 I know some of you comes from your other tutors, a WhatsApp group and because they've requested that you also participate in today's session, so that you are ready to submit because the closing date of the assignment is tomorrow. If you are from another tutor, you are more than welcome. Feel free to ask any questions that you may have. Today's session is question and answer. That's what we're going to be doing only for today. I expect today you bring all your challenges to the front. Like you tell me what you're having challenges with, and then we find questions and we help you understand the content better. Because we already did two sessions relating to content material for study unit 6 and content material related to study unit 7, and then we also did activities relating to study unit 6 and study unit 7. Today is just to make sure that you are ready to submit. Are there any comments before I start opening up? But before the question and answers and all that, remember that we're doing the question and answer today. Next week, those who are in my E-Tutor group, next week remember we're starting with new chapters. We're going to do study unit 8, which relates to your assignment 4. That will be next week Sunday. Other than that, are there any questions? Comments? Let's go back to the screen. Know that you can also post on the chat if you have any question. Because if there are no questions or comments or anything, because I didn't prepare for the session today, because there is nothing to prepare for, so most of those who are in my E-Tutor group, you have the questions already, so it means we will just be going through the same questions that we shared over the last two weeks. This month, I see that you are talking. I'm not sure if you want to say something. Probably it's my mic, not on mute. Let me just mute, it's my sound. All right. Any comments, any questions, anything? How can I help you? Because so far we've covered so much, I don't know. Where are your challenges? Talk to me. Where do you find problems? Do you find them because you don't understand how to handle a question or answer the question? You don't know how to use your calculator to calculate this. You don't know how to use the table. Talk to me. How can I help you? No one. No, sir? Yes. Hi, I was Elizabeth. Yes. So my challenge from my side, I'm a bit confused when it comes to the table. So when they refer to the table, do they refer to the table in our study guides? Yes. So it's either in your study guide, if they didn't give you statistical tables, then they will be referring to the table in your study guide. There is a table called, I'm just gonna open my, in your study guide, it should be called table E2. I don't know what page number will that be on your study guide, but it needs to read S, the accumulative standardized normal distribution table. The table has the negative side of the table, and also has the positive side, right? That is the table that you should be using for, for you to answer normal distribution and also sampling distribution and the same table. Next month, which starts tomorrow, you are still going to be using it. And the following month probably, no, yes. And probably the following month, you will be using a different table, but next month you will still be using the same table. And we will be introducing a new other table, but this is the table. Okay, thank you. And then on the calculator, like I'm having a very difficult time with this module and it's my second year doing it, but I think it keeps getting bad, bad, bad. On the calculator, like for example, your shop, what's this? Not a shop, a cashier. When you put it on stats, is it supposed to, like when you put in the figures on it, when you put your setting on stats? Okay, so at the moment, when you are doing these calculations, your calculator should just be on a normal calculator mode. So if your calculator is on state mode, make sure that you change it. You just go to, now I've opened a different case, your calculator, but you just go to your mode function and then you set it up. Sorry, let me do this. I need to press shift first. Oh, gosh. But you will need to go to your normal mode. It will say norm, N-O-R-M, and then you will press that. So the calculator that I have now it functions differently to your case, your calculator, but make sure that it's not on state mode. That's all what I think. Yeah, make sure that it's on a normal mode, not on a state mode. Okay, thank you so much. Any other person? Or if you have another question, is that all that you have? Okay. So since there are, we have almost one hour, 30 minutes together. So I'll just continue to show you some of the questions, because I don't know where your challenges are, but we can just continue with the session. Unfortunately, today's session, I won't be, probably why am I also even recording it because I'm not going to upload it and share with, or maybe I can, I will see, I will upload it and share with everyone. Okay, so let's go to you. Yes, we have a question. Lizzie, sorry, can I ask a question? Yes, you can. With regards to attempts that we have, we have three attempts. So do we have to do all the attempts or let's say you did the first one and the second one and you are satisfied with your marks? Do you have to do the third one? Is it like compulsory or how does it work? Okay, no, it's not compulsory. So you just need to do one. So let's say you do your first attempt and you get 100%, you are done. You can just close your books, but that doesn't stop you from doing again another attempt because I think when you do more attempts, you get more different questions, right? So it gives you a practice. What will happen is the highest mark between the three attempts will be recorded as your final score that you will get. So if the first attempt you get 100, the second attempt you get 70, the third attempt you get 30, you will still keep your 100. So if your first attempt you get 39, you have the second chance to do again, you get 60, 60 will be captured. If you do your third one and you get 80, 80 will be captured. So it's up to you. If you feel that 50 is getting a score of 50 is enough, then it's enough for you. But at least because these are assignments, you can use them to practice, yeah. Desmond? Asli, I was wondering if you could start with some of the standard error type of examples because I think it's either I need more practice or I still don't get how they actually work. Okay, so the standard errors, right? We can talk about the standard error. So let me go to the presentation itself. I'm not sure if I have a question with standard error. Okay, this one has some standard error question. And this one also has some standard error. We can use exercise 11. Oh, sorry, I don't know what's wrong with my throat now. Okay, so what you also need to remember and know because you're dealing with sampling distribution. Sampling distribution is split into two, right? Remember that? That when we talk about sampling distribution, it is split into two. It's either the sampling distribution of the mean or sampling distribution of the proportion. That you always have to remember that. How do you identify that? Now you need to be calculating things relating to sampling distribution of the mean. It's because they will be giving you like things like standard deviation and the mean. When they give you the mean and the standard deviation, you know that you're doing sampling distribution of the mean. If they don't give you the mean and the standard deviation, they will give you proportions. And then you will know that you're doing it for the proportion. Now, when it comes to standard error, what do we mean by standard error? Standard error for the mean, your standard error for the mean will be your standard deviation of the sampling distribution of the mean. Or we call it the standard deviation of the sample mean distribution. Or we can call it the standard error. So it is your standard error. And it is given by your population standard deviation over. It will be given by your population standard deviation divided by the square root of your sample size. And that is the other thing that you also need to be mindful of because normal distribution and sampling distribution, they are almost close to one another and they look almost exactly the same in terms of questions. The difference is with sampling distribution, you will be given the sample size as well. All right? So your standard error is just your population standard deviation, which will be given in the statement divided by your sample square root of your sample size. When they talk about the proportion, then it's different because it will be the standard deviation of the proportion or the sample proportions. And that is given by depending as well in terms of what other information they have provided you. If they have given you the population proportion and the sample proportion, the default calculation will be on the population proportion. So to calculate the standard deviation of the sampled proportions, we use a sample proportion distribution or what we call the standard error. We use the population proportion times one minus the population proportion divide by n. You just need to know how to distinguish between the two. If, for example, if they didn't give you the population proportion, right? But they give you observations that satisfy the proportion, which the sample proportion, which they give you your observation and your sample size, then you can calculate the sample proportion. Now, in the absence of your population proportion, you can use your sample proportion to calculate your standard deviation of the standard error of the sampling distribution or the standard deviation of the sampled proportion or the standard deviation of sampling distribution of proportions, which then you will use the formula, your P times one minus P divide by n. That is only when you calculating standard error. Standard, I don't know how to write today for some reason. When you calculate the standard error, you use that. So based on question 11, we are given, you need to be able to go to the question and read what you are given. Sometimes in the question, they might be giving you more information than you supposed to be receiving. So for example, a normally distributed population with 1,000 observation has a mean of 100 and the standard deviation of 10. This 1,000 is just there to confuse you. You can just ignore that. It's just another information just to tell you that you are given. Your capital letter N is which is your population. We know that we're going to take, because we're dealing with sampling distribution, we're going to select multiple samples from this population. And from this population, n of 1,000, when they calculated the mean, which is your mu, your standard deviation, which is your sigma, you get that it's 10 and the mean is 100. You are also told that a sample size, which is your small n, your sample size of 200 is randomly drawn from the population. Automatically now you know sampling distribution, we've got two mean and proportion. We know that this is not the proportion, it's for the mean. Calculating the standard distribution of, the standard error of the sampling distribution, it means we're going to use this formula, right? Because I'm given the standard deviation and I'm given the mean, I'm not given the proportion. So it's not the proportion. So to calculate, you just substitute into the equation. Substitute the values given. Your standard deviation is 10. So you just put the 10 divide by your mean. It doesn't really matter because we don't need the mean for calculating the standard deviation. We only need the square root of the sample size. Square root of 200. And that you can calculate and tells me what is the answer. Also, I've also noticed from the previous activities and for people who are always quiet, they don't know how to use their calculator effectively in order for them to get the correct answers. So worry not, I'm here to help you with that as well. So on your calculator, especially for those ones who don't have the cashier calculator, if you have a cashier calculator, please try by all means to use this fraction function. It will save you time. So we go into that with the cashier, with the fraction. So you just select that fraction thingy, function, not the thingy, it's a function. A fraction function, you select that, you put the 10 to move to the lower block or box, you use your arrows. So I just go to the bottom one, I using the arrow that goes down and because at the bottom from this calculation, it has a square root and you do have a square root, you just press the square root and in the block when it's flicking, you just put the 200 and then you press equal or you can press answer. I prefer to press equal. And now you get an answer that looks like this and you panic and you're going to panic. Do not panic, panic not because on your calculator, you do have this arrow change which is changing to decimals. So we can press that and change and we can see that our standard deviation or the standard deviation of sample mean or the standard error is 0,7071. And looking at the answers, if we're looking for only three decimals, therefore the answer will be option two. And that is if you're using this kind of a calculator. If you don't have this kind of a calculator, you are using an HP calculator or you are using a case, even a case of calculator that does not have this function or you are using a sharp calculator, then you need to calculate these things manually and by manually I mean step by step. The first thing you can do is you can press 10 and press divide by on all your calculators, you do have this function called the square root. You just press that square root function and then you press 200 and then you press equal. You will get 0,7 on your calculator, you only get this and fractions. You will get your answer automatically as decimal and then you will be able to answer your question. Alternatively, if you don't know how to use your calculator effectively like that, you can do step by step by first finding the square root of 200 and press equal and get the answer of 14.14213562. You write the whole number as you see it and once you have written all that number, then you go to your calculator and you press 10 and you press divide by and I can just recall that number that I have stored and now I cannot recall it. You know why you didn't wanna recall? My stored number doesn't wanna recall it, but anyway. So you would have recalled your number, which now I forgot I'm just gonna put the 200 again and you recall your whole number and you should be getting the same answer. So you will do square root of 200 is equals to 14.1421, you will just need to remember the whole number that you have there, that whole number and you will use it. Do not round off, do not take only two decimals and then say 10 divide by the two decimal, you will not get the answer. So use the whole number only round off when you get to the final answer and that is how you will answer the questions on standard error. So let's look at more questions unless if there is another question from the group. Busisiwe, any question? Do you have a question? Yeah, I do. Well, by the way, hi everyone. While we are on the question of standard error, if you're not given the standard deviation, do you have to calculate the standard deviation first before using that formula that we're using now? Yes, if they've given you the values. So let's assume that they didn't give you the standard, they didn't give you the standard deviation but they gave you all thousand observations, right? So because if they didn't give you this 10 or this 100, they would have given you the observations like maybe one was 20, one was 100, one was 20, one was 30, one was 150 and so on and so on until thousand which they will not give you that because they know that in the exam or in the assignment, you don't have the whole day to calculate thousand records so they will provide them to you. So only in the proportion way, they will give you your observation and you can calculate the proportion but it's highly unlikely that they do not give you standard deviation and the mean and ask you to calculate them and then use them. It will be highly unlikely, right? I see Desmond, your hand is still up. Do you have another question? No, maybe it's my headache, I might have not even dropped it. Okay. I wonder how, I don't know how to drop the hand, if ever. No worries, it's fine. As long as you have clarified that, that is 100% who you are, who you are, who you are. So I just want to go to the other questions while we still have the time. A question like this. The STA 1610 module student exam marks are normally distributed with the mean of 80 and the standard deviation of 20. So they gave me the mean, they gave you the standard deviation of 20. They gave their way 2,969 students with a score higher than 95. So they give me, they tell me that if I call this an X value and they tell me that those number of students have scored higher than and higher is greater than, right? Higher than 95. So my X will be greater than 95. And if they say how many students took the exam, so I need to know of my X plus my Y, how many took the exam will give me, how many took the exam? This plus the unknown other students whom I don't know they probably both combined will give me the number of students who took the exam. The other way you can look at it to say, if there's only those ones who scored 2,969, those 2,969 scored more than 95, then those who scored less than 95 will be represented by one. So these are those who scored higher and these are those who scored less than 95. So combining both of them should give me the total number of students who have participated in the exam. So there are two ways of doing this. So I'm going to show you both of those ways of calculating the same, right? The first way that we can find is to first find the probability of this 2,969 students. But to find the probability of these students, we use our formula, the probability of X greater than 95 and we know that we need to standardize this score because this is the original score. So in order for us to standardize that, we use the Z score of greater than your sample, your, not your sample mean because yeah, I'm still using scores, your X minus the mean divided by standard deviation. And this is a normal distribution because they didn't give us a lot of other things that we need to worry about, right? So now we can calculate our probability. So the probability of Z greater than our X is what we are given in the, sorry, in the question, which is 95. So it will be 95 minus the mean of 80 divided by the standard deviation. We found that it was 20. And I will ask you to go and calculate and give me the answer. That will be Z of greater than, what is the answer of 95 minus 80? 0.75. That will be 0.75. And once you have this, because you need to make sure that you understand how to use the table as well. This is greater than, what you need to know and remember is that from the statistical table, this table contains all probability of Z less than an A. So it means if you come to this table and you come to look for a probability value or to look for a Z value, you must know that you are looking for a Z value of A less than that value, right? It contains whether in the negative side or positive side, all these values and you can see from the shaded area, it tells you that all these probabilities within the table are those probabilities that are in the shaded area. And the shaded area is the less than because if it's this side, it's less than. If it's that side, it will be greater than. So everything in the white area, it's greater than anything in the less than area is less than. So then it means if I have a probability that is looking for a Z value of greater than, then there are a couple of things that I need to always remember. So I need to remember that the table contains the probability of Z less than. So anything that is Z less than a value, I will go and find that value on the table. If I'm looking for Z of greater than a value, then I need to say one minus the value that I'm going to find on the table. And remember, if it was between, if Z was between two values, A and B, then we would have found the probability of the second Z value, which is Z minus, the probability of the first Z less than A. So it means we're gonna go to the table, find the value for B and come to the table and find the value of A and subtract the values from one another. So we're going to do the same. So based on this, what we just learned, it means for this one, we need to go and say, we need to go and find the probability of Z. We change the sign to less than 0.75 because that is the probability that we're going to find on the table. So we go to the table and this is the table, right? Then on the table, we're going to look for 0.75. This side, it's negative values. So it means we're going to go to the positive side. And yeah, we're looking for 0.75. On the left-hand side of the table, we need to find 0.7. At the top, we need to find the last digit of five. So let's do that. 0.7 will be on the left. So the first two digits, one before and one after the comma, we find it on the left. The last digit, we find it at the top of the table, always. So our last digit is 0.05. So we both meet and that will be our Z value. If I make it bigger, that is our Z value. For those who can see, that is 0.7734. So going back to our question. So we say one minus 0.7734. And what is the answer that we're getting? What is the probability of those that are? 0.2266. 0.2266. Now, this is the probability or the proportion of students who received higher than 90. If in a fraction, I can say the same answer of those who received, I can calculate them by saying, those who received more than 95 divided by the total number, the proportion will give me 0.2266, because that is the proportion. It's the same thing as if I had 100 students and out of 105 of them passed the exam and, oh, sorry, five of them failed the exam. And if I want to calculate the proportion of those who failed, I would say five divided by 100 and it will give me my proportion of those who failed and that will give me something like five divided by 100 and I will get something like 0.05, right? 0.05. So at this moment, I don't know this number, but I know now because I went and I calculated and I found that proportion is 0.2266. I know the proportion and I know how many students have failed. So if I say this is my N of those who have failed from N number of samples that I've selected, five of them failed and the proportion is 0.05, how many wrote the exam? That's what I am currently doing. So in order for us to get this, we can just represent it as a proportion like this and we know what our X is because we were given our X, our X is 2969 divided by N is equals to 0.2266. Now, the other thing that you need to remember, this is math, we can cross, we can multiply N there and we can divide because if I want to make N the subject of the formula, it means I need to change my subject of the formula. I need to make N the subject of the formula. I'm going to multiply by N on the other side and when I multiply by N on this side, it will be N multiplied by 0.2266 and on this side I will be left with 2969, 2969, but I need to leave N on its own. I'm going to divide this side by 0.2266, but I also need to do the same on the other side. 0.2266. So here I was using the simple way you cross or whatever I do on the right, I must do on the left, so those two will cancel out and you will be left with N and just give me the answer of 2969 divided by 0.2266. What is the answer? 13102.38. 13102.38. 13102.38. And therefore, if I look at the answers here, they are in all numbers, they are integers, so I just need to remove the decimal and the answer will be 13102, which is option C. Now, this is one way of answering the same question. The other way of answering the same question, I just want to also give you, because there are many ways to skin a cat instead. So I'm going to leave everything that I have here because I'm not going to repeat the same. So we know that the proportion of those who are above 95 are those, we still need to know that your higher plus your lower or less, they give you N. So X plus Y should give us N. So if I know what the proportion is for this, right? I know what the proportion is. I'm going to remove this because I don't have enough space. Let's delete that and apply the second option. So the second option, I can take, I can say what is the proportion? What will be the proportion of those? If I don't use X, I can use Y. Y will be those who are less than or equals to 95 because remember that the probability of X greater than 95, the complement of it plus the probability of X less than or equals to 95, which will be the complement of it, should always be equals to one. So the sum of both probabilities should always be equals to one. So if I know that and I've got this probability here, therefore it means the probability of the probability of X less than or equals to 95 will be the same as one minus the probability of X greater than 95 or vice versa. We can do it vice versa. Vice versa means I can also say the probability of X greater than 95 is the same as one minus the probability of X less than or equals to 95, which is what we have calculated some way. We found it there because we said this probability of X greater than 95 is 0.2. If I take this and put it there, it will still give you the same because 0.2266 plus 0.7739 should give me one if you wake it out. So then it means the probability here will just be 0.7734, but that doesn't help me because what we're looking for is the probability of, are we not looking for the probability we're looking for N? So now how do we then find N? So if, let's assume, if my X is equals to, so I know that my X is equals to 0.2266 and my Y proportion is equals to 0.7. 7-7, because we did calculate or find it on the table, 7-7, 34. Now I know what X is. So if I know what X is and I replace X with the actual number of 2969 equals 0.2266 and my Y, which is still, I don't know what that is and I know that the proportion of it is 7-7, 34. I can also apply the mathematical function by crisscrossing. So I can multiply by Y there and when I multiply by Y there, it will be Y on this side and I need to divide by 0.22 on the other side. So 2969, I need to divide that by 0.2266 and I need to multiply by 0.7734. So when you multiply both sides, let's write it properly since my screen is short and I'm making it squash. So we're going to take 2969 and I'm going to multiply so we're going to take 2969 and multiply it with 0.7734 and we're going to divide by 0.2266 and we're going to multiply by Y on the other side because we are crisscrossing, right? Because we want to make Y the subject of the formula and when we do that, we will find that Y is equals to, if you solve the entire equation, what do we get? Which is 2969 times 0.7734, close bracket, go down, 0.2266 equals and we get the answer of 10133. I'm just going to ignore the 38 for now, 10133. 10133, now N is equals to, we said N is equals to X plus Y. That's what we said, right? What is our X? Our X is 2969, 69 plus. Our Y, we just calculated it, it's 10133. And the answer will be, you just add them together. 10133 plus 2969 equals 18102. 18102, which is the same as what we got, yes. So this one is the longest method than the first one that I showed you. So you've got two ways of answering the same question. Okay, let's go back to question and answers. Since the session today is your session, not my session, you should not be talking a lot. Okay, are there any questions, comments? Hi, Lizzie. Yes. There is a question that I posted on WhatsApp and I need it. On WhatsApp? Yes. And I'm going to scroll my screen right now, tell me if it's one of the questions that is on here, because I've also posted some of them here. Is that this one? Not yet. When did you post it? Is that this one? A few minutes ago. Oh, no, when I'm online, I don't get WhatsApp. Okay, it talks about emotional intelligence. If you have a question. Okay, sorry. You are giving me your assignment questions. Your assignment questions, I don't deal with assignment questions. And I think we've dealt with that one. The emotional intelligence one, are you sure? It's not something similar to this. Because I don't deal with, I don't answer. I don't understand the question because it talks about the mean, it says that the emotional score is between X and the population mean is 0.4641. I don't know, it was just confusing. Okay. Where did you get that question from? Is it from the notes? You can just say from the notes. Yeah, from the notes. There is no problem with that. Okay, I'm going to stop sharing and then I'm gonna come back because you didn't give me all the information as well. I'm gonna copy it and paste it here and then we can do it. I have to discount and I need to stop sharing because then when I go to my WhatsApp, I don't want it to be on YouTube. Stop sharing. I'm gonna copy it the way it is, but you gave me half of the story, but it's fine. Don't worry. I hope you copied it correctly the way it's supposed to be. And empty some way in between. Is it the one that says the EQ score of grade eight class is normally distributed with the mean? Is that the one? Okay. Hence this type of questions I want. I prefer to have them before the session starts so that then we have them on the slides. Okay, so your question looks like this, all right? And I'm gonna hope and pray that you copied it correctly as it was supposed to be just fixing some of these things. So this is X bar, right? Yes. Okay. So we're gonna, and that was an X, right? And we need to determine X, right? Let's get to this. Let's understand what we are given on this question so that we are able to answer it correctly. The EQ score of grade eight class is normally distributed with the mean of 80. Sorry, Lizzie, we can't see anything. I didn't share my screen. Sorry, my bad. Now I hope you are able to see the screen. The EQ score of grade eight class is normally distributed with the mean of 80. So you always have to, when you answer a question, right? Notes for yourself so that you identify what you are given, right? And the standard deviation of 20. So the standard deviation of 20. A random sample of 36, which is our N. Eight learners, grade eight learners is selected. Let X be the EQ score of grade eight class. It is further known that the mean EQ score is between X and the population mean is 0. Yeah, your numbers are not matching. 0.4641 and 0.64642. They need to be the same. Let's keep them consistent. Which one is it? Is it two or is it one? It's one. It's one. So it means discuss. I'm going to keep them because I need to just go here and change this. It's one. So now the first question or the first thing that you need to ask yourself is this. If they say, because I think probably here they wanted to say the probability of is this because this is the probability. The answer of this is the probability. Instead of saying the population mean, they made a mistake here. They should have said the probability. So if we know that and because they give you this statement, which reflect what they have written there. If they say that you know that the probability of X, or the mean, the sample mean line between two values, X and 80 is 0.4641. What needs to come into your mind is the following. Before you start even calculating, it means in order for them to find this probability, they went and they said the probability of Z. I'm going to use not Z. I'm going to use the mean. The probability of the sample mean less than 80 minus the probability of your sample mean less than X gave them 0.4641. What does that mean? It means when they went to the table, they found this value here. And when they went to the table, they found this value here. And when they subtract those probability values on the table together, it gave them this value. Now, because we know that we can manipulate this equation in order for us to find what this probability of X is, we can manipulate the equation by leaving the probability of X on its own and then go and find the probability of X less than 80. But before we go and find the probability of X less than 80, we need to standardize that. First, OK, now how do we do that? Step number one, step. This is how you're going to answer your question. If something like this similar to this comes into your exam, step number one, you will need to standardize this first one first. So let's go find the probability of X mean, the sample mean X of less than 80. To find it, we're going to standardize that sample mean by using the Z score of X less than and our mean minus the population mean divided by the standard error. Remember that, because this is sampling distribution, because it's sampling distribution because they gave us sample of 36. You must also pay attention to information provided in the statement as well. So to do that, we're going to find our Z is our mean. Sample mean is what is given in the statement, which is 80, minus our population mean is 80. Our standard deviation is 20. Our square root of, so I must expand my bracket. It must include everything. Square root of my sample size is 36. And that is equals to the probability of Z of 80 minus 80 is zero, right? And anything divided by zero divided by any number will just give you zero. So I'm just going to assume that this is 0,000. So now we need to go and find this probability. So to find this probability, you need to go to the table. Can you give me the value for this probability on the table? It's 0.500. Yes, it is 0.500, because it's on the positive side. You go to the positive side of the table. For those who don't know how to get there, let me delete this. You come to the positive side. We're looking for 0,000. You will notice that the answer that you would have found is 0, but I added two decimals, because I know that the first two decimal, one before and one after comma, I'm going to find it on the left. And then at the top, I'm going to find the last digit. So and that is 0,500. So we're going to go there and find the probability is 0,500. Now that's step number one. Step number two, we still need to find this, right? But I can come back to this and substitute back into the formula, the probability that I found there. So coming back to our formula that we wrote there, based on this formula that we were given, we can substitute that with 0,500 minus the probability of the x bar less than x is equals to 0.4641. What else can we do here? We need to make this probability of x a subject of the formula, because at the end of the day, we need to go and find x. So because it's minus, I'm going to bring it to the left-hand side. So this to the right-hand side. So 0,500. And I'm going to bring that over. It will be negative 0.4641. And I'm going to take the probability onto the other side, which is the probability of zx. Remember this x is the one that we are looking for. We are not yet done. Don't get excited. We still have a long way to go. We're not done yet. I'm going to break it down again. So now we can find this. What is the probability of x bar less than x? I can interchange them. It doesn't matter, because it's an equal sign. It balances off on the left and the right should be the same. So even if I move them around, they should also be the same. So what is the answer? 5, 9. Sorry, repeat again. 0,0359. 0,0359. Have I written it right correctly? Yes, that's correct. Now, don't get too excited and say, oh, yes, we now know what we need to do. No, we still have a long way to go. So now, step number two, complete, done, kaput. You are done with step number two. We move to step number three. We move to step three. Step three is to go and find the z value. So let's go find the z value. That corresponds with that. So it means we go to the table. So we need to go to the table, because this probability is the same as the standardized probability of z less than a value. We were given that, less than a value. And I'm going to use that x as a value. No, I'm going to use a as a value. Or x doesn't matter. I can use x as well. x as a value. So I need to go and find the z value. Or maybe let's not use x. Let's use z. Let's use a small z, z value of 0,0359. So let's go and find this value on the table, inside the table and go find the z values. So we need to come inside the probability table. It's not going to be in the positive side. It will be in the negative side of the table. So go for 0,0359. That's what we're looking for, right? Negative 1.8. Negative 1.8. Negative 1.8. And I know that at the top it's 0, right? So it will just be negative 1.80 or negative 1.8. It depends on you how you want to write it. So that we found that. So we now know that this probability z is negative 1.8. That's step number. Step number three completed. Now comes step number four. Step number four, we need to calculate or find x. That is our last step. We need to find x. So if I know that this was that, then it means if I take my z value, remember the z formula is z of your x mean, right? Minus your population divided by your standard error divided by the square root of n. So if I need to find my x and I know that my x is the mean because they told us h and h should be sample, not the mean, but the sample sample. X is your sample mean, right? So it will be this value that I need to be calculating. That is my x that we need to find. So in order for us to find this sample mean, we need to make this subject of the formula. So I can cross. Remember that first I can multiply z. z multiplied by the standard error because I'm criss-crossing. Then I will be left with x bar minus the mean, the site. Then I can take the mean to the other site and then it will be positive. If I add the mean to the site plus the mean, and I add the mean to the site, it will be plus the mean. Therefore, on the other site, on the right-hand side, it will cancel out. And that will be the formula that I'm left with. And now I can calculate this x bar. So the x bar will be given by the population mean plus your z times the standard error. And that will be our mean is 80. Our z, we did go and find it. It was minus 1.8 times our standard deviation is 20 divided by the square root of your sample size, which is 36, and go and calculate that. If you are using your cashier calculator, it's easy. If you are calculating manually, I will suggest that you work from left, also from right to left. So you will first calculate 20 divided by the square root of 36 multiplied by minus 1.8. And then you can do plus 1, plus 8. So now, if you're using your cashier calculator, it will be easy because I can write the whole thing on the cashier. And if you are not using the cashier calculator, make sure that you use brackets properly. Plus, and this is the negative value. So I'm going to put it into the bracket, then use that. Don't use the minus. Use the negative. Those on using a sharp calculator, your negative number is somewhere at the bottom. So you must know where your negative values are at. What's wrong with this? OK, so minus 1.8. Close bracket, open bracket, and fraction, because that is the fraction. 20 divided by the square root of 36. And I must use my arrow to go out twice. You click it twice so that it creates a long bar flicking cursor, and you can then close your bracket. If you only tap once with your left arrow, it will not do what it's supposed to do. And when I press E equal, and the answer is 74. So therefore, it means this probability would have been, it lies between 74 and 80, which is equals to 0 comma 4641. But your value of x that you are looking for is 74. And that's how you will answer the question. You just need to remember to do it step by step by step by step. Thank you. Is it possible that instead of them giving us that x bar is less than 80, they make them both less than x, and they give us the answer? It's not going to be possible if it's between, because how would you know? Because there can be multiple scenarios that can be between. That gives you the same answer, right? It can be 0.3, some number, minus 0.9, some number. It can be understood. So there can be multiple scenarios that that can be. So no. If it's between, they will give you at least one of those values. So like, for example, only when it is one number, they will give it like that. Because the only thing you need to worry about is just finding that one number. If it's between, OK, so also this is one number. Let's see if we do have another example like that. No, I don't have. Nobody don't have. Nobody don't have. Nobody don't have. OK. Any other question? Is there someone who is still unsure of what they need to do with their assignment? What we didn't do most is what we didn't do a lot. It's on the proportion. So maybe let's do one example on the proportion. So let's say, for example, this is one of the question. Autism South Africa knows the true population proportion of children with ASD in special needs school is 0.7 and 4. So the reason why I post that focus, I missed where I placed my pen. So the population proportion is given in the statement. Assume a sample size of 100, which is your N, with ASD is selected. What is the probability that the sample proportion of children with ASD in special needs school will be at least 0.7? So the other thing, when it comes to sampling distribution and normal distribution, it's not very strict to use the signs as opposed to when you were waking out binomial distribution, Poisson distribution and distributed probabilities. It's not really, really that strict. Let's say you don't have to conform to the inequality sign and equality sign because it doesn't really matter whether you write your sign, like at least you write it like that when you answer the question. The answer that you will get will be the same as if you put the equal sign to it, right? So usually I don't, in the normal distribution, I don't even worry about the at least for now. But you need to know that, that at least means greater than or equal. That is very strict. You need to know that, especially going on, even when we do hypothesis testing, you will need to know the signs. It's very, very important because the minute you miss write the sign, you're going to get the answer wrong. So but for sampling distribution and normal distribution, it will not matter that much, but you need to know that at least is greater than or you can replace it with greater than or equal. So let's find this probability. So we know that we're looking for the proportion. Sorry, that's proportion. Proportion is P, P of 0,7. So what the question is asking, probability. So we need to find the probability of P greater than 0.7. I can put the equals in there as well. And that is given by the probability of Z of greater than. Now we need to always remember because it's greater than. There are certain things that you will need to remember when you wake it up. Why am I putting X? So it's P minus the population proportion divided by the standard error, which is population proportion times one minus population proportion divided by N. You need to remember that. Always, always remember that if I'm going to find the probability of Z less than A, then the value I will find on the table is the probability that I'm looking for. If I'm going to find the probability of Z greater than a value, then I need to say one minus the probability of the value I'm going to find on the table, which is one minus the table value. Right? So this is greater than. So it means the answer will not just be the answer that we find on the table. We'll need to subtract that from one. So we go and say the probability and we substitute the values. Our sample proportion is always given in the question and also in the statement, as you write it, you have it there. So you just substitute that 0.7 minus population proportion 0.74 divided by the square root of 0.74 times one minus 0.74 close bracket divide by 100 is our sample size. Probability of Z greater than and we need to go and find that probability. We put my calculator there and I use my fraction 0.7 minus 0.74. The reason why I'm not putting 0. The calculator knows you can just start by putting the point number point number. It's 0.0 point. So I normally don't put the 0. Square root of because this under the square root is a fraction. I'm going to also replace it with a fraction and 0.74. Remember that is the population proportion. Right? Not the sample proportion. You must substitute correctly. One minus the population proportion. Which is 7.4 close bracket down the 100 and press equal. I hope you get the same answer as I did. It's minus 0.9. You need to keep two decimals after the comma. So you also need to make sure that you run off correctly. So it will be minus 0.91. Minus 0.91. And then we go to the table. On the table we go to the negative side of the table. We looking for minus 0.9. 0.1, 0.2, 0.9 and 1 at the top. Where they both meet. I know that that is the second column. Where they both meet. That will be the answer of less than 0.1814. 1814. So then we need to, we're not done. One minus. We went to the table to go find the probability of Z of 0.91. Which is the same as 1 minus 0.1814. Which is equals to 0.8186. 0.8186. And that's how you will answer the questions. So you just need to pay attention to the statements given. So the majority of the questions. The way they ask them, they are exactly the same. The only difference is the context that they are giving you. For example, like this one it uses autism. South Africa. Next time they can use schools. They can say, like we did with one of the examples. Yeah, they can say the great eight. Learners in South Africa knows the true population of children with whatever they are talking about is this much. So you just need to know how to identify the things given in the statement. And write the correct symbol relating to those things. And then understand the question asked, whether are you calculating X? Are you calculating the probability? That is very important because this tells a lot. If you are asked to calculate the probability, it means you need to go to the table. And when you go to the table, you need to know the size. So it means in the statement of the question, they will give you whether it's greater than or it's less than. It's at most all those signs that the mathematical signs that you need to know and remember if they give it to you in a sign. It's great if they give it to you in a weight format. You just need to know how to interpret those signs like at least at most less than greater than exceed minimum maximum higher than all those things. You need to know what they mean in a mathematical format in order for you to know the symbol because with the symbol tells you how you're going to go to the table to find the values. If you place the symbol wrong, it means the value on the table you're going to get wrong because you're going to go to the wrong value on the table. And that is not good. Okay, so and that is seven minutes before the end of the session. Any other thing that you want me to discuss with you before we leave, we pack ways, any question, any comment, anything that you still unsure of and you want me to get to it. Nothing. Nada. Are you good? Talk to me. Are you ready to tackle your second attempt, third attempt for those who want to do that attempt? Are you still unsure? If you are unsure and you're not from my group, right? Let me do this. Those who are not from my steps group, since I know that most of you are not from there. Let's do this. Open a new Google thing in my bubble. Now you're reading my messages. That's why I don't like sharing my screen. YouTube. So I'm just going to show you where you can find. Let's just only share that one screen. Okay, so when you go to YouTube, you just Google me. By now you should know who I am. If you don't know who I am, then I am so sorry, because I didn't introduce myself so you can write it out. You need to go to this channel. I know some of you, your lecturers have your channels, their own channels. Possibly, but this is the channel where I post most of the recordings and the recording for today will also come here. So you can come and look at it from this site. So you go and Google my channel and then you click on my name. Possibly because I'm registered. I'm logged in as myself. It will come here to where I am. What I will suggest you do, don't go to videos. Go to playlist. And when you get to the playlist, you will get the created playlist. I don't know how it will appear on your site. There is a playlist called STA 1610 Group 1 E 2022. You going to find the recording under that. And this is also for those who are not following me on my UNISA, especially the ones in my internet group. You can skip the ads. I don't want to give them money. So you also don't have to watch the ads. You can skip them. Okay, that's me. Anyway, so the first video that will pop up will be the first video we did together. But that's not what I want to show you. I want to show you the following, especially those ones who are still lost and you feel like you are still struggling. So on the left panel, I don't know, on your phone, it might look different. So if you scroll, these are all the recordings of all the videos. We only had 10 sessions. So this is the 11th session. Again, let me make myself pass. They don't get my face. Okay, so you go to the channel on the side panel. If you go to the last one will be today's session. But if you go to the 24th of July, especially those who haven't submitted their assignment. Right now, I mean, I'm referring to those who haven't even started those who are even way behind. But if you have started it, then you have done. You can watch other videos. We do have the normal probability video where we discuss the content. We discuss the sampling distribution there. And I also did some activity just to show them show some students who are struggling with how to use the calculator to calculate certain things because the formulas are complex. As you can see them, but just to show them how they can tackle the questions using their calculator, whether it's a case or a shop calculator, how to calculate those and answer the question. You can also watch that. And then last week, we did question we did activities. So I will suggest that the mid. Okay. I hope the lecture is not on this group that we are on right now. But if you can watch this recording. Right on the 24th of July. And do all the practice activities that we did. Possibly, you will pass your, your, your assignment with good marks. Like, if you studied, you will get 100% if you. Yeah, if you are struggling, you might still get 60%. It's up to you. But all this recordings are related. So but I'm saying for to those ones who are not yet there. I will say concentrate on doing the activities from here and tackle your assignment. From the 24th of July and tackle your assignment. And then also watch today's recording and then tackle some of the, the activities and do your assignment and see where you end up. But other than that, there is not much I can give you. I've put myself out for four weeks now. So it's up to you. You use the platforms. And the resources that we are providing you. And don't forget to subscribe. Thank you. Okay, so I need to close this. I don't know how to get back now. Are there any questions before we had ways. If there are no questions or comments, I'm just gonna stop the recording. Yes.