 Click on that link and complete that register. Select basic statistics for human science, and probably it has your module code next to it. So every week you come in, make sure you complete the register and if you forgot, I will remind you closer to the end of the session or during the session just to keep track of everyone so that everyone is able to complete the session or the register. If you have any technical issues with the recordings, with the notes, or with anything relating to the platform that we are using, send an email to CTNTAT at unisa.ac.za. If you have content related issues like you've got comments or questions, you are struggling with certain topics, you can send an email to eboye.unisa.ac.za, alternative and copy CTNTAT at unisa.ac.za. There is also an opportunity for one-on-one consultations, but those we prioritize based on the people who attend the session. So if you come in every week and complete the register, you will stand a chance of your request being prioritized as well, because we also want to save those who participate as well. Not that we're not going to save everyone, but I mean we've got limited hours, so make sure that you attend the sessions, and if you have any questions, you can request for one-on-one session with me. So today we're going to deal with calculating the Z-score, and how we find the probability of a Z-score. That's the only thing that we're going to be doing when we are given the mean and the standard deviation, and how we interpret the Z-score. So therefore it means you need to have your tables with you, your probability tables, and I'm going to show you what it looks like. And if you don't have it, probably you need to find the old past exam paper or some way. Your lecture should have given you at some point, some way in one of the tutorial letters, the normal probability tables. But if not, then I will show you, and I can always share with you also where we post the notes. So then next week we're going to learn about the basic probabilities. It's very important to learn all this, because most of the things that we're going to be doing in your module, you're going to use probability. So if you know the concepts of probabilities, you will be able to sail through your module as well. So, and I must just apologize. I'm recovering from flu and fever, so fever and flu or something like that, similar to that. So my throat is coarse and itchy, and I will keep on clearing it. So please don't get irritated by all the noises you hear. Yeah, so we can start with this week's session. Remember, we're going to also always apply the problem-solving and humans prompts where we ask ourselves what is that we are given, what is the question asking us to do? What are the effects given in the question? Is there a formula that I need to use? Is there a table that I need to know how to use to answer this question? And then I can start doing the calculation and then giving feedback. We're going to do that throughout the session as well, because our sessions are skills sessions. So in order for you to complete this session, sorry, you require formulas and you require a calculator, and I forgot to put there, you require a table, a statistical table. So when we talk about the Z-score, we want to actually normalize your data or standardize your raw data. And with that, we talk about normal distribution. And when we talk about normal distribution, it's a belly-shaped calf, and you will notice that sometimes in your assignments and in your exam, you will be given questions with a belly-shaped calf, and they will ask you to answer certain questions. So it's a normal distribution that they have given you and they will expect you to answer questions based on that. And a normal distribution, it's a distribution that is symmetrical. Therefore, it means the mean, the median, and the mode are always equal for a normal distribution. And the location of your normal distribution is determined by the central tendency measure, which is the mean, which is always equals to zero. And the standard deviation, which is one of the measure of dispassion, which is denoted by one, which will be equals to one. When we talk about normal distribution, we also talk about the area underneath the calf, because like I said, it's a belly-shaped calf. So we always concentrate on the values that are underneath the calf. And the calf will never touch the X-axis if this is our X-axis and the Y-axis is our probability or our function, our density function. So if here we're representing probabilities, therefore we're talking about the area underneath this calf, we should be able to calculate the probabilities of those area. And that is what most of the time when we do probability questions, we will be calculating those probabilities, which are your areas underneath the calf. And that area underneath the calf, it's always going to be between zero and one. It can never be negative, and it can never be, no, it can never be negative, but it will be between zero and one. And here I'm talking about the probabilities. Next week, when we deal with probabilities, we will still cover some of these concepts as well. So today we're not gonna cover probabilities in too much detail, but we just brush it off. So the probabilities should always be between zero and one, and the sum of all probabilities should always be equals to one. It can never be more than one. So the sum of all probabilities. So if I add all the probabilities underneath the calf, they should give me a sum of one. The other property that you need to know is if you increase the value of your standard deviation, which is denoted by the sigma, you will notice even when we do the z-score, we're going to be using some of these letters. So this is called sigma. So when we increase the sigma, or when the value of your sigma changes and it increases, then it means your spread or your care also increase in terms of the shape. So if this is one standard deviation, because we set a normal distribution, it's one standard deviation away from the mean. Another graph, if we have a two standard deviation, it will look like somehow like that because the distance between the mean and the outside of the calf of this red will be two standard deviation. And if it moves again, and it might look like this, it might look like this, because then the distance also, you can see that it has increased by x amount. So when your sigma, when your standard deviation increases, your graph either becomes narrower or it becomes flatter. So the smaller your standard deviation, your flatter the calf will be. And it will be like that and like that all the time. And when it comes, and when actually I was drawing it, yeah, so the bigger as it increase, your graph will become flatter. And as it increase, the graph becomes flatter. And when the standard deviation decreases, the graph become narrower. As your mean changes, it shifts only the graph from left to right. So the mean shifts the graph from left to right. So it just moves left to right. And that is your normal distribution. In order for us to standardize the raw data, we use the formula, the zed score or what we call zed distribution formula. And your zed score is just your raw score minus the population mean divided by the standard deviation. Your population mean, you will be given in the statement, your standard deviation, you will be given in the statement. In the question, they will give you your raw value and they will ask you what is the zed score or they will ask you whether there are differences because you can use your zed score to determine whether there are gaps or differences between groups as well because you can look at the bigger the score, the bigger that your zed value, it means your data has more of the outliers. The smaller your zed score, it means your data is more closely to a normal, it doesn't have a lot of outliers as well. And an outlier will be the value that is far away from the rest of the other values as well. Then always remember that your zed score is distributed with the mean of zero and the standard deviation of one. These are some of the properties that you need to know in order for you to answer some of the questions in your assignment and in your exam as well. So how do we standardize the score? Let's look at this example. If x is distributed normally with the mean of 100 and the standard deviation of 50, the zed value for x of 200 would be. So we need to go back and ask ourselves what is it that we are asked to do? So we are asked to calculate the zed value. So it means we need to calculate the zed score. What is it that they have given us? So we need to identify our fact. The first fact we can identify is that the mean is equals to 100. So therefore our mean we know that it's denoted by mu. We did discuss this last time. Our mean is denoted by mu. So we have our mean, the standard deviation of 50 and our standard deviation is denoted by sigma. So since I've identified all the facts, but not all in the question, the zed value for x, so they gave me my rose score x. So I don't even have to identify it because they already made it easy for us by giving us the x value as a letter x. So I need to identify what are the things that are going to help me answer this question. The first thing is knowing the zed score formula. So I know that I'm going to use the zed score. So yeah, it is my zed score. I just need to substitute already if I identify my values. So I just substitute into the formula. Your x is 200, your mean is 100, your sigma is 50. I just substitute, then I take my calculator and start calculating. Okay, I do have an online calculator. Let's hope it's not going to disappoint me and it's going to work. And there is my calculator. Depending on the type of a calculator that you have, I use a Casio calculator. If you have a sharp calculator or a normal calculator, the calculator on your phone is still fine. You can still use that. You don't need a fancy calculator to do your module. But it's very important that you invest in your studies. So one of the investment is going to buy, go and buy a Casio calculator. Because when we come to the regression and the correlations, you will need something that will simplify your life. But I will share that with you in no time when we get to that point. So my Casio calculator, it has the fraction. So if you go to the shop and you want to buy a similar calculator, you need to look for the one that has a fraction button, even though it's Casio. Sometimes it's expensive. Don't buy a too expensive calculator. Any calculator, even if it's not a Casio, it can be Casio. In shop right, they sell a Casio. It also has a fraction button like this. They also have non-brand calculators that also have the same. You can buy those ones. You don't have to buy a 500 or 600 calculator. So because my one has a fraction, I can use it to do the calculation. 200 minus 100, divide by, then you need to also use the arrows, divide by 50 to go down. And the answer is equals to two. And that is my answer. And that's how you find your Z score. If you don't have a Casio calculator or you're using a normal calculator from your phone anyway, always remember to use your calculator profile. So 200 minus 100, because it's the top part, I need to say equal. If you don't do equal and just go ahead and do divide, the divide will divide 100 by 50, not the answer. So you need to say equal so that you get the answer for the top part. Then you can say divide by 50. And the answer that you will get will still be the same as two. Alternatively, I'm not gonna do this for all the slides. I'm just showing you at the beginning. Alternatively, you can use the bracket for the top part because there are two values, put them in the bracket, 200 minus 100, close the bracket, divide, because the sign here is a division. It says divide by 50, then you just divide by 50. What the brackets does, it will solve whatever is inside the bracket first before it does the division. So if I click equal and I will still get the same answer. And that is one of the skills that I can show you on your calculator. Coming back to statistics. How do we then interpret this? We can say that our X of 200, it is two standard deviation or two increments of 50 units above the moon of 100. And that's how we can interpret the Z score. Are there any questions before we look at more examples from your module? Remember, I cannot see the chat when I am discussing. So if you have any question, please feel free to raise your hand or sometimes I won't be able to see my hand. No, you need to look for, on the register, please look for basic statistics. Basic statistics, what is it called? Basic statistics for human science. If it's not there, probably it is called the rich search analytics. But if it's not there, it should be called to research analytics because I remember the last time we were calling the session, research analytics. Okay. Yeah, so you just need to find the right one. Okay, I see that there are no questions. So let's move on to the exercise. So this exercise comes from one of the tutorial letters. I forgot to take the name of the tutorial letter, but I think it was 2021. It was 2021, tutorial letter. So it reads as follows. The max in the different subject of Patrick and high school people are represented in the table below. His marks for different subjects are given along with the mean and the standard deviation of each of the subject for all the learners in his class. So they took the class mean or the class average and the standard deviation of the class marks. Use the information in this table to answer the following questions below. In which subject did Patrick do best relative to his class? So we need to calculate the Z score for Patrick for all the subjects. There are four subjects. I'm gonna do one. You're gonna do two, all three. And then we're gonna look at the scores afterwards, right? So I'll start with the mathematics one, so that I show you how to do that and then you can do the rest. The Z score for math. I'm just gonna put the math. It's given by, we know that the formula is X minus the mean divide by the standard deviation. Our X will always be, our X will be Patrick's score. Our mean for every question. I'm sorry about that. I'm just wiping my noise. Our mean and our standard deviation. So we just going to substitute the values. Because these values are in percentages, it doesn't really matter whether you convert the percentage to decimals, but since I'm a mathematician, I would prefer to convert them. So a 56% is the same as 0.56. You can use 56% if you want. The mean of 42 minus 0.42, because I just divide 42 by 100. My percentage, if I don't want to use a percentage, I just use, I just divide the percentage by 100 to convert it to a relative value. Divide by my standard deviation of 0.06. Because 6% if I show you, 6 divided by 100 is 0.06. So if you have a case show and you get answers like this in a fraction format, sorry, you can just press the SD, it changes your answers to decimals. Can you please mute yourself, whoever is unmuted and they are busy? Thank you. So now we can just calculate. Just gonna move my calculator this side this time. 0.56, 0.56 minus 0.42, not that. 0.42 equals, and I can just divide the answer by 0.06 equals. On your calculator, you don't have to put 0,0, you can just put 0.0 some number, it's also know that that is a zero. So yeah, the answer is 2.33, 2.33. So do the rest, do for science and for geography and then we will come back to it just now. I'm gonna give you five minutes. Or if it left, just try to do it. There is no right or wrong answer. I'm just going to show you because practice makes perfect. The more you try, the more you do, the more you learn. Don't just wait for me to give you the answers. Are you winning? Yes. That's good. Are we done? Are we still busy? I'm done. Okay, two. Are those? Yes, I'm done. I hear no one saying I'm still busy, so I'm going to assume that everyone is done. So then let's calculate for science. Who wants to do that? I'm not putting you on the spotlight if you want. Okay. It's 0.54 minus 0.54 because it's 54 and 54 divide by 0.04. Now, 0.54 minus 0.54 is equals to zero. Zero divided by any number will stay the same as zero. So it's not gonna change. So we know now the Z is zero. For geography, anyone who wants to try? Nobody. No, I'll do it. Okay. Okay, I need to be speaking. Okay, so I have geography. I didn't use a zero point. I just used 62. Yeah. 62 minus 50. It gives me 12 divided by eight. It gives me 1.5. 1.5, it will still give you the same whether you use the zero point or you use percentages will still give you the same. And for these papers, for this one, not converting can still get you there, but in a proper, if I need to teach you the right way of doing things, you need to convert it through a decimal. But for these papers, it's still gonna give you the same. Okay, so who wants to do history? Next time. Yes, who wants to do history? Okay, history will be 0.68 minus 0.75 divide by 0.05. And so I'm just gonna use my calculator, 0.68 minus 0.75 divide by 0.05 equals and change. And that is minus 1.4. So we do have all our z's. I'm just gonna put them close to one another of 1.5 and z of minus 1.4. So looking at this, how you will interpret this, remember how we interpreted the other one, the previous one, we said it is two standard deviation, which is two increments of the 50, away from the mean, right? So with this one, it's the same thing. So it will tell you how far is Patrick's score from the rest of the class. The close side of z score is, it means he's got almost relative to the class. If this z score is bigger, it means he's got better than the class. That is how you can interpret this. So if we look at this from the process of elimination, history, he's got less than the average class, right? But it's 1.4 less than the average class. Geography scored 1.5 above the average class. So it's 1.4 and 1.5, almost similar. Zero, it means they scored almost the same. His scores and the average class are exactly the same because there is no distance between the two. Mathematics, he's got 2.33 away from their class. So it means in order for us to answer this question, which subject did Patrick do best relative to his class? The answer would be drumroll. Which one? Maths. It will be maths because he performed way much better than the rest of the class because his scores was two standard deviation away from the rest of the class. And that's how you will answer the questions. So the challenge also with multiple choice questions is that sometimes you have to work three times or four times as much before you can get to the answer. And here is one of those cases. You spent almost five minutes to more than five minutes to answer this question. So now let's look at how we can find probabilities because probabilities help us also to give us the proportion. He tells you what is the exact proportion of the differences or the proportion of those raw scores. And we use that by finding the Z score because if we find the Z score, we can use the Z score to go find the value of your probability. So for example, like I said, the probability is the area underneath the kef. So I'm gonna run through this as quickly as possible and then we're gonna get to the exercises. So since we know that the area underneath the kef is the probability, if we need to find the probability of Z score less than a value, which will be the value that we get, which is these values are what we're going to call A. All these values are what I'm referring to as A's, right? So if we need to find the Z score of less than a value, it's very important to pay attention to this sign. If they say less than or equal, less than or equal, less than or equal. We can also use equal, less than or equal. You need to read your table as follows. If the answer of your Z score, colleges, if your answer of your Z score is negative, let's look at our previous question. Like for example, for history, it's negative. If the answer of your Z score is negative, the question was, find the probability of Z less than. Then we're going to look at the probability or we're going to locate the probability on the larger portion, oh, sorry, on the smaller portion. What do I mean by smaller portion? I'm gonna show you the table just now. If the answer is positive, like 1.5 is positive, 2.3 is positive, then we're going to look at the value from the larger portion. What do I mean? This is your past exam paper, 2018, May, June. There is a table called appendix probabilities associated with standard normal distribution. On this table, it has the Z score, which are your Z value. It has mean to Z. It has the larger portion and the smaller portion. The mean to Z, we're going to talk about it later on. The larger portion is what I was referring to. So, going back to the presentation, if the score is negative, we go to the smaller portion. So it means if we're using this as an example, history, the probability of Patrick's score for history, we can go and find it where Z score is 1.40. We need to add zero at the end, so 1.4. So it means we need to go to the table. We need to look for the smaller portion. We need to look under Z values. We are looking for 1.4. And sometimes it can be 1.40, depending on your table. As I can see here, your table has two decimals. The space is a decimal in between. Sometimes UNICEF's papers, when you print them out, because they are scanned, they miss the decimal point. You can see decimal, it appears on some, but not on all. So when you see a space, know that there is a decimal point in between. So we're looking for 1.4. 1.1, 1.2, 1.5, 1.4, there is 1.40. So we're looking for the last block, which is that one. So that's how you will find the probability. So the score is just the probability of Patrick's scoring less than the rest of the class is just 0.8% because we're looking at the smaller portion. So what about the probability of scoring less than 1.5? You do the same. Because 1.5 is positive, remember the thingy. Let's go back to the presentation. It says, if the Z score is positive, we go to the larger side, right? So we know that we're looking for 1.4. So we need to go to the larger, just need to go back to my tax, 1.5. So this is 1.53, so it should be on this side. So 1.5 will be 1.50. And we're looking for the larger portion. Therefore, it says they would have scored 93% above the, less than the rest of the other classmates. So that's how you read the table. So if the score is, if the sign is less than 1.5, and the answer to that sign is negative, then we're going to look for the larger portion. If it's positive, then when it's positive, then it means it's on this side, because remember in the middle it's zero. So any value this side is negative, and any value this side is positive. So if it's positive, like the 1.5, then it will be somewhere, the shading will be from all this. That is why the probability will be bigger than the rest of the other side. So that's how you find the probabilities. So if I'm not gonna come to this table, I've already showed you the table, and I'm not gonna do the example. We've used that one. Like I said, we're going to run through it quickly as possible. So you can see that for x of 18 to find the probability, we go and find the z value. The z value is 0.12, and we go and find the probability because it's positive, and it is less than. We go to the larger side, and we find the probability. So I want to come to when it is greater than. So when the question is, if it's bigger than or greater than or equal, greater, greater than or equal, which means it's got greater than or equal, the value, if it's negative, we go to the larger side. If it's positive, we go to the smaller side. So the only way to remember this is by drawing a picture for yourself. As you can see on this, if I'm talking about greater than means the bigger side, right? So if the answer I get is negative, if it's negative, it will be here and it will go this side. As you can see, the shading on this side will be larger than the shading, the remainder, the thing that I didn't shake. Can you see? If the answer was negative, I will be using the larger portion. That is why we go to the larger portion of the table. If the answer is positive, like you say, you see here, the answer is positive. So it will be positive from this side of a zero and we know that it's greater than, right? So then the shading will be a smaller shading. So it means we go to the smaller portion. So how would you know whether you need to calculate greater than or the less that use your hand? Left means less. Right means more, right? So you can use your current for the left to point the direction. So left will always stay less. Right will always be more because when you are right, you weigh more. When you are wrong or left, you have left current. I don't know how else I can tell you how to determine which one is which. But if you use your arm, left less, it will also guide you in terms of how you find the answer. And always remember the belly shape. It's always cut in the middle and in the middle, that is your mean, which is always equals to zero. So therefore it means anything, this side will be negative of zero. That side will be negative, it will be positive. And that's how you will determine which area you need to shade and which area you leave out. And that will guide you in terms of how you're gonna get to the table. And that is for the greater than. Now we also have the between. With the between, it's different. To find the probability of between, we can do this two ways. We can use, depending on whether we find the answer is negative and positive and things like that. Or we can just use the mean to z. So for greater than and less than, we use smaller and larger portion. For between, always use the mean to z column. So we're going to use this column, mean to z for the between. Anything between these two, they use less than or greater than or less than or equal or greater than or equal. The between, we will always use mean to z. So let's look at how we do this. So let's suppose that a normal, x is normal distributed with the mean of 18 and the standard deviation of five. And we need to find the probability that x lies between 18 and 18.6. Now, in order for you not to get confused with this, you can calculate the z value for the first value and calculate the z value for the second value, like I did here. We calculate the z value for 18, which will be 18 minus the mean of 18 and the standard deviation of five, because this was our x and we get that it is zero. And then we do the same, we calculate the mean of 18 points, oh, sorry, x of 18.6 minus the mean of 18 divided by the standard deviation of five. And we get that it is zero comma, one, two. Somebody is not muted. I don't know who or the Polini. So now we did find z lies between. Can I ask Polini, can you please mute yourself? Because I tried muting you and thank you. Please remember when you come to the session, please try to mute yourself all the time, especially if there will be some noises in your background. If your noise are like your radio playing and they play music and we are explaining certain concepts, it affects what we load onto the platform because music's most of the time is copyrighted, right? So it will say we need to cut out the copyright information before we load it. And we don't want to cut out the chunk of the explanation. So please reconsider it. Every time you come to the session, make sure that you just mute yourself, especially if you're not sure whether your environment is conducive enough for you to participate with your mic on. Thank you. Okay, so let's look at the between. So between, we know that we calculated our z lies between zero and one, zero point one two. In order for us to go find them, the probability remember we said we're going to use the different, not the difference, but we're going to use the two values that it's located between. So we know that it was located between zero and 12, ignore the top one. I was explaining this to another group of students who do not do psych. So for you, you just need to concentrate on the mean to z. So we go for the first one. We know that the z value was zero. We just go to the zero table and find the mean to z value because this is also another table from another module. Let's use the one from your module. So our one is zero. So the A is zero, mean to z is zero. The other one, so we got the probability of z lies between zero point one two and two. So we need to go to zero point one two and the A is zero point one two and we find the mean to z value there. You just add those two values and they will give you your probability of between. Just adding the zero plus zero comma zero four seven eight will give us zero comma zero four seven eight. And that's how you will find the probability of between. Now it's not as simple as this. Let's look at an example from your module. Look at the graph based on the frequency distribution of measurement of variable x from normally distributed data, which is produced below. Using the information provided in the graph, calculate the probability that the measurement made at random on this particular scale will fall in the area under the graph colored in gray. So already they are telling us that we need to calculate the probability of the between. Because we need to find, so they gave us the normal distribution table with the z values, which are Roscos, but they also gave us the mean, population mean and standard deviation. The question says, select the answer closest to the calculated probability from the options below. Now, based on the information on the graph above, if there's not much, but what we can deduce from there is that they asked us to calculate the probability, which is what we know. And they also told us something that we need to find that probability that falls within the area underneath the cap and they shared that area. Now, with that area underneath the cap, they highlighted certain things. One of them is 25. So it means we need to find the probability that our x variable lies between 25, which is the smallest value, this side. But also on this side, the graph ends on 50. So it lies between 25 and 50. So because it lies between those, we can go and calculate the z value for 25, which means we need to use the formula. And that will be 25 minus, because your x will always be given in the question, so here is my question. We need to find this probability. It's my question. So 25 is x. Our mean is 40 divided by the standard deviation is 10. And that is equals to 25 minus 40 equals divide by 10 equals change minus 1.4, minus 1.5, sorry, 1.5. That is z of 25. Now, I'm not gonna do all the work by myself. I'm just gonna give you work as well. Go and calculate z of 50. The z score for 50, your x minus the mean divided by the standard deviation. Calculate that and let me know when you are done, even if it's one person done, so that I can also rest my voice. I'm done. Okay, so help us. So then it's 50 minus 40 divided by 10, that equals 1. So it's 10 divided by 10 equals 1. So I'm gonna put here zero, zero because the table has two decimals. And also on this one, I'm just going to add the zero at the end, so that it is two decimals. So we do have our probabilities. So we know that our probabilities of z lies between minus 1.5, zero, and 1.00. So we can go to the mean to z. So let's go to mean to z. Let's make this smaller. Mean to z, we're looking for mean to z of z, z, 1.5, z, 1, okay. So in the meantime, I can also identify because we're looking for one as well. So the a is one. And I know that the first column is mean to z. So we have three, zero point, zero point three, four, one, three, zero point three, four, zero point three, four, one, three. I'm just gonna use plus, and then go look for 1.5. So 1.5 we will find on this slide. 1.50, so we just need 1.50, mean to z. 1.5, mean to z, we'll read that one. Plus zero point, four, three, three. Zero point, four, three, three, two. Zero point, four, three, three, two. You're the probability of between this right here. That will be 0.4332 plus 0.3413 equals. And I just need to change this. And the answer is 0.775. If I round this correctly, because the number to the right of where I want to leave my digit, it's bigger than five or equals to five. Then I must add one to that number because I just need three decimals, as I can see here. They are, they are three, three decimal answer and one decimal answer. So even if I look at one decimal, it will not be that one. So the answer is zero comma, seven, zero comma, seven, seven, four, four, five. And if we round up, we get the answer of zero comma, seven, seven, five, which is option two. You need to, you need to be able to also round off correctly in your module. So you need to learn about rounding off as well. So if the number to the right of where you want to round off to, it's greater than or equals to five. You add one. If it's less than five, you do nothing. So if the answer here was zero comma, seven, seven, four, three, we will just leave the answer as one. Seven, seven, four, three, we will just leave the answer as zero comma, seven, seven, four. Okay. And just to recap before we do extra other activities or exercises from your module, we have learned so far about the basic concepts of normal distribution, especially when we want to standardize the raw score by using Z score formula, which is your X value minus your population standard deviation. Sorry, minus your population mean divided by the population standard deviation. We also learned about the basic concepts of normal distribution that a normal distribution is distributed with the mean of zero and standard deviation. I'm just writing in eight. And the standard deviation of one. And we also learned that in order to find the probability, we first need to find this or standardize our values by using the Z score. And if our Z value, we need to find the probability of Z less than eight. If the answer is negative, then we use the smaller portion. Right. If the answer is positive, then we use the larger portion. That's what we've learned. We also learned that if we need to find the probability of Z greater than a value. And if the Z value, the answer is negative, then we're going to use the larger portion. As you can see that it's an opposite of that site. And if it's positive, then we're going to use the smaller portion. If we find there between the probability that Z lies between two values, A and B, then we're going to use the sum of the mean to Z. Mean to Z values. Right. That's all what we have learned. That's just that you are ready for you to answer any question relating to Z scores. Okay. So let's look at more exercises. So some of the questions that comes in the exam looks like this. We just dealt with this. A standardized normal distribution has the mean of and the standard deviation of dot dot. Which one is the correct answer? One. It's number one. It has the mean of zero and the standard deviation of one. Number two. Study the following figure. As you can see, there is our figure. The figure is representing the distribution of an X value. So the X values are lying around somewhere here. On this, they've given us the standard deviation. They've given us the mean. They've given you your X value. Because this is your X value and this is your X value somewhere there. They just did that. And they have shaded this area. So you must just pay attention to the shaded area as well. So this I'm just deciding what I'm seeing in front of me. What is the probability that a specific number drawn purely at random from the variable distributed like this would fall in the gray area. So anyway, I don't have a gray pen. I have a red. So we're going to assume that this red is your gray area. That is that it would be equal or greater than. So equal or greater than remember than the X value of 55. So it means they want you to find the probability that X is greater than or equals to 155. But since they have identified this and they've shaded the gray area, it should give you some insight in terms of where you're going to find that probability when you answer the question. Okay. Choose the answer closest to the correct one in this form. So the first thing that you need to do. You still remember is to calculate the Z score, which use X minus the mean divided by the standard deviation. Now you will tell me, oh, but I don't see the things that you are talking about. So here they use the sample mean. Remember the sample mean is X bar. So they have used the sample mean and they use the sample standard deviation. So I can also go back and change this. In state of using population mean, I can use X bar. And in state of using Sigma, I can use the sample standard deviation. So we're still going to give me the same answer. This is for the population. This is for the sample. Sample population. So depending on what they have given you, you can always interchange. Doesn't matter that much. Because if they've given you the information, you just use that. So our X is always given in the question. So this is what we want to find the probability of the shaded area. Our X is 155 minus our mean. I have given you it's 150. Your standard deviation is 5. What is the answer? 155 minus 150. Divide by 5 equals. Do you also get 1 as your answer? Yes, 1. So tell me which site? Is it the smaller site or the bigger site? It's very clear. The smaller side. It will be the smaller side because the shaded area shaded the smaller area, which makes it easier for you to see which site. So we're going to go to the smaller site and look for one on the Z score. So there is our one. And our smaller site is the last column. So we go to one. I hope you are able to see because I made the table smaller. You go to one and you go to the smaller area. So the answer is 0,15. So the probability that Z is greater than or equals to one is 0,15. What did we get? 1587. But the answer here is in two decimals. So two decimals, the number to the right, there is the right to where we want to stop. We want to stop there. So it means we need to add one to the rest number. So the answer will be 0,16. What is the answer? Easy, right? Easy, easy, easy, easy. So let's look at more examples unless if there are more questions. Let's see. Let's go to the chat. No questions. Any questions? Do you have any questions? Yes, Mrs. Boy, I just wanted to ask you. Okay, I get stuck now. You have a sample on the right and then a population on the left. And if I look at the 150, there's an X and there's like a stripe above it. So I'm just a little bit confused now. Why is there an X with a stripe at the top? What does it mean when you say the sample? The sample mean? Okay. So it means you didn't watch the recordings of last week. We did speak about the statistics and the parameters. We spoke about this. So a population parameter uses the mu. So we use the Greek letters to describe the population. And then we use the Roman letters to describe the sample statistics. So these are called population parameters, the mu, the sigma. These are your sample, your population mean, your population standard deviation. For the sample, we use the normal letters that you know of. But also some of them, like the mean is not just the normal letter. We put the bar at the top. X bar represents the sample statistic, which is called the sample mean. S, you can see there is an S. And S represents the sample statistic called sample standard deviation, whereas this one is population standard deviation. So it doesn't change anything. So in the previous section, we were using the population mean and we were using this population standard deviation. But on this one, they didn't give you the population parameters. They gave you the sample statistics. Okay. Sorry, can I just ask you which section you did last week? No, we just covered introduction to the psych. We just did at the high level overview. What are the variables? What are the constructs? What is the population? What is the sample? We just did the introduction to basic statistics at a high level for your module. I can go back and still go listen to that. Yes. And then we looked at the sampling methods and all that. So yeah, it's just basic things that we covered. Okay. I'll go have a look and listen to that. Thank you. Aris. Okay. We left with 10 minutes. I have about nine questions in this. The notes will be uploaded at the front and you can go through the activities. If you have any questions, you can always ask. Yes. So another question. Joseph scores 60% in history test class and the class meaning 65 and the standard deviation is 10%. And 50% in biology. The class means 53 and the standard deviation is 12%. Use that score to decide which statement is true. Relative to the rest of the class. Joseph does. So we need to choose whichever statement. Better suited for that. So it means we go into need to calculate the Z score for history. Z score for history, which will be X minus the population mean divide by the standard deviation. So what are we given for history? Peace score, which is our X class mean, which is our new standard deviation, which is our Sigma. So we can just substitute. So as you can see, most of the time I've applied the Newman's prompt problem solving framework, right? What is it that we are asked? I identified what is it that exactly they want us to do because I said use the Z score to determine which statement is through the relative rest of the class. Joseph does and what is it that we need to choose in terms of this and what are the facts given? I've identified my effects in terms of my X, my mu, my Sigma. What is the formula that I need to use? I selected the formula and I'm just going to substitute before I take my calculator. So those are the steps that all the skills that we want to impact to use. So before I take my calculator, put in the values here. So 60% is 0,60 minus 65 is 0,65 divide by 10% is 0,10. And you just calculate the answer there. So to my calculator, 0.6 minus 0.65 divide by 0.1 equals. You will notice that I don't put the 0.60 and 0.10 as 1 because the calculator knows that whether I put it or not. And also I don't use 0. And if it makes you feel it is, you can use 0. You need to do that. Okay, so the answer is minus 0.5. So that will be minus 0.5. And I can just put 0 at the end or not. I don't even have to put 0 at the end. You can just leave it at that because we're working with Z score. So the next Z score is for biology. So the formula I need for biology, it's X minus the mean divide by the standard deviation. Biology test, the score is X, the mean and the standard deviation. Already I have identified the fact 0.5 minus 0.53 divide by 0.12. Take my calculator 0.5 minus 0.53 equals divide by 0.12 equals. And change will be 0, minus 0.25. So now before I even look at my answers, I need to make a decision based on what I see here. So history has a standard deviation of a Z score of minus 0.5, which means it's 0.5 far away from the rest of the other class mean, right? Biology has the score of 0.25, which means it's almost closer, but still below, but closer to the rest of the class. So it means which one, because this is minus, minus means below. So the far away, it means the far away from, it's even worse, right? So a 0.5 score will mean worse than a 0.25. Let's go to our statements now. Since I've determined that a negative means worse, because negative means below the class average, therefore it means it's worse. But the bigger the number of the worse will mean that that subject is even worse than the other subject, regardless. So now let's look at this. So our statement says Joseph does better in biology than in history. Is that true? Biology has got 0.25, history has got 0.5. Remember they are both worse. Which one will be better than the other? Is it biology or history? Biology is better than history. Biology will be better than history, right? Because number two says better in history than in biology. That won't be true. The third one says equally well in history and in biology, but we know that they did worse because they just got less than the class average anyway. So the correct answer will be number one. Relative to his class, Joseph does better in biology than in history, because the score in biology is almost closer, even though it's below the class average, but it's closer to the class average. And that's how you will interpret your Z scores. This is just another property that you just need to know. The area underneath or the area under the normal kef equals to, is it equals to its me? Is it equals to its standard deviation? Or is it equals to one? The area under the kef, which means the sum of all the values or the probabilities under the kef? It's one. Number three. It's number three. Hey, you guys, you were listening to me. I can believe that because I said this right in the beginning at six o'clock at 10 past six at 20 past 28 past seven. And you still remember this? Oh, you guys. You make my heart home. You make me home. You make me even no longer feel sick while helping you. Okay, so we've got one minute left, but let's see if we can squeeze one minute of this one. Joseph. I go. Where do I see Joseph? John received 45 marks for his psychology test. The average mark for this test is 35 and the standard deviation is 10. What is John's score? Now Z score. Now we need to just calculate John Z score. So we know that John's score will be calculated by X minus the mean divided by the standard deviation. So what is it that they have given us? They've given you the X because that is the score that John received. They've given you the average mark, which is your mean, your standard deviation, which is your sigma. Let me just come and substitute into the formula, which is 45 minus 35 divided by 10. And this one I'm giving it to you to answer. The answer is one. The answer is one, which is number one. Okay, so just for the sake of kept chatting everything on the video in case you don't get the chance to download the code. The notes, you can come back to the recording and look at the rest of the questions. So this is one of the other questions. Also, they are almost similar. So if you must answer one type of a question, you can do the rest of the question if you are asked. Because this is the same as the one that we just answered. Compare whether they do better in this one or the other. So you just calculate this for the Z scores for English and geography and compare and say which one they do better in. This is almost similar to the one that we did as an example. But now here they're asking you to find the probability and we used some of them as an example during our session. When we were demonstrating the table, so you will go and calculate this. Remember the probability if it's less than. Are we using the answer? Is it negative or positive? You need to know whether you're using the smaller portion or the larger portion as well. Things like that. If it's greater than because they say it's better. This is better. Better means greater than or equal or better than then means it's greater than. So you just need to make sure that you calculate that correctly. The next other question also asked you to calculate the Z score. They've given you the information at the top. What is going to confuse you with this is the following. At the top, they gave you a whole lot of other numbers. Like they say they are 100 kids or 100 learners. The questionnaire has has 50 questions that has nothing to do with what you are asked to do. What you are asked is they gave you the statement, the mean, the standard deviation and they say respectively. So when they say respectively, so it means this corresponds to the mean standard deviation is eight because. First they mentioned the standard, the mean and then the standard deviation. So as well, in terms of the numbers, the first one mentioned with the value of the first one mentioned corresponds to the value or the one that was mentioned first. And yeah, they gave you your X value and you just calculate your Z. That's how easy it is straightforward. So some of these questions come from your past exam paper. Some of them comes from your tutorial letters and especially this one. So where I didn't have some tutorial letter, they come from your previous tutorial letter like 2021 or 2019 tutorial letters as well. And in this one, they are just asking you to find the probability and yeah, you just need to make sure that you understand the question that is asked. It says less than so remember that less than is the sign less right left left less less left. And then you can go and find the probability remember to apply the correct way of finding the probability by either using the smaller portion or the larger portion. And that concludes today's session. Are there any questions, comments, theories, anything you want to know? I have a question. It's not done because speaking I would regard the attendance register. I'm using my phone so I can't seem to find it. It's in the chat. Thank you also for reminding me about the attendance register. It's in the chat. I just posted it on the chat. And just want to see. So let's see. I can see why there are so many confusions. I've got three, four. One, two, three, four. Why four? All of them are not even and that that is the one that you need to select. You need to select the one that says, I don't know. They should have changed it, but that is the one that you need to select. Research analytics literacies for psychology students on the UNISA platform. It's something else. It's called basic statistics for human science. I hope it doesn't confuse you. Let's see if they didn't add it here as such. No, they didn't. But that is the thing that you need to. So that is the one research analytics literacies for psychology students. That's the one that everyone needs to sign up to. This one's for statistics, pure statistics. And this is the basic numeracies. I hope you are able to get to that. I just need to go out of this. Are there any other questions? If there are no questions, I'm going to stop the recording now. Sorry. Just give me one minute of your time again. I know that we are over time.