 Today's session was meant for probabilities and let me know if you can see my presentation. So since we are left with very few sessions, oh no, you don't have a lot of time for me to carry on and do more sessions because you are writing on the 10th of September. So I've decided to combine the probabilities, so we'll do the probability and the normal distribution, which is the Z score together. And then I also just want to extend it in terms of how you use your Z test statistic, which is the Z score, I'm just going to show you how to read your tables for when you're looking for the probability, especially when you do the hypothesis testing and you're looking for the probability of greater than for a one-take test or a directional test and so for a two-tail test or which we call it a non-directional test. So I just wanted to show you that process as well. So without wasting any further time, do you have any question before we start with the session? Anyone with a question or comment? None? None? All right, and we can start looking at basic probabilities. So by the end of the session you should be able to explain your basic probability concepts, you should be able to know how to calculate your probabilities, looking at the basic rules of probabilities, which are your multiplication rules, your mutually exclusive event for rules, your multiplication rule and your independence. And when we deal with the independence we're going to look at conditional probabilities, calculating a probability given that something else has happened and then later on we look at how we calculate the Z scores. So in order for you to know the probabilities you need to understand the concepts around them. Probability section in your module probably is the easiest one because you normally use your probabilities on a day-to-day basis, but in a nutshell when we talk about the probability we're talking about a study of chances. Here we look at a chance that a certain event will happen and that when we calculate the probabilities they should always be between zero and one. When you calculate and you find that the answer you get is 1,6 something, you must know that that is not the probability, there's something wrong that you did when you would calculate. Or when you are calculating the probabilities and you get minus 1,4 must know that that is not a probability. Probability value should always be between zero and one. And when a probability has a value of one we say that when we calculate in that probability the event that okay they were certain so it means that event was said to have or to always have to happen or it has to always okay like for example we always know that the sun will come out so the probability of the sun coming out will always be equals to one and also the probability of the sun going down it will always be equals to one. We also have an instant way that probability of an event will be equals to zero therefore it means that event has no chance of happening. Now I'm lost, I don't know which example I can give you in terms of where an event will have a probability of a zero. You will have a probability of a zero, let's say for example the probability of living on the planet sun. That probability is a zero, nobody can live there, you will fail. So we know that that probability will the event of living of that person living on a sun will be equals to zero and that is the probability of an impossible event that is an event that can never happen. So when we assess the probabilities there are several approaches that we can do, we can use what we call a priori which means the likelihood of an event happening will always have a finite number because then that event will happen and we also know about it, that it from experience or from the past that that event will happen and that is your priori and to calculate the probability of that event happening we use the number of events or the number of outcomes from that event divided by the total divided by all of the events that could happen. We can also assess them by using the empirical method which also looks at the likelihood of an event occurring based on historical data. So this one usually we use this when we calculate the probability like for example how will I know that the probability of a sun coming out and will always come out will be equals to one is because he's still told us that the sun comes out every day. So I know about that and that is empirical probability. Also they say we calculate that probability by taking the outcome satisfying that event divided by the total outcome so by all some of all the events together. For example also for priori let's say for example a priori we can use a coin. If I toss a coin I know that a coin can land on a head or a tail so when I toss a coin my number of events that a coin will land on a head it's one but the total outcome is because the a coin has two sides to it has the head and the tail so there are the total will be equals to two so it will be one divided by two. Then the last one the last approach that we can use is what we call a subjective method and this is when the researcher uses their own objectivity or subjectivity when it comes to calculating that probability. So it uses it's based on a combination of individuals past experience so there are some biases that will okay when you do the subjective probability. So individual past experience personal opinion and analysis of a particular phenomena or a situation and that will be a subjective and the probability that you will calculate there will always be a biased one because if you bring in your own feeling as a researcher into the research you're going to bring in your own biases into it. Okay so let's I've been talking about outcomes and events now let's define what those things are that I've been talking about. So when an event happen like tossing a coin I can call that process an event when an event happen it produces some outcome and an outcome of an event can either be a head or a tail if it's a coin so when I toss a coin I'm creating an event and the outcome that can happen from me to see that coin will be either the coin will land on a head or a tail those are your outcomes. So we use events to calculate the probabilities because we look at the outcomes and we sum all those outcomes and use them to calculate our probability. So a simple event is when you do one thing at a time a simple event will be tossing a coin I'm doing one coin tossing it that's a simple event and with a simple event it also have what we call a sample space and I will describe what the sample space is but in nature a sample space is the total of all the outcomes that can happen from that event we call those sample space. So when I create an event which is a simple event of tossing a coin and that event can land either on a head or a tail so if I say I want an event when I toss a coin and it lands on a head I'm going to assume that that event is my simple event because it's only one variable that I am looking at or one characteristic that I'm looking at and also I have what we call a joint event so a joint event is when you do two things or two things happening at the same time that we call it a joint event a joint event of going to school let me put it this way a joint event will be when I have two or more characteristics so it means from event let's say my event is rainy days so it's I can either record whether today is it rainy yes tomorrow it's not rainy no so I have two outcomes for the rainy yes and no days so yes for rain no for no day no rain on that day then I can also have if I work as an HR practitioner I can check the absence absenteeism of employees at the at the workplace so I can check so my my event will be absenteeism and I can check my outcome can either be yes they were absent no they were not absent so there are two events that I'm talking about I'm talking about a rainy day and absenteeism so both of them they can happen at the same time because someone can be absent because it's rainy and they are absent because it is rainy so those two are joint events happening at the same time and that is your joint event then we also have what we call a compliment event so if I have I'm going to go back to the scenario of using a coin if I have a coin and it has a head when I'm looking at the probability of a head or an event head the compliment of an event of a head will be the other one that is not in a head which will be a take so when we talk about probabilities we always also have what we call a compliment event a compliment event is another event that is not part of the original one but they all form part of a sample space remember all probabilities should be between zero and one and later on I will tell you that the sum of all probabilities should be equals to one so therefore it means if I add the probability of a head and the probability of a tail I should get one because one divided by two is 0.5 and one divided by two will be 0.5 and 0.5 plus 0.5 they equal to one so a compliment event will be the event that is not part of the original event or not part of the first event that you were looking for but both of them needs to be part of the sample space then we also have what we call mutual events so we also spoke about joint events I said absenteeism and rainy days those are two events when we talk about mutual exclusive events we're talking about events that cannot happen at the same time so those three events cannot happen at the same time and later on we'll learn what the probability of that is so for example I'm now I'm going to go back to my slides for example if I have a calendar for 2014 and I choose a day that is in January and I also choose a day that is in February so a day in January becomes my A and a day in February becomes my B so a day in January and a day in February cannot happen at the same time because one happens in January and the other one happens in February and that is what we call mutually exclusive events we also get what we call collectively exhaustive events and a collective exhaustive events are all events that make up your sample space so also if I have a day a calendar of 2014 let's assume that I have an A as my weekday and B as my weekend C as my January day and D as spring all of them can happen in my calendar because a weekend will be in 2014 and a weekday also exists in 2014 January is also another month in 2014 and D will be spring in 2014 they all can happen so all of them are together are collectively exhaustive but now if you look at A and B alone if I take a day on a weekend a weekday and a day on a weekend only A and B are also collectively exhaustive for the days in 2014 because all old days in 2014 either falls on a weekday or falls on a weekend and a day on a weekday and a day on a weekend are mutually exclusive because they cannot happen at the same day at the same time and A, B and D are not mutually exclusive and also A, B and C are not mutually exclusive because a day on a weekday and a day on a weekend can also be in January and they can also be in spring so they are not mutually exclusive but A and B are mutually exclusive and they are also collectively exhaustive because they all form part of 2014 calendar a sample space like I've been talking about the sample space is a collection of all events all of them so for example if I roll a die it has six sides to it so all six sides are part of my event they form part of my sample space a deck of cats has 52 cats in them 52 cats are my sample space because an event will be me drawing a king of hat or ace of diamond or drawing a ten of dice that will be an event that I will draw from 52 cats so and we can use the events and outcomes sample space which will be our total to create probabilities but before we do the probabilities and one you also to understand that you can also visualize the probabilities and we can use different methods to visualize probabilities so when you answer the probability questions try and use one of the methods because one of them might assist in visualizing your problem that you have and you might get the answer easier that way so you can use what we call a VIN diagram which is the diagram that we have been using where I was doing the illustration of joint events simple event compliment event what do I mean by that so you can use a VIN diagram oh sorry so way of VIN diagram your outside will be your sample space inside your sample space you can either have an event one event which will be a day happening on a Wednesday and have another day happening in January and we can see that they intersect at some point so they do combine because a day in January will also be there and a day I'm not drawing it right but you can see so the blue area is where they both intersect and that is our joint events because Wednesday is your simple event January is a simple event and where they both meet is what we call a joint event and we can calculate the probability of this joint event because if I want to find the probability of a day being on a Wednesday or in January so if I want to find the probability of a day in Wednesday or in that false on January that will be given by the probability of a day on a Wednesday plus the probability of a day in January minus because I don't have to recount double count those ones because they those values they will be in January and they are also in on Wednesday so it means I must take away one of them so that will be the probability of Wednesday and January which is the joint probability I should be able to calculate that but if a day does not fall on in January so this is for the normal one so for mutually exclusive events let's say we want to calculate let's say this is our sample space we have a weekday let's use weekday and weekends weekday and weekend I'm separating them with an nd and nk weekday and weekend so this is our sample space so if this is the scenario then we have a day on a week weekday and a day on a Wednesday day do not meet anywhere therefore that probability will be the probability of a day on a week weekday or the probability of a day on a weekend weekend will be given by the probability of a day on a week day plus the probability of a day on a weekend because then that does not exist because the probability of a joint probability of a weekday and and a weekend sorry and a weekend that will be equals to zero because it does not exist so then that will be your probability of a weekend or a week okay later on we will do this in more detail don't worry about it now then we can also visualize it by using a decision tree and with a decision tree for example if I toss a coin it's either going to land on a head or a tail so let's say for example let's keep one let's say I toss and the first time I toss it it lands on a head the second time I toss it it can either land on a head or a tail let's say it landed on a head we record that if it landed on a tail we record that at the end after we have collected all the information we want we can calculate how many outcomes do we have so for example now in terms of this one it's head and a head therefore the outcome will be a head and a head and this will be a head and a tail a tail and a head and a tail and a tail and in nature they are one two three four five six seven eight outcomes when you toss two coins we can also visualize it which is one of the tables that I like to use which is a contingency table because then on my contingency table later on I will show you you can calculate your simple event which means the probability of men can be calculated from there so this will be your simple event simple event women and a simple event promoted and a simple event not promoted and that is your sample space joint event of promoted and then you can find it on there and so and so on and so forth so I like I like to use the contingency table because it's easier to represent the probability and also it's easy to represent your probability using a vign diagram as well okay now let's learn how to calculate these probabilities so to calculate a simple probability we use a simple event because there is only one of them so let's say if I want to calculate the probability of a simple event I use an outcome satisfying that event divided by the sample space which is my grand total which is my N how many they are so let's look at an example like I said I like using a contingency table you will see that I use it more often so if I have I work at this HR department then I have the number of men and women who were employed at this company and I have a record of how many were promoted this year and how many were not promoted and I I know that in this company there are 1200 employees so if I want to calculate the probability of being promoted regardless of whether I mean in this company I want to know whether is a male or female it doesn't matter all I want is to know how many employees in this company were promoted and there are 324 employees that were promoted but I need to know the probability outcome satisfying the event divided by the grand total the event is promoted which is 324 divided by the grand total or our sample space which is 1200 it then gives us 0.27 number of employees that were promoted in this company that is simple event stop me if you're not understanding anything that I'm talking about so that then I don't just go go go go we will have some exercises later on as well joint probabilities and joint probability refer to probability of any occurrence of two or more events remember it comes from when we have two events happening at the same time so in order for us to calculate the joint probability we can use the same because if the outcome satisfying the event of a joint probability regardless of whether the other thing doesn't happen or not we know that there is a joint event there are two events that happen so we just use those outcome of those two events divide by the grand total or the sample space if I need to calculate the probability of men being promoted using the same data that I had previously so now it means I must go and find this joint probability for this joint event so to calculate the joint probability there I'm going to take the event satisfying the joint event is there are 288 I need to divide it by the grand total which are there are 1200 members in a nutshell if I use a contingency table I should be able to calculate my simple event and my joint event where inside the table inside my contingency table I should be able to calculate my joint probabilities because those are my joint events and I should be able to calculate my symbol probabilities because my simple probabilities are calculated from the total and in a nutshell up to now so far what you have learned are the following you have learned that a probability is the likelihood of an event happening and it is also calculated as a numerical measure and it is between zero and one remember that the your probabilities are between zero and one and if it has the probability of one we say that is a certain event if it has a probability of a zero we say it's an impossible event and if it's a 50 50 chance we say it's a moderate or a 50 50 probability it's a 50 50 chance what we also learned is all probabilities are between zero and one but we also learned that the sum of all probabilities should always be equals to one so if I add all events all probability all probabilities of the events happening I should get to one also have learned that a compliment event is an event or it is an event that is not part of the original event but they all form part of the sample space so it means since we say the sum of all probabilities should be equals to one therefore it means if I take the probability of a plus the probability of a compliment then a compliment we always use a subscript so that it becomes or it shows as a compliment if I take the probability of a compliment this site then it will be probability of a plus the probability of a compliment will be equals to one which is what we just said the sum of all probabilities should be equals to one so if you have a probability of an a and they ask you to find the probability of a compliment then you just say one minus the probability of an a it will give you the probability of a compliment what we also I need to remember as well is the probability of a joint event a and b will be equals to zero and it only happens if a and b are mutually exclusive if they are mutually exclusive then the probability of a joint event will be equals to zero and that is probabilities thus far so now let's look at how we use the rules we already introduced some of the rules but in terms of the addition rule which is one of the rule that we already introduced is we say the probability of an event a or b happening it's given by the probability of event a plus the probability of event b minus the probability of event a and b happening in the same time and that is for a normal circumstance if a and b are mutually exclusive because we know that when they are mutually exclusive the probability of a joint event a and b will be equals to zero then the addition rule states that the probability of a or b will be equals to probability of a plus the probability of b then sometimes you will have what we call the conditional probability that is the probability of an event happening given that another event has already happened and that probability is the probability of a given that the probability of b has already happened and that is the probability of a and b which is the joint probability of those three events divided by the probability of the given and we can also rewrite this if we do b given a we can write it as the joint probability of a and b divided by the given which is the probability of an a if and only if events are independent if and only if events are independent so event a and b if they are independent it means a has no bearing on what happens to b or b has no influence on a or a has no influence on b if that is the case if the two events are independent then our conditional probability if we're looking for the probability of a joint event because we know don't get confused with the formula as you see it because our joint probability remember it was a given b is equals to the probability of a and the is equals to the probability of a and b a and b divided by the probability of b you remember that all what we're doing is multiplying the probability of a and b by b and making a and b the subject of the formula so what we're saying is with multiplication rule for independence we're going to state that if we're looking for the joint probability of a and b and we know that the probability of a and b are independent when event a and b are independent therefore the conditional probability remember now we're talking about that the conditional probability will state that since b has no influence on a so therefore it means the probability of a given b will be the same as the probability of a because b has no influence on a then the multiplication rule will be the probability of a and b will be equals to the probability of a multiplied by the probability of b only for if they're asking you to find the probability of the joint event a and b then you do the multiplication rule pay attention the first one we did where we were talking about addition rule addition rule states that the probability of a or b either a or b are happening here we say a and b are happening so both of them are happening the the addition rule says either one of them is happening so that will be different because for a or b we know that it will be the probability of a plus the probability of b minus the probability of a and b or it will be if they are mutually exclusive then therefore it will be the probability of a plus the probability of b for mutually exclusive so two things multiplication rule you know that if a and b are independent how will i know if they say there are 11 boys in the class and eight girls in a class and they're asking you what will be the probability that we select one boy and one girl the key with there are those two what is the probability of selecting a girl one girl and one boy therefore we're looking for the multiplication rule what is the probability of selecting one girl and one boy multiplication rule what is the probability of selecting one boy or one girl then we're talking about addition rule you need to pay attention to the question okay so later on when we do the exercises from your from your past exam papers then you will learn how to do the um differentiate between the two which one do we use multiplication rule or addition rule and with that uh this is just an additional one for the independence we already covered the first one so if it's probability of b given a we know that b um sorry a has no bearing on what happens to b so therefore the probability of b given a will just be the probability of b because if a and b are independent then one event does not affect the other event in closing and in summary independent events happens when an event has no effect on the other event mutually exclusive event occurs when one event precludes the occurrence of the other so they cannot happen at the same time exhaustive events or collectively exhaustive events are events that has all sets of possible a set of events representing all possible outcomes so it means they should be part of your sample space then we also have what we call a law of disjuncture which is your editing a addictive law addition law i'm just going to call it that way it's the law when events are mutually exclusive and we also have a law of conjunction which is a multiplicative law which is the law regarding independence so remember the two so this first one is the probability of a or b and this one is the probability of a and b okay so let's look at the exercise in a population there are 450 people room 150 smoke what is the probability of randomly selecting a non-smoker so now here they say we have smokers and and we know what our sample space is so this is our sample space and this is our x in this instance so you can do these two ways remember smokes and smokers two sides like a head and a tail take it like that so you have two so you have smoker a smoker and you have a non-smoker so if they all are equal so we have a total day if i know that there are 150 smokers and they told us there are 450 of the people in this population so how many smokers are there 150 how many non-smokers will be there that is the compliment of so if this is a this will be the compliment so what they're asking you is calculate the compliment of smoker but because we're looking for the probability so we need to know how many smokers are there we can just say there are 450 minus 150 there are 300 so therefore in order to calculate the probability of non-smoker i'm a lazy writer so i'll just abbreviate and smoker is non-smoker there are 300 remember the probability of a is your outcome satisfying the event divided by how many they are so the outcome satisfying which is my x in this instance now will be 300 divided by how many there are which is my total of 450 and that is how much 0.66 are there no other numbers 0.6666 is recurring at the end yeah so you must pay attention because there you have two decimals and they say you must have only two decimals so they are carrying so if i keep those two this is six to be 0.67 to add one to this value and that answer will be 0.67 yes and that is option number one so remember we just take it up until the third decimal so six six six and do we stop at the third decimal to make it yes okay the law of rounding off says they look at the last digit after the value that you want to round off to and decide whether if that value to the right if the value to the right if i'm leaving two decimal i must look to the right the value to the right if it's greater than or equals to five i must then add one to the value i'm stopping at or i'm rounding off two i know that it's complex but that's that's how it is so you just look to the next door number and you add one if it's more than five five or equals to five more than five four equals to five okay which one of the following does not represent a probability number one two or three so in this instance if we were talking we said there are between zero and one and because there are decimals because this will be decimals so if i multiply the probability by a hundred then i'm turning it into a percentage so this will be so if i had my remember my probability line and if here at the middle it's 0.5 so this will be 0 percent this will be 50 percent and this will be 100 percent so now which of the following does not represent the probability of say two zero three no no i think it's three it's three because three has a negative the probability i i thought when i'm writing this i'm giving you so remember that the probabilities are between zero and one so and if it there is a probability of zero because for mutually exclusive event the probability of a and b not a heading is equals to zero because it does not exist is there so anything outside of that will not be a probability if if here they would have said 10 no 10 percent will still be no let's not say 10 percent if they would have said 10 and this was 0.5 without that in the minus number two will be correct because it's also outside of one it's bigger than one so it will be number two so you must just pay attention probabilities are between zero and one and we can represent them as percentages as well because they can be proportions and when they are in a percentage form they will be 100 zero percent to 100 percent thank you if 10 000 students were if 10 000 students rolled exam admission test 7 000 passed and they obtained a 50 percent or more and 400 obtained exactly 50 percent what is the probability that a randomly selected student will fail so now the same thing we know that student pass or they fail now the tricky part with this question is because they talk about pass and obtain a 50 percent or more and then they say pass with 50 and they also say you want you need to calculate those who have failed I know someone on WhatsApp did ask this question um and I tried to explain it in a quicker way so but now what you can also do is we know that the total all of them who have passed or failed got less they are 10 000 of them in this so we also know that 70 of them passed so we can put 7 000 7 000 of them have passed because this is where the tricky part on this one is it says how many of them have failed it be the same as the one that we just did so this 400 are included in the 7 000 in the 7 000 because here it says they obtained 50 percent or more so it means those who from this 7 000 if I split them so I can find out who got exactly 50 percent and who got more than 50 percent because I can take the 400 for the for the 50 percent exactly and the balance of 6 600 will be those who passed with more than uh 50 percent both of them will be 7 000 so now regardless of this confusing one we just use the 7 000 we calculate how many have failed how many have failed will be the 10 000 minus the 7 000 3000 which will be equals to 3000 therefore to calculate the probability of fail we say 3000 divide by 10 000 and 0.3 and that will be a probability of an event or caring which depends on something else or caring such as passing a test when you do not understand your cost can be described as a probability of something else or caring depending on something else happening can be described as number one which is conditional probability remember mutually exclusive events independent events events have no bearing on the other mutually exclusive events they cannot happen at the same time multiplicative events same as independent if we want the probability of the joint probability but for independent events then that will be our multiplicative probability that we can calculate this was not supposed to be there okay I will we'll come back to this when we do the normal probability questions if 10 000 students wrote a university admission test 7 7 000 passed and 3 000 obtained a 50 percent oh this is exactly the same as the one that we just did um I'm not going to ask you to calculate it or do you want to calculate it on your own let's see what is the probability that the randomly selected student will fail a test it's still number three because we're just going to take the 10 000 subtract the 7 000 to get the fail and that will be 3000 of them and we just calculate the probability of a fail which is 3000 the event satisfying a fail divided by the grand total or the sample space which is 10 000 which is 0 commentary still the same what I've noticed as well with your past exam papers are almost your questions are the same year after year after year the only thing that they change is the values like you remember on the other one it was 400 on this one they put 300 I like your teachers I like them very much they don't like to complicate them their lives and your life okay so we're done with basic probabilities now let's look at normal distribution it's also going to go as quick as possible because then we're not doing any exercises we'll only do the exercise late at the end of the session so a normal distribution is what we call a belly shaped calf and the distribution is symmetrical because the mean the median and the mode are the same so they all equals to zero and we also say normal distribution is distributed with the mean of zero and the standard deviation of one even though I'm saying it's distributed with the mean of zero and the standard deviation of one it does not mean all normal distribution will have a standard deviation of one because you will not you will see later on at the next when I do the next slide so this is for a standard normal distribution we say it has a standard deviation of one because we say the distance between the mean area and the outside of the calf should always be one standard deviation away so or the value should be one standard deviation away from the mean but it doesn't stop the calf it can still be a normal distribution calf if it's like that where the mean and the median are the same is symmetrical but the the distance between the mean and this the outside area of the calf will not be one standard deviation it can be half standard deviation it will be if it's if it's flatter than this let's use the same and let's draw another one there so you can see that there's distance there and they are different so it's the distance there is bigger than the distance there is bigger than this one there it's smaller than that one there is things like that also what you need to know about normal distribution is the area this whole area underneath the calf this hill underneath this belly shape we call it the probability it is always equals to one so the sum of all this area underneath the calf it's equals to one therefore if I split this normal distribution calf in half where the mean is this area there will be 50 percent or 0.5 and this area will be 50 percent so the probability and then which is the area underneath the calf so the area underneath the calf which we also call the probability probability is equals to one so half of it will be 50 percent this side 50 percent that side that is why when we do hypothesis testing you remember when you do hypothesis testing for the mean and you need to find the p-value which is the area underneath the calf and we use the value of our z-standardized score to go find the probability and that's what I'm going to show you right now so we will be finding these areas these probabilities underneath the calf so we also say when we look at normal distribution you need to also know that when the value of your mean increases if your value of your mean increases or decreases it will shift your graph from left to right so your graph will just move not like that it's not not with the point it will just move from left to right if it will move from left to right or some way it might move there it will move from left to right when you increase or decrease the value of your mean I know that I said the mean of a normal distribution is equals to zero so this normal distribution will not have a mean of zero because it has shifted it will have a mean somewhere there a standardized mean and why I'm saying this is later on we will learn how do we standardize all this information so that we can create a normal distribution calf so the mean here won't be zero because zero is here so let's say the mean here will be one for this standard deviation or for this normal distribution and the mean here will be two this will be minus one and this one will be two because this side is minus because if this in the middle is zero then that is a minus then when it comes to the standard deviation remember we already spoke about this I said the standard deviation either will be tall or short and it will be one standard deviation so when you increase or decrease it will spread your care so the bigger you you will see if the distance is bigger I'm not drawing it right but you can see that the the flatter the bigger the standard deviation the flatter your careful being the smaller your standard deviation the smaller your standard deviation the taller your care will be so when you move increase or decrease your standard deviation your spread of your graph will grow taller or flatter and that is in relation to the mean and the standard deviation now how do we standardize the value so we will be standardizing the value by using the z score or z standard normal distribution we can also use the z score to compare the values as well to see which one is better than the other if you want for the better weight of it to see which which module or subject perform better than the other is doing better than the other all the students are doing better than the other because with normal standardized normal distribution z score it will tell us in terms of the units how far apart are your values from your mean from your original from your from each other how far apart how far do your students score or differ within the group as well so we use the z where we take the observed unit minus the probability population mean divide by the population standard deviation and this is a normal distribution because then it is distributed with the mean of zero and the standard deviation of one so I'm not saying then go there and put one and go there and put zero no you will be given the population mean and you will be given the standard deviation and in the question that will give you your x and then you will use your x to answer the question like for example if x is distributed normally with the mean of 100 and the standard deviation of 50 what is the z value or the z score of x is equals to 200 so we know what our population mean is our standard deviation is and our x is given in the question we just substitute into the formula we know that it's z is equals to x minus the mean divided by the population standard deviation our x is 200 minus the mean of 100 divided by the standard deviation of 50 200 minus 100 is 100 divided by 50 we get 2.02 that is how far apart your observations are from each other or from the mean so we can say that the x of 200 is 2 standard deviation above the 100 mean that we're given and if we calculate another one let's say we find that the z of this was 1.5 we would say this one is much better because it's closer to is closer to one and it's closer to the mean so the smaller your z value the better the outcome okay so we'll look at the z score but I just also wanted to because I don't think we will have time to do to repeat the hypothesis testing because I think we started with hypothesis of two samples we never went with one sample or something like that so but I wanted to revise this part because I think it's very useful to know this so if I need to find the probability remember anyway yeah we're talking about the z score this is just for the z units converting them from the standard the units to a normal distribution yeah I'm talking in general when you do hypothesis testing you're going to calculate the test statistic which is the z score remember that you will calculate your z of x your mean minus the population mean divide by the standard deviation divide by the standard divide by the square root of n remember that that's what you use in that test statistic so I'm not using this I'm using the normal distribution which is the z score that we did but the logic is the same so if we need to find the probability of the less than you need to pay attention if you need to find the probability of a less than then you need to look at what the answer you got for the z score if your value for the z score is negative because we're looking for z less than a value that you got if that a is negative then on your table you will go to the smaller portion if a is positive you will go to the larger portion why I'm saying that so if the value of z is negative therefore it means we're looking for this red side which is the smaller portion so that is for that if your answer of z is positive therefore it means we're looking for that value which is in the larger portion you need to pay attention to that on your table remember this will be your z score that you would have calculated so let's say it's minus I'm going to use the value I see in front of me minus 0 comma 66 and we are looking for say we're looking for z less than minus 0 comma 66 so you're just going to go you ignore the negative because that negative tells me I'm going to my smaller portion so you go to 0 comma 66 which is that and you look there that's mean to z larger to portion and so that will be my probability that is if z is less than if z oh sorry if z was greater than 0 comma 66 then you will be going to the larger portion remember that let's get this example let x present the time it takes in seconds to download an image file from the internet suppose x is normally distributed with the mean of 18 and the standard deviation of 5 find the probability that your x is less than 18.6 so we go and calculate because we know that this is not normally distributed we want to standardize this we go and calculate our z value what we're given the mean which is the population mean and the standard deviation which is the sigma our x is given in the question the x 18.6 minus the mean of 18 divided by 5 when you calculate this we get 0 comma 182 now I need to look at this it said if my answer is positive I need to go to the larger portion now going to the table because I have standardized my normal distribution to my mean of 18 and standard deviation of the mean I've standardized it to a normal distribution now I can go and find this probability and finding this probability you will go to the table okay so yeah I've copied another table which is so yours looks different to this so we're looking for z of 0 comma 12 so you go 0 comma 12 and we're looking for the larger portion so you go to larger portion which is that so if I let me see if I go back to your table the way it looks 0 comma 12 so we do have it 0 comma 12 and the larger portion would be 0 comma 5478 and that's how you will find the probabilities so that was for the less than if it's greater than you also need to look at the sign if the value of your a is negative you go to the larger portion okay so if it's make sorry if it's negative you can see that for a larger z we put the sign the red shaded light on the side of greater than so if it's negative we know that you're looking for the larger portion for negative if it's positive we use the smaller portion let's get an example now find x the same probability now we just change our sign to a greater than so we know that when we calculated all this we found that the probability was 0 comma 12 I'm not gonna go back and calculate z of the mean again because we did calculate that and we found that it was 0 comma 12 and that is the answer we got so to get to the table you go because it's positive let's write let's redraw that again because we're looking for the greater so it means we're going to have our shaded side in this side the answer is positive therefore we're looking for this smaller portion so we go to 0 comma 12 and we look at the smaller portion and that will be the answer that you are looking for that is for the greater than if it was negative we will go to the larger portion now the other thing you need to take into consideration is if they ask you for the between if they say it lies between the two values then you need to know that you need to take either the difference between the two values so if we need to find the probability that x lies between 18 and 18.6 we can go and calculate for 18 we find that our z value is 0 comma 00 if we go to the table it will just be 0 I guess and we need z of 18 we calculated as well individually and we get that it is 0 comma 12 so now we can go and find the probability so but we're going to be using mean to z so when we use mean to z you've got two options you can either use the larger portion and subtract one from the other so yes subtract one from the other or you can use mean to z if you use mean to z therefore you go to the probability of oh sorry the z of zero and the z of zero comma one two you take both of them add them together and that will give you the same so if you go to the larger side you will have to take the bigger one minus the smaller one but I will prefer that you use the mean to z because it's easier you just go to the mean to z for the first one and mean to z for the second one take the mean to z values add them together and that will give you the probability of between and that is in a nutshell how you use your tables including also how do you find the normal distribution z scores so now let's look at examples of questions you get from your exam papers relating to only okay I must also clarify here we only looking at the z score the normal distribution z score not the probability I just wanted to show you the probability part of it okay so I know that we had one question up there but we can always get back to it once we're done with the other one a standardized normal distribution has a mean of and the standard deviation of option one option two or option three sorry number one it's number one it has the mean of zero and the standard deviation of one study the following figure representing the distribution of a variable the probability that specify the number drawn purely at random from a variable distributed like this is would fall in the gray area oh then here we can use also what we just learned now they say it would fall in the gray area which this would have been the gray area that will be equals to greater greater than so greater than x is equals to 155 choose the answer closest to the correct one from the points so yeah they're asking you actually to find the probability what is the probability which is what what we we just said so pay attention we're looking for greater than so yeah we say probability of x greater than 155 we're looking for that so since it's greater than remember if it is positive we go to the smaller side of the table if it's negative we go to the larger portion you always need to remember that so now we need to go calculate x is equals to the mean divided by the standard deviation and I think for your subject they use the x bar which is the same so we can use x bar here instead of the population mean we can use your sample mean and your standard deviation which is s mean one and the same because this is your sample standard deviation so your observation is 155 minus your mean of 150 divided by your standard deviation is five calculate and tell me what the answer is answer is one answer is one so we need to go to the table do you have your tables up you can open one of the past exam papers yes do you have do you have them up if you scroll right at the back of your exam paper there is a table called mean standard normal distribution table and okay let me open let me open it for those who don't have the paper in front of them stop sharing and show my entire script so you have this table can you see it I can open a better one okay and this better one this is the this is it okay so this the first part don't get confused this is zero comma zero zero this is zero this is zero comma zero one this is zero comma zero two because when you scroll I can't say anything I can't say anything you can see and share again and now nothing just a blank screen others say I can see yes I can see I can see yes I can see I can see okay so those who can see look out and log in quick quickly I can see it now I can see it now I'm just pulling things okay all right so what I'm saying is all this z goes here on your table these are zero comma zero comma zero zero zero comma zero one so if you're looking for one this is not one and this is not ten it's zero comma ten so you need to go to the bottom and there is your one so we're looking for the smaller portion so when we look for the smaller portion we go to that value there which is zero comma one five eight seven eight seven if I go back to the presentation zero comma uh what did we get zero comma one five eight seven five eight seven zero comma one five eight seven now around it up to two decimal zero comma one six zero comma one six and that's how you will find your answers sorry miss boy to distribute but this is also on page 165 on our study guides okay so you can go find the table on page 165 on your study guides appendix d okay for the video okay let's go on Joseph scores 60 percent in a history test the class mean now the class mean and the standard deviations are ten so here is one part for history test 50 percent in a biology test and they also give you the mean and the standard deviation use the z score to decide which statement is true relative to the rest of this class John does so we need to calculate for history and we need to calculate for biology what we are given like normal what is our x what is our mean and what is our standard deviation let me not even go there let me not write it here you can write it there so here we can write the formula we know that we will calculate z of x the mean of the standard deviation and we do the same you will need to calculate that so he scores 60 percent you can either change this to a decimal because it's 60 percent or you can use the 60 percent so this is your x this is the mean this is the standard deviation for by which let's start there so if I convert this to decimal 60 percent is 0 comma 6 you can also use it as 60 percent there is nothing wrong you can say 60 percent on your formula and when you calculate it will still give you exactly the same thing as when you convert it to a decimal and the mean is 65 which is minus 0 comma 6 5 divided by the standard deviation of 0 comma 1 0 calculate that 60 minus 65 divided by 10 what do you get history is 0.5 and wait don't do biology because we haven't got history 0.5 it is 5 it cannot be 0.5 it has minus 0.5 0.5 okay and then we need to do for biology biology he scores that was our x our mean is 53 and our standard deviation is 12 so we substitute 0.5 minus 0.5 3 divided by 0.12 minus 0.25 negative 0.25 so now you need to check the answers they say number one he did better in biology than in history number two they're saying he did better in history than in biology number three they say he did equally well in history and in biology I will go with number one number two uh number two number two so if let's say this is one or zero let's put it zero there this will be minus 0.25 and this will be minus five there so if this is the mean of the class if it's the mean if your z score is zero there so therefore this is closer to there therefore it means there is no difference in terms of performance between the class and Joseph because the other thing you can do also to check the problem is the probability you can go and find the probability of each one of them if we use the probability of more than or less than or less than we can check whether you did which one says the probability is more than the other and you can check on this one as well so if for example you're not sure about your answer in one of them which one is open we have 0.25 and 0.5 0.25 uh this is smaller let's go to the smaller it's 0.40 0.5 on this side 0.5 smaller is that so that is 40 percent and that is 31 percent so going back which one relative to his class did he do better in biology than in history or did he do better in history than in biology it's one better in biology he did better in biology than in history no it cannot be one because this we say below uh if we use below then it means he performed 40 percent below the class there and this says he performed 30 percent below the class so he did better in history where is it better in history than in biology that is that one that's number two the area under the care can i just ask a question yes on the previous yeah um if you just look at the the mean of the class for history there's five percent difference between the two and then with the biology test there's three percent difference between what he got and what the class mean is so that would be that there was a five percent difference in the history and a three percent by which means that the history would be better no remember you're not only looking at the difference between the means if it was like that straightforward then would have said he did better in uh in biology because the mean of biology is closer to the mean of the the class okay but we also need to take into consideration the standard deviation because the standard deviation tells you how far apart everybody in that group was so in biology there is a 12 percent standard deviation on which means 12 far away from the mean for each one of them from the class on average all everybody their mean was 12 standard deviation standard deviation away from the mean whereas in history the difference is 10 percent okay cool thanks and when you look at the probability as well remember he did 40 percent below average done the rest of the class remember we're looking at relative to the rest of the class so he actually did better in history because it's 30 percent better below the average class the area under the standard deviation calf equals it's mean it's standard deviation oh zero one the area under the standard normal calf equals number three number three it's equals to one Joseph's received 45 marks for psychology test the average mark for his test is 35 and the standard deviation is 10 what is Joseph's score so you need to calculate z we get x minus the mean divide by the standard deviation one i get one all the answer number one how do we calculate what is our x it's 45 and our gene is 35 and our standard deviation is 10 and then we say 45 minus 10 minus 35 divided by 10 and that will be 10 divided by 10 and that is one john's got 15 in english the class mean is 12 the standard deviation is three and 18 in geography the mean is 18 and the standard deviation is five use the z score to decide which statement is true relative to the rest of the his class john doesn't better in english than geography equally well in english and geography better in geography than in english so you need to go and calculate the z score for john english and for geography the answer is two for english what is our x it's 15 it's 12 standard deviation is three three divided by three which is one and in geography what is our x 18 minus 18 and our standard deviation and that is five divided by five which is one and the answer is number two that's well both ways the all and that concludes today's session but since we have 10 more minutes i just want to go on to one of the exam papers uh all this i send you the link this all exam papers i downloaded from from the um let's open one which is really one of them okay question 23 select the correct notation for the option below for the statement the probability value is larger than one and a half what will be the notation that you use what is one and a half to start with um yeah we're looking for decimal what is one and a half 0.5 0.5 so if that is 0.5 it should be clear which one they are talking about in this instance but a larger will be greater than so therefore it means is number three because then that is incorrect and that is incorrect i think this we did did we the in the population there are 450 people 150 do not smoke what is the probability of selecting a smoker we did that we did that because it was 300 no yes it was 300 divided by 450 we did do this as one of the exercise okay this is one of those things that we didn't do it talks to the central limit uh but when we do the exam preparation we'll go into detail on those ones so let's look for the questions that we did here is one a class of 10 boys and 11 girls and her friend elizabeth not me ah first time that they don't use john or joseph chooses a class representative by writing the names of everybody in the class on a slip of a paper putting those in a box and asking their teacher to draw one name blindly so there are 11 and 10 11 girls and 10 boys what is the probability that mary will be selected so it means what is the probability that one girl will be selected number three it will be one over 10 plus 11 because they make 10 plus 11 makes up your sample space so that will be one over 21 which is number three what is the probability that either mary or elizabeth will be selected number one now here is the thing because mary and elizabeth cannot happen at the same time then they it means they will be mutually exclusive and because this is mary or elizabeth therefore we use the probability of mary or elizabeth remember that was the probability of mary plus the probability of elizabeth and since the probability of a simple event is one over 21 over 11 they meant they are not 11 they are how many 21 plus one over 21 so we need to apply a little bit of meds because the probability are mutually exclusive therefore the probability of mary and elizabeth will be zero they cannot happen at the same time because they are both girls so doing a little bit of meds we're doing addition they have common denominator so our common denominator is one we just add one plus one and that will be two over 21 a college student claims that he can identify types of cheese by taste an experiment is set up to test his ability he is blindfolded and given three pieces of cheese and each representing a type what is the probability that he will incorrectly identify one piece of a cheese there is one out of number three so that will be number three suppose a height military recruit is normally distributed with the mean of 1750 and the standard deviation of 50 drawing a sample of 25 each we expect the standard deviation of the mean samples to be about okay so we didn't do this because this talks to the sampling distribution of the means we're not going to do this one now because I don't want to introduce you to a new concept that we didn't cover we'll do this when we do the exam prep let's move to the ones that we did cover and normal distributions are three two three two so I hear three two so spell shapes one or three or three or two okay those who said three do you still stick with your three what have I said about the standard deviation I repeated it so many times I even forgot now it's between zero and one it's yeah but we know that the mean is between the mean is zero and the standard deviation of one that is the normal distribution but I also said you can have a normal distribution for the mean so there's all of them are normal distribution but they have different means do you understand so this one might have a mean of 100 day this one might have a mean of 200 and this one might have the mean of of 12 similar happens with the standard deviation now I cannot clear the space and I have to delete the whole thing with the standard deviation I said one can have a standard deviation the the different standard deviation because if this is your mean the standard deviation for this one is there the standard deviation is there so if this one standard deviation is 0.5 this one is one and this one can be 1.3 different standard deviations even though they are barely kept so yeah it says they have the same standard deviation so the answer must be one Parin so the answer must be one the answer is number two it's symmetrical with the mean of zero okay number two is incorrect this is a that one that one we did okay during the interpretation of a psychological measurement the normal distribution is uh is this relating to something else yeah they just want you to tell them what normal distribution is is it an adapted to fit the observed frequencies of scores is it used as a theoretical model for interpreting the observed distribution of scores or three is used to calculate the relative frequency of the observed scores and that should be number three we did this one that one we did and I think times up so I will let you know when we're going to have our exam preparation probably I'll have to negotiate the time for next week so you're writing on the 10th so we can we can check on WhatsApp if you are available on Friday and Saturday we can do the two sessions one on a Friday and one on a Saturday um because next week's session on the 7th it's supposed to be the states session but I can negotiate that with the states 1502 then we can have a session again there because then I will have them for the rest of the month every day but I will communicate that because then uh the 7th is another group uh session for another class so I need to negotiate that with them first otherwise is there any questions or comments or input if they are none then we are done for the day just to recap what we've done there was a hand up oh I can't see the hands up let me stop sharing I can go to the chat as well yes you can know know my timber yes good evening ma'am how are you I'm fine thanks and how are you um good um I just want to say thank you so much um I was scared I thought I was even thinking of postponing this module to write it next year but um it's getting better and better every day every time after I attend the session however I just still have a little problem maybe next time just go forward or send me to download the calculator that calculator the one that you showed us the last time the last session okay on your phone yes ma'am because I'm using a phone most of the times we don't have electricity so we use I'm using your phone okay thank you so much you can contact me on whatsapp then I can help you okay ma'am all right thank you so much I've posted the the the register again on the link on the chat please make sure that you complete the register before you leave any other question or comments if they are none then enjoy the rest of your your evening we'll chat further on whatsapp thank you thank you bye bye thank you I will also post it on the on the whatsapp if anybody can't find it on the chat