swap oh okay I'll send the link on the on the um I will re-send the the link for the recordings again thank you sis Liz so for some reason the table doesn't want to open it closed I don't know how oh it was just like somewhere on the side my bed okay okay so we were looking for 2.8 which is 0.0030 which is option one okay let's look at the between between minus 1.1 and 1.9 so remember the value we will find on the table for 1.9 minus the value you will find on the table or minus 1.1 and because this are one decimal you can just add a zero to the end so that you know that you're going to use the top first column you will use that column so since I'm on the minus side we can look for 1.1 and zero so just write that value it's 0.153 for the second one 0.13571357 and for 1.9 we go to the positive side and we look for 1.9 1.9 and zero which is 97130.9713 and you take the difference what is the answer 0.8356 option 3 and that's how you will find the value regardless of whether can you see now whether the value is negative or positive you should be able to go to the table and write the same answer this the A is any value it can be negative or positive remember the negative or the positive tells you how uh sorry it tells you where because it tells you whether are you going to the positive side of the table or are you going to the negative side of the table the how you're going to find the value it's based on the sign the negative the positive it doesn't matter so don't get confused with the bigger side than the less the and the smaller side because this tells you that this is the bigger probability of the less than value so if the value of z is less than this this is the bigger side of it because if it's positive because it's on the positive side it will be those values if it's negative z value because you can see that it's always the if it's negative it will be the smaller side of the thing but also this doesn't also distance this it's the smaller side from the greater than hands you take one minus that value when we deal with the greater than to get the smaller portion and the same thing if I need to deal with the greater than or the negative value you will take one minus the this value to get the bigger portion so don't get caught up with this is the smaller side this is the bigger side just remember the negative or the positive just tells you which which table are you going to be using the only thing you need to worry about is the greater than side or the less than side or if it's between okay so moving on we almost 30 minutes almost to the end of the session so let's see if we can get through some other study units so now let's move to study unit eight which is confidence interval so with confidence interval you have three things that you need to worry about the confidence interval for the mean when the population standard deviation is known or when the population standard deviation is unknown or for the proportions so when the population standard deviation is known therefore we use the point estimate plus or minus the z critical value of alpha divided by two times the standard error which is standard deviation divided by the square root of n when the population standard deviation is unknown we use the point estimate plus or minus the critical value of alpha divided by two and the degrees of freedom which is n minus one and your standard error for the proportion we use the point estimate plus or minus the critical value which will be alpha divided by two times the standard error which is given by the certain proportions now if your p is not given we know that we can use your value and the number satisfying divided by n how do we find the critical value you find the critical value on the table now this is where it's very very important to keep note of the following a z value of zero let me not do it that way a confidence interval of 95 percent interval you need to know that the critical value of z alpha divided by two for the for it is the same as 1.96 for a 90 percent you need to know that that is the one exception that we have the critical value will be 1.6 for 8 for a 99 it will be 2.33 for a 98 it will be 2.58 i might have the numbers vice versa let me double check i think the 90 percent is 45645 yes 645 i wrote 8 why did i go where do i get 8 from uh these two i might have the numbers vice versa yeah you got it swap 99 is 258 and 98 is is one 98 is 2.33 and 99 is 258 yeah there we go yes so you you just need to remember that that is for confidence interval it's easy to always remember all this but usually they use that all this those are the two that they will use mostly in the exam but the majority of the time this one with the big dot 95 percent they love it love it love it um so therefore it means you should be able to find that and remember the plus or minus is the same thing as finding the lower limit and the upper limit so if they ask you to find the lower limit you just use the minus the critical value times the standard error and that's that's all you need to to always remember just to substitute the values and know that you calculate in the lower limit or the upper limit and find that and always remember the critical value the only critical value that it's not going to be easy be easy to memorize and know that you need definitely you need to go and calculate or find that critical value it's for t so for for z for the proportion and for when the population standard deviation is known those two it's fairly easy if you can remember that this you will have to go and find it on the table so if they gave you a 95 percent confidence interval and your n was 24 or 25 let's assume that 25 so that then i can show um a 0 comma 0 5 divided by 2 which is 0 comma 0 2 5 therefore your t value of alpha divided by 2 will be 0 comma 0 2 5 and your degrees of freedom which is your n minus 1 which will be 25 minus 1 which is 24 will be given by you just go to the table and look for that critical value so how do we find the critical value you go to the t table and always remember on the t table not to use the cumulative value only use the upper tail values and we're looking for 0 comma 0 2 5 always remember your alpha value is divided by by 2 0 comma 0 2 5 and you go and look for 24 where they both meet that this way your critical value is which is 2 comma 0 6 3 life also remember that a 95 percent confidence interval is the same as uh 0 comma 9 5 is equals to 1 minus alpha and what is important is to find your alpha value which is 1 minus 0 comma 9 5 which is the same as 0 comma 0 5 hence i use 0 comma 0 5 there so you just need to know and remember all those things as well how to find your alpha value from given the confidence interval so let's look at the example at least one of them so now we were in the normal distribution we didn't do a whole lot of activities on that but for example if you want to know how do you move from a normal distribution to a sampling distribution in your question they would give you things like the samples because this is still the normal distribution where they will ask you to find the probability but i want to go to the one way they give you your n there you go so they say the mean is that the sample size is that and the mean is between and also keywords like sample mean and all that so that is what we did previously so but we want to go to the confidence intervals so with confidence intervals with confidence interval they will tell you that a confidence interval this or confidence interval that things like this they might give you to use so that you can compare and say which value is rich and we did this in most of the activities but i want to go to the one way you need to be calculating um see me as a flight here we go there we go so here is a question mava to a scientist to get a random sample of 30 adults and found that the reading time is normally distributed with the sample mean and samples standard deviation that is the other thing that you always need to remember how do you identify whether the population standard deviation is given or whether the sample standard or the population standard deviation is unknown is t words like this sample standard deviation then you know that you're going to use t if they gave you sample mean and population standard deviation then you need to know that you're going to use z how do you identify whether is the proportion sometimes they will tell you that a sample proportion you will know those things so now let's let's find this confidence interval the mean plus or minus we know that we doing t alpha divided by two and degrees of freedom times the standard deviation divided by the square root of n so you can go and find our t alpha and yeah they told us which is 99 percent do you still remember 99 percent is 2.58 2.58 but that is not the case because this is t you will do that in the exam you will get it the wrong don't do that to yourself so it's 99 if it's t remember it's not something that you can always remember we need to well find it which 99 is the same as 0 comma 0 1 divided by 2 and our degrees of freedom would be 30 minus 1 our t of 0 comma 0 1 what is 0 comma 0 is 0 5 or 0 0 5 0 comma 0 0 5 0 comma 0 0 5 and 29 let's go find this critical wave so 0 comma 0 0 5 and 9 and 29 0 comma 0 is the last column and 29 2 comma 7 5 6 4 2 comma 7 5 6 4 then you can substitute into the formula our mean which is our x bar our s and our n so our mean of 90 plus or minus our critical value 2 comma 7 5 6 4 times standard deviation 18 divide by the square root of 30 can also just expanded it 90 minus 2 comma 7 5 6 4 times 18 divided by the square root of 30 and 90 plus 2 comma 7 5 6 4 times 18 divided by the square root of page and you go and calculate the answer and that is the confidence interval unfortunately let an open mind online do you have an answer for the lower value it's 80.94 0 5 8 9 9 0 5 8 I'm gonna assume that that was that should have been the answer and that would be how you would have answered you would answer the question and identify certain things so I just want to get to the other sections and then we should be done because some of them we just covered them not so long ago okay so the next last two chapters let me just remember now we dealt with confidence interval and the next one is hypothesis testing hypothesis hypothesis testing and with hypothesis testing as well three things when the population standard deviation is known and when it is unknown when the population standard deviation is unknown all for the proportions so with hypothesis testing remember all seven six steps state your null hypothesis and your alternative hypothesis always remember the null hypothesis always has the equality side the alternative hypothesis it can state that it's less than or it's greater than or it's not less than all this are what we call a one tail test they help us develop which side of the rejection area will that be this is a two tail test right so those are the signs that can go on today and always use the population parameter so for the population for where the population mean is known we use the mean we use the mean and for this one we use the proportion when we state the null hypothesis right this is when we state the null hypothesis we always use the population parameter step number two we need to determine what is given what is given in the question you need to be able to identify your alpha value your n and whether the population standard deviation is no no no so things like that and if for example is for the proportions if they didn't give you your p you need to be able to calculate your p over n there um if it is um yeah only that and then step number three we need to state what type of a test statistic we are going to be doing so yeah you just say it is z it is z test or it is a t test or it is a z test that's all what you need they think the state the test statistic step number four is to find the region of rejection which is also called the critical values so remember the critical values also dependent on the side so for a less than your critical value or your region of rejection will be on the left hand side anything that falls here you reject the null hypothesis for a greater than it will be on the right hand side anything that falls here you reject the null hypothesis whether it's for the t or the z you do the same for a not equal the rejection area it is on both sides anything that falls here you reject h naught anything that falls here you reject h naught that is the step number four you need to find the critical value and remember now finding the critical value for one directional for a z for a less than or for a greater than the critical value will be z alpha or z alpha for a two-tailed test it will be z alpha divided by two for a t it will be the same t alpha t alpha and t alpha divided by two and don't forget the degree of freedom and for z it will be the same as the first one step number four is to calculate the test statistic you need to be able to calculate your test statistics statistics so therefore it means here you will need to calculate the sample mean minus the population mean divided by the population standard deviation divided by the square root of n or for t you need to be the population mean minus the population standard deviation of the sample mean minus the population mean divided by the standard error which is your sample standard deviation for the proportion you need to be able to calculate your z p minus the population proportion divided by the standard error and last but not least you need to make you need to make a decision and conclude how do we make a decision making the decision based on the critical value and the region of rejection if it falls in the region of rejection on the left hand you reject otherwise you do not and you stated back in relation to your null hypothesis right the other thing you need to always remember as well is the p value which is the other thing that you guys asked about how do we make decision based on the p-value now also with the p-value also remember the following for the less than we'll start there remember you calculate first the test statistic so the same concept that you have applied on the other side if your z value so for the less than we start there if your alternative hypothesis was less than then it means the test statistics that you are calculating there is the list of statistics now if it is the less than statistic therefore the value you find on the table that is your p-value irregardless of whether you find it on the negative side or on the positive side right the value you find on the table that is your p-value that is for a one-sided test what about when it is greater than irregardless as well if irregardless irregardless the value you find on the table will use the value you find on the table will be one minus the table value and that will be your p-value so you will subtract the value you find on the table and that will give you your p-value irregardless of whether it's on the negative or on the positive the only difference it is when it is not equal now when it is not equal so you will have to pay attention to what the sign you put on your alternative this is very important it's very very important this step the alternative so when it is not equal now this is where the negative side of the table and the positive side of the table comes into play when the sign says not equal and your z-value is negative when the z-value is negative and the alternative hypothesis is not equal therefore your p-value will be equals to two times the table value so it means the value you will find on the table you will use that value two and multiply it by two to get to the p-value or you can say table value plus table value you can add it to itself twice if your z-value is positive then your p-value will be equals to two times one minus the value you're going to find on the table because remember the table the the p-value contains the probability of greater than right of sorry of the bigger portion of your less than side so if already your answer is 0.999 if you multiply that by two you will get more than one so we because we're talking about the not the two sides we need to be able to split the bigger portion into the smaller side the smaller proportional side so we're going to take two times one minus the table value so we need to get one minus the table value that will give you your p-value and remember the decision okay the rule the decision the decision rule state if the p-value is less than alpha we reject the null hypothesis that value will make use of the hypothesis testing so these are the things that you always need to remember and put a bay in mind when you deal with hypothesis testing um what else nothing else from this side i'm not going to go into the activity and look at how but always remember that every step of a hypothesis testing can be one of the options on your statement as well that you need to to answer uh so when you look at hypothesis questions like actually this one they gave you the hypothesis testing in different so yeah is one example so you need to be able to know how to state your null hypothesis you need to be able to know how to find the critical value you need to be able to know how to calculate the test statistic you need to be able to know how to find the p-value and make decision and conclude which are all the steps that i just explained so you just need to be able to know all the steps but also the key things how do we do certain things how do i determine what my null hypothesis and my alternative would be based on the information given can i put a greater than or equals to in the in the alternative no you cannot because in your alternative hypothesis there can never be an equal side to it how do you get the the the region of rejection is it a one tail test or is it a two tail test do i need to say alpha divided by two those things are very very important especially when you do your critical value you need to know and be able to look at that so for z the critical value when it's alpha divided by two you already know all those things it's 1.96 it's 1.645 but for z alpha it's totally different so you need to be able to know how to find the critical value on the table on the table what we didn't touch now is that finding this critical value of z alpha alpha is the probability within the table if you go to your z table these are your alpha values in terms of the critical values so if you need to find z alpha of zero comma zero five you just come inside the table and look for zero comma zero five and you will find that and because remember this is one of those exceptions where we use the 1.645 it applies to the same thing as what we just did if you want to find z value of it will still apply to the same so it don't get caught up with how do i find z alpha divided by two is it different to z alpha over two the effect of the matter is the minute they put the z alpha the answer and you get it zero comma zero five it will be the same as if you would have calculated z alpha divided by two and you find your critical value by using the answer which would have been zero comma zero five it's the same thing but you just need to make sure that you know that you're using your value of your alpha correctly when to divide it by two and when not to divide it by two because you can make a slightest mistake on that you will use the wrong value you will get to the wrong answer so one slight mistake it changes everything so that is hypothesis testing the last two things that we discussed that we know and learned about are those two important things that we did with the template you just need to know your chi-square and your regression i'm not going to touch on both of them because it's something that we just did them two weeks ago one we did the regression we did last week as well so in terms of chi-square remember that if they give you a contingency table without the totals calculate the totals they are very important know that you need to calculate your expected value by using your row total times your column total divide by your grand total and to calculate uh somebody said they are they don't know how to calculate the expected value unfortunately we ran out of time but to calculate your chi-square test we use the sum of your observed minus your expected to square divide by your expected so if in your table that looks like this the values in here are called observed you use your expected to calculate your expected value formula to calculate your expected you just substitute them so let's assume that our observed values table was like a b and l and m was one and three and five and four you're expected you would have calculated it the same and found the expected right and you will have found a and b and l and m i'm just making rough things here let's assume that the expected year was 1.1 and the expected year was two and the expected year was six and the expected year was three right all what you do with this formula is just say your chi-square is your observed is one minus your expected 1.1 you square the answer you divide by 1.1 plus you go to the next say three minus two squared divide by two unless you go to the next five minus six squared divide by six plus you go to the next one four minus three squared divide by three you can do this manually by saying one minus 1.1 it's 0.1 square that answer which is 0.1 squared divide by 1.1 and get the answer plus you get the answer plus you get the answer and that will give you your your test but also remember you need to also go find the critical value so it means you go to the chi-square test find the critical value which is also similar to how you find the critical values 14 you will fuse the degrees of freedom and your upper tail area and not the cumulative to go find the critical value and also you can tell what type of a test it is it is a left skew test it's not a symmetrical test the other thing that you need to always remember with chi-square is to make a decision as well with regression we deal with two numerical values the scatter plot tells you the relationship you just need to know how to interpret the correlation of coefficients and how to calculate the coefficient of determination and interpret the coefficient of determination also you need to know how to use your a is equal a x let's go b0 plus b1 b0 plus b1 x which is your regression life how to estimate the value how to calculate the slope and interpret the slope how to calculate the intercept and interpret the intercept those are the things that we need to learn and practice and know how to do them and on that note thank you for coming are there any questions before we leave today's session does it just a question with regards to the class that you have tomorrow will we do revision in that class as well or will we be back in the beginning uh that's no you see you're asking me about our e-tutorial with e-tutorial we have our own plans so you need to check the schedule right we are still so this is different to to that how I handle the truth so with the class for tomorrow we deal with assignments what's assignment that we assignment five right we still we're still busy with assignment five okay yes thank you thank you so I will see you tomorrow yep and then for this one we are today was our last session and you will have to wait for the correspondence from UNITA to see if we're going to set up additional sessions and especially the exam preparation sessions I know that the timetable provisional timetables might be out it might change or might not if it's a final timetable so everyone would have received one you know when you're writing your exams when is your exams kick starting you know that October it's mostly like exams and UNITA might not offer additional classes for that but that does not stop me from supporting you like I said those who have been attending tutorials or academic sessions I might ask that they send you a link where you are able to book for one-on-one consultation with me um I don't have a lot of days and a lot of times but that hence I am only allocating to those who have been attending because then you have spent time and dedication for me to assist you therefore you will get priority in terms of the one-on-one consultation if I have to hold one other than that we will have an exam preparation based on your modules separately STA 1610 on a separate day STA 1510 on a separate day and STA 1501 on a separate day where we only look at your question paper and answer questions and deal with your own module content thank you for coming I will see you when I see you I don't know when happy let me enjoy your day bye thank you bye bye enjoy your day as well bye everybody