 Good afternoon, everyone. How are you today? Afternoon, Lizzie. I'm good. Thank you. How are you? I am good. I can't wait. Thank you, Lula Mille. I'm also sharp sharp. Hi, Lizzie. How are you? Thank you. Okay, we're going to start in one minute. Yes, this month. High five as well. I'm good. Or do you want to raise your hand? Do you want to say something? No, no, no. I was just waving to say I'm also here. I'm waving back as well. All right. Okay, we're going to start in one minute. One minute. Okay. Don't want to leave anyone behind. Okay. So before we start this session today for Mali, I'm going to use this opportunity. Let me share my entire screen, which has so many other things. I'm just going to close some of these things that we're not going to use now. I'm just going to close, close. And I had some questions that many people are asking, oh gosh. Let me stop again the sharing. Let me just, I had it open. I hope it's going to be open to where I want it to be. Okay. Yes, it is open now. Okay. So I had several questions. People asking, how do I know where to find what? And I think we can start there today. So when you log in onto your My Unisa, you get to this platform and then you get to select My Modules and you will have a whole list of them on listed there. Our one has an E1 to it. So if you have 10 modules that you are doing this year and all of them have tutors, then you will have 10 modules. Like I have only three here and then one E-tutor. So you will have 10 modules from your lecture site. And if all those 10 modules have E-tutorials as well, you will have other 10 E-tutorial modules as well as part of that. So your list here will be 20 modules. So you just need to know which one and where to select. So when you go to our E-tutor site, which is the one with an E1 at the end, you will scroll on your right. Is it right or left? Left. On your left, when you scroll down, there are two things that I need to mention for today. So there is this E-tutorial content. Let me change my view to student to view so that it makes it easier. So you will be able to see it in this way on the left hand side menu bar. There are two folders. So you can see the two folders, E-tutorial content and the online sessions recording. The E-tutorial content are the ones that I will always post three days, four days before Sunday so that you can go through the content. And these are videos from the previous year's tutorials. If you click on that, you will get your lessons in terms of your study unit. When you open one of them, it will have recordings. If there are multiple recordings that I shared, there will be multiple of them. It will also have some questions and you can answer those two or three questions depending on how many questions I have loaded, right? These are the things that you need to, in order for you to prepare for the Sunday session, you need to go through this. Some of the work that is done in the previous years will help you to prepare for the Sunday session. And I'm going to also repeat some of these things in the Sunday sessions. Okay, then the Sunday sessions, you go to the online session and how I broke it down. I broke it down by assignment per month because every month you have an assignment that you need to submit. So now we are in July, I expect everyone to catch up and watch everything that was done in July. So you go to July and you will find all the recordings that are relevant to the assignment three that is due in July. So last week's session is loaded there. This week's session will appear here and so on and so forth. Every week you will have that. That is the recordings. Now, the notes that relate to both of these recordings are under additional resources. You scroll down and you get to where it says additional resources. When you get there, there are different folders. You can see there is the statistical table, which is the table that we're using every week. You can use that table so that we all use the same table. Then we have the summary notes. So the summary notes, you will see that it has all the study units from study unit one up to study unit 11. So there is no excuse to say there are no notes. Here are the notes that I have shared with you. And in each and every study unit, there are some activities as well that you can practice. But this study unit note correspond to the video recordings because when I compiled this note, that's when I was offering those tutorials for the previous years. So those previous years, which is the tutorial content, these are the notes for them. The Sunday notes, every Sunday I can only upload it on Sunday because I can only do the notes on Sunday. I don't have any other time during the week to do the notes for today's session. I only do it in the morning or in late morning, every Sunday. The notes are under the weekly online and you will see that it goes according to the sessions. So every session, every study unit, the notes that we have gone through every Sunday, they are here. Today's notes is also uploaded. So there is no excuse to say there are no notes. Okay, so that is how you navigate the system. So if you missed any session, you must know that you come to the weekly online session, you download the notes, you go and watch the recordings from the online according to the assignment. Ignore the others because then you're not going to catch up with the week. Only concentrate on the videos and the notes for that month that we are working with. Okay, that is one thing that I wanted to share with you and I'm going to close that and go back to our session. So let's start with this week's session. We're doing study unit seven, which is sampling distribution. In terms of the session plan, I think I'm going to also adjust it a little bit because your lecturer said your assignment is due up until the 11th of August, I think, somewhere there. So I'm going to relook at this and adjust it a little bit, but that doesn't change next week's session. We're still going to do more questions and answers. And if I think that we have done a lot, then we keep the session as they are. Right. Okay. And you must notice the Sundays that we don't have any session. I've also tried to remove some of those meetings online. So we should not have any problems this time around. So this week, we're going to be looking at sampling distribution. And before we start with this week's one, I need to recap on the last week's one, because whatever we learned in the last session, we still need to apply it in this session. What have we learned in the previous session? We learned the following that from a population, we are able to find the Z score. Oh, if we're not looking for the Z score to standardize the X unit or the observation unit to make it normally distributed, then we can be asked to find the probabilities. So in terms of the Z score, what we learned is that we need to calculate the Z score by using the Z of your observation minus the population mean divide by the standard deviation. And that is the Z score to standardize your X unit. Using the Z score, you can then also go and find the probability. And finding the probability, there are three ways that you are able to calculate the probabilities. You can find the probability of a Z value less than a value. And we know that when we go and find the probability of Z less than a value, then we're going to be using the value that we see on the table, right? Remember that. Since our table has the negative and the positive side of things, all these values have the value of Z value of a less than, regardless. If the question asks that we need to find the probability that Z is greater than a value, then we need to say one minus the value we're going to find on the table, right? What about the probability of between? If we need to find the probability that Z lies between two values, a and b, then we need to go find the value where Z of b of the second value minus the probability of the Z of the first value. So we're going to go to the table, to the table to go find the table value of the second value, which is b minus the table value of a. That is what we have learned. And we also learned that we can work it backwards. But if they gave us the probability, we can go find the X value by finding, sorry, not X bar. If we are given the probability, we can go and find the X value because your X value will be your mean plus your Z times the standard deviation. And this is based on similar to when we were doing the, I'm not going to say it, but, and that is one thing that we have learned. So based on this, what we have learned, we're still going to continue from that with today's session. Now, what I also said, but I didn't stress too much last week is that remember in study unit five, we were saying something like greater than or equals to it's at least, it's called at least, at least. And now when we are working with normal probabilities, when the question is asking you about at least, you can just use the sign greater than. So whether here we put greater than or equal or greater than, it doesn't really matter that much in this study unit as well. If the question says, what is the probability that at least so we know that it's greater than or equal. And when we go find the value on the table, it's the same thing, the probability of a greater than is one minus the value that we find on the table. So we're not going to worry too much about the inequality side to it, but you just need to know the signs in order for you to be able to answer the question. And those are the things that we plan past time. Okay, so now moving on, I'm not touching on all the signs right now later on we will catch on them. I just want to introduce sampling distribution for us to move on. By the end of the session today, you will learn the concept of sampling distribution. You will learn how to compute the probabilities to related to the sample mean. Whether we calculate the probability that a z value for the sampling distribution is less than knowing that we're going to find the value on the table. Or we're going to find the probability of z greater than a value, which will mean that we're going to say one minus the value we find on the table. If we're going to find the probability of a probability of z lying between two values, we're going to find the probability of z less than b minus the probability of z less than a which means I'm going to find the table value minus the table value. For a table value for b minus the table value for a the same concept that I just told you that we used last week we're going to use it today is just that we're going to use a different z formula which almost look exactly the same as the one that we have been using. And that is for the sample mean. Now we're also going to do for their proportion. So we need to be able to calculate the probabilities of the sample proportion as well and also the same method. When we go find the probabilities, we need to know that the probability of a less than is the table value. The probability of a greater than will be one minus the table value. The probability of between, we will subtract the second value that we find on the table from that second value minus the value of the first value. That is how we're going to work in this. So it's just a continuation of what we did last week. Okay, so when we talk about sampling distribution, so last week we were talking about one population. Even today we're going to be talking about one population, but in state of just only focusing on one population and calculating and looking at the measures for that one population. Here we're talking about sampling distribution and a sampling distribution is a distribution of all possible values of a sample statistics for a given sample size selected from a population. So what do we mean? In your section or in your chapter or in your module, you do not need to know the sampling method. However, we also spoke about this in chapter one, remember chapter one where we spoke about the terms that you need to know before you start doing statistics. Like there is a population which is elements, all elements of interest. And we said when the population is too big, we create a subset of that population. And when we create that subset of that population, we are sampling. Now, with sampling distribution, we create in those samples. Instead of only doing one sample, we go to the population, we create multiple samples by using any of the sampling methods. It can be a stratified sampling method, a cluster sampling method, a random samples, all of them. You do not have to worry about knowing those sample methods. All you need to know is from this population, we're going to create now multiple samples. So for example, here we create a sample, the first sample we go to this population, we draw 10 people from that. We go again, we draw 10 people again, we go again and we draw 10 people again. So we create multiple, multiple, multiples of samples depending. So in this instance, I created three samples from the same population. So I take my first sample, I calculate the central measures and central measures of location. I calculate the measures of variation as well. And with the measures of central location and measures of variation, I'm only talking about the mean and the standard deviation here. So I'll calculate the mean, which is the average of this. And I will have the average of that sample. And I go to the next sample, I calculate also the mean. I go to the next one, I calculate the mean. All of them will have different means, right? Even though they all have a different means from the, in terms of the sampling distribution, we want to work with only one mean of the samples. So we know what the sample, we can also go to our population and calculate our, no, not x bar. We can go and calculate our mean. Here we calculate our average, which are our sample means for sample one, sample two, sample three. So with sampling distribution, our population can be not normal. Like for example, the population can just look like a uniform distribution. And once we created all the samples and we take all the samples, we add them together and divide by how many they are, they are three of them. And that will give us the mean of the samples. And if we take the average of all these samples, we can convert a uniform population into a normal distribution. Right. What do I mean by that? So let's assume that I have this population within size. So I've got this four people that I've selected and I said this are my population of study. And in this population, I've got their ages. One is 18 years. The other one is 20 years. The other one is 22. The other one is 24. And I can go on and calculate the population mean. You know how to calculate the population mean is the sum of all observation divided by how many they are. I can go on and calculate the sum, the standard deviation, the population standard deviation. You will not be required to do all these activities at gate. You're not going, in sampling distribution, we're not going to ask you or your lecturer will not ask you to calculate the mean, calculate the standard deviation. They will be given to you. You will be told that from this population, the population mean is 21. The population standard deviation is 2.22. But yeah, I'm just demonstrating that how we calculated the mean and the standard deviation based on what we know from chapter one. Okay. Oh, not chapter three. If I take this population of mine and I create a visualization of the data that I have, because I only have 11111111. And if I calculate the probability of selecting 11A or B or COD, I will get that they have the same probabilities because therefore it will be 0.25. So the force, those that are 18, there will be 0.25, 0.25, 0.25, 0.25. And that creates a what we call a uniform distribution. You can see a uniform distribution is a distribution that looks like that. That is a box, a rectangular distribution. We need to take this rectangular distribution and convert it into a normal distribution. And how we do that is we go back to this population of ours. We create samples of two. Let's do that. So if I go back and I create samples of twos, then I can say if I go and I create, I pick two people from there. Let's assume that I'm taking two 18-year-olds or I go again and it will create an 18-year-old and a 20-year-old and I go again. I pick a 22-year-old and 18-year-old or I go again and I get a 22-year-old and a 20-year-old. So you can see how many, all my samples that I'm creating. So with the data that I had of those four people, if I go and create more samples, I can create 16 possible samples from there. Because if I need to create a sample of two, if there were three, then the information here will change a little bit. But here we're working with a sample size of two. So from those sample size, I need to go and calculate the mean of this sample size. The mean of 18 and 18 will just be 18 because 18 plus 18 divided by two is 18. The mean of 20 and 18 will just be 19 because 18 plus 20 is 38, 38 divided by two is 19. And I can calculate the mean of all these possible samples and it will create the sample means. I will have 16 sample means and this are my sample means. And if I take all these sample means and I draw a visualization of them and because they are 16, how many 18s do I have? One divide by 16. It will give me something like zero point with my calculator. What is one divided by, because 18, there is only one 18 out of 16. One divide by 16. It will be 0, 0, 0, 0, 0, 0, 0, 0, 0, 0. And that is why it's there and there are 2 19. So it will be 2 divided by 16, 2 divided by 16, and that will give me 0.13, and 0.13 it's there. And how many twenties? One, two, three twenties, and twenties there are three, so three divided by 16, and that is 0.19, 0.19 is closer to 0.2 and so forth. And there are one, two, three, four, four, twenty-one. So it will be four divided by 16, and that will be 0.25, and that is why that one is there. And you do, again, for the three, it will be 0.22, the way it will be 0.19, and then twenty-threes, it will be 0.1, the number that I got the previous time, I forgot now, and twenty-four, there is only one that will be 0,06. And there you can see that I have converted, let's go back there, I have converted, I need to go back there, I have converted this uniform distribution, I have made it a normal distribution. And that is what sampling distribution is all about, it's about taking your population which is not normally distributed and converting it into a normal distribution, okay. So in terms of the measures that we just did right now, we calculated the mean and we took the sample mean, so in order for us to calculate the means of a sampling distribution of the means, we take all the means that we have, remember all the means, all of them, all of them, we add them together and divide by how many they are, so there are sixteen of them, so we will take all of them, add them together, and that will give us the sampling distribution mean of the means. So this is what we call a sampling distribution mean of the sampled means, why are we saying sampled means? Because of that sampled means, it's sampling distribution of the means of the sampled means. We can also calculate the standard deviation of the sampling distribution of the mean and the standard deviation like the normal standard deviation that you would have calculated, you will say your mean, your sample mean minus your population mean squared, the sum of them squared divided by how many they are, they are sixteen and you will get the standard deviation of one point five. Now if you remember, let's go back, do you still remember we got the population mean was twenty one, the standard deviation was two point two two, so now we calculated the standard deviation and the mean of the sampling distribution of the means and we found that this is the same as your population mean, but your standard deviation of the sampling distribution of the means is not the same as your standard deviation of the population. And that, yeah, I just want to also explain that we took the, it's the same thing that we did there, we took a uniform distribution and converted it into a normal distribution, the mean of the population is 21, which is the same as the mean of the sample means distribution and the standard deviation of your population is two point two, whereas the standard deviation of the sample means distribution or the sampling distribution of the sample mean is one point five eight. Okay, in a nutshell, what we learned is that the population mean of the population mean is the same as the sampling distribution of the mean mean or the mean of the sampling distribution of the mean or the mean of the sample means distribution. I know that it's very confusing, but that's the other way of reading it. So in terms of the sampling mean of the sampling distribution, we also have the standard deviation. Remember that. So if we select from different samples of the same size from the same population, it will yield different sample means, right? So all the samples will have different sample means. And if we need to check the variability of it, so therefore it means the variability of those sample means is what we call the standard deviation, right? We know that if we look at how dispersed or how far apart all the means are from the mean, the sample means from the means we are calculating the dispersion. In this regard, a measure of variability for the sampling distribution of the means is called the standard deviation of the sampling distribution of the means and it's also called a standard error. So this is also called the standard deviation of the sampling distribution of the sample means, sample means, or it's called the standard deviation of means of sample means distribution. Sampled means of distribution. So if they ask you calculate the standard error, you know that you are calculating the standard deviation, which is given by the population standard deviation divided by the square root of N. So the standard error, which is also called the standard deviation of a sampling distribution of means, which is also called the standard deviation of sample means of distributions is given by your population standard deviation divided by the square root of N. What you need to also take into consideration is that when, let's remove all the ink. When your mean decreases, then the sample size will also decrease. So when your standard error of the sampling mean, so when this value decreases, it means also the sample size will increase. So the bigger the sample size, the smaller the standard error. So if this is big, this will be small. So it works in opposite. If this is big, then that will be small. Right? As your N increases, your standard deviation will decrease. So in summary, from what we just learned right now is that if a population is normal with the mean and the standard deviation, the mean mu and the standard deviation sigma, then the sampling distribution of the means is normally distributed with the mean equal the sample, the mean of the sampled distribution is equals to the population mean. And your standard error will be equal to the population standard deviation divided by N. You just need to remember these two formulas. You need to know that the sampling distribution of means, mean is the same as your population mean and your standard error or the standard deviation of the sampling distribution of means is also the population standard deviation divided by the square root of N. Okay, we're not going to talk much. We can also apply the same concept using the central limit theorem. As the sample size gets larger enough, your sampling distribution becomes almost normal. So if you look at our blue sampling, our blue sample visualization, it shows us or it demonstrates that this distribution, the blue distribution, it is right skewed because the tail is to the right. But as we increase the sample size, if we increase the number of sample sizes that we pull from this population, then your distribution tends to become normal and it will take this normal distribution shape. And that is the central limit theorem. The larger the N, the more your distribution becomes normal. And that is what I just explained. I don't have to explain it again and again and again. Let's look at this example. Suppose a population has the mean of eight, the standard deviation of three. Suppose a random sample size of N of 36 is selected. Now, with sampling distribution, so the previous one, they didn't give us the N. When we were looking at study unit six, you are not given the sample size. You only given the mean and the standard deviation and you are asking, you are asked other questions to calculate the Z and all that. With sampling distribution, the other thing to recognize that this is a sampling distribution is that apart from you be given the mean and the standard deviation, you will be given the sample size N of 36 or they will say the sample size is this much. So you always need to remember that. So the first one says calculate the mean of the sampling distribution. So this is for you to calculate. I am not going to be calculating. So the mean, we know that it is the same as that. What is the mean of a sampling distribution? What is the answer? Calculate the mean of a sampling distribution. Anyone? I think it's equals to eight. That is equals to eight because it's the same as the population mean and you are given the population mean. So there is no need for you to do any calculations here. Number two, calculate the standard deviation of a sampling distribution of means and that is given by the standard deviation is given by your population standard deviation divided by the square root of N and here you need to be calculating. You are given the standard deviation, the population standard deviation. You are given the sample size. You just substitute and calculate. Sometimes they might say calculate the standard, the standard error. They mean one and the same thing. So our population standard deviation is three. Our N is 36. And what is the answer? 0.5. That is 0.5. And that's how easy it is to find the sampling distribution of means, standard deviation and the mean. Now, if we need to calculate the probability of a sampling distribution, then we need to use this formula, which is the Z value or the Z score of sampling distribution. In normal distribution, chapter 30, you need six. We learned that the Z is X minus the mean divided by the standard deviation. As you can see that this almost resembles that. So what is different is the following. Instead of using the X unit, we use the sample means because we created multiple samples. So we're using those sample means, which is why we have the sample means. And we know that our population standard deviation is the same as the sampling distribution mean and our standard error. We know that it is the same as the standard deviation divide by the square root of N, right? So if we convert the normal distribution formula that we had before into the sampling distribution, this is how it will look. So our Z will be given by the sample mean minus the population mean divided by the standard error, which is the population standard deviation divided by the square root of N. And that is the Z value that we're going to be calculating or using. So this is study unit six and this is study unit seven. In study unit seven, the formula looks like that. In study unit six, the formula looked like that. Remember, in study unit seven, the only thing that is going to help you know whether are you calculating Z or Z from study unit six or Z from study unit seven is just that in one of the sentences, they will give you the sample size. Whereas in the normal distribution, they don't give you the sample size. And next week, we will look at the three differences so that you are able to identify the questions correctly. So now let's calculate the probability. Suppose the population has the mean of eight, the standard deviation of three. And suppose a random sample size of N is equal to 36 is selected. What is the probability that a sample mean between seven point the sample mean is between seven point eight and eight point two? The other key thing that we need to recognize when we read the sentence is that in your sentence, it will give you the word sample mean. So we know that this is sample mean. This is sample mean, right? So in the question, they will tell you that this is a sample mean. And if that, then we're going to calculate Z of sample mean minus the population divide by. The population standard deviation divided by the square root of N, which we can first start with the seven point eight. Our mean is eight N is 36 standard deviation is three. So we just substitute. We start with the seven point eight. So it will be seven point eight minus our population mean of eight divided by our standard deviation of three divided by the square root of 36. And you calculate that. And we do the next one, which is the mean minus the population divide by the standard deviation over the square root of N. And this is for eight point two. So it will be eight point two minus the eight divided by, and we can repeat the same thing, square root of 36. And you calculate both of them. And once you have calculated, you will find that. Seven point eight minus eight divided by three divided by the square root of N is minus zero comma four. And eight point two minus eight divided by three divided by the square root of 36. You'll find that it is zero point four. And we're going to use the table to go find the probability that Z less than zero point four minus the probability that Z. Less than minus zero point four. And we go to the table. Remember to go to the table. So let's go to the table. So go into the table. We're looking for two values minus zero point four and zero point four. So we have to go to the positive and the negative side of the table. First, we go to the positive side because we're starting with zero point four. So zero point four and at the top. So if your answer was zero point four, just a zero at the end. Remember the last digit and the two digits, one before the comma and one after the comma, we find them on the left. So here we're looking for positive zero comma four. And that is zero comma six five five four. Right. And you write it down zero comma six five five four. Then you go and look on the negative side for minus zero comma four. And we know that the first column is zero. And that is the value that we are looking for, which is zero comma three four four. And zero comma three four four six. And we calculate and we find that the probability is between zero comma three one zero eight. The probability that X, the sample mean is between seven point eight and eight point two is zero point three one. So we took a distribution and converted it into a normal distribution. We standardized it. And that's how easy and straightforward it is to calculate the probabilities similar to what we did last week. Okay. There is your exercise. We can do this together. So a sample of n is equals to 16. Observation is drawn from a normal population with the population mean of thousand. And the standard deviation of 200. Calculate the probability that a sample mean is less than 10, 1050 now. They gave us the sample size of n is equals to 16. That's one of the key things that I will know that I need to go and use the z of your sample mean minus the population divide by the standard error, which is the population standard deviation divide by the square root of n. So now everything is identified. It's easy to substitute, but the question says it is less than. So we need to also be careful about what we're doing. So I'm going to change my equal sign from here to a less than sign. And I'm going to put that into the probability. So that then I can calculate my probability because it says calculate the probability that the sample mean is less than 1050. So the sample mean is given in the question. So we're going to just substitute, which is z of less than our sample mean is 1050 minus our population mean of a thousand divide by the standard deviation of 200 divide by the square root of our n 16, our probability of z less than a value. And that will be equals to if you have calculated it, you can give us the answer. 1000 let me know if you get the same. 1050 minus 1000 is 50. Yeah, so you also get one. So just put 00 at the end because one is the same. Remember, you need to always keep it to two decimals, right? Even if your calculator doesn't do that, you just need to keep it to two decimals. So do you all get one? So now we go to the table. It is positive. So it means we're going to go to the positive side and remember, it's a less than. So when is the probability of z less than a value goes directly to the table? So we go directly to the table to find our probability. So let's go find the probability. It's positive. So we go to the positive side and we look him for one and 00 at the top. And the answer is 0,84130. And I don't have 0,1483 here. Oh, no, no, no. Sorry, my bad. Let's see. Maybe I went to the wrong. Let's double check that. Let's double check. Is it one? Yes, it is 1,0. It's 0,84130. Yeah, so we can also scratch that one out. 0,84130. Maybe next time I will double check this assignment questions. The tutorial letter questions before I post them on here. OK, any questions? Any questions? Any questions? Are we all good? Are we good? No comments? Now let's move on to the next question. Sorry? OK. So now let's move to the population proportions. So in terms of the population proportion, the same thing that if what we did with study unit 4, if they didn't give us the probability, they would have given you the observations that satisfies that probability. And then you can calculate the probability. Similar here. So they will give you the population proportion, which will be the proportion of the population for some of these characteristics that you have or you are calculating. And your sample proportion, if they don't give you the sample proportion, which will be the proportion calculated from the sample, then they will give you observations that satisfies the sample proportion. And you can calculate it by taking those out cap divided by n. And that will give you the sample proportion. And always remember that your proportion can only be between 0 and 1. If you calculate something and it's bigger than 0 and it's bigger than 1, then it means there is something wrong that you have done. You need to double check that. So our population proportion will be denoted by the pi. And our sample proportion will be denoted by a p. Similar to what we just did also, the mean of the sampling distribution, the mean of the sampling distribution of the proportion will be similar to your population proportion. If they ask you to calculate the mean of the sampling distribution of proportions, know that is the same as the population proportion. If they ask you to calculate the standard era or standard deviation or standard era of a sampling distribution of proportions, then you know that it's given by the square root of your population proportion times 1 minus the population proportion divided by n. Everything that we do now, you need to always remember it. Even next month when we do confidence interval, even the following month when we do hypothesis testing, the following month when we do confidence interval and hypothesis testing, this is the base of what you will learn next month. So you need to get this right. You need to know this section. You need to know these things that we are busy with right now. Right. The mean is the same as the population proportion. This standard deviation or standard era is the same as your square root of your population proportion times 1 minus the population proportion divided by n. Right. Let's look at this example or exercise. From the past knowledge Africa check, from the past knowledge Africa check knows that the true proportion, and there is our key weight, the true proportion of ghost profiles on Facebook is 0.2. And that's another key because then it's not a whole number. It is a proportion. So they have given us our population proportion. Your population proportion is always stated in this statement, not in the question. So that will be your population proportion. Suppose that we take a sample size of 200 and there is our n of Facebook profiles and found that only 34 to be ghost profile, and that is our x, because they didn't give us the sample proportion. The question is, what is the value of the population proportion and the sample proportion? So they are asking us, what is the pi and what is the p? What we know is the population proportion is given in the statement. The sample proportion, if they haven't given us the sample proportion in the statement, we can calculate it because it's x divided by n. So what is our population proportion? 0.2. Population proportion is 0.2. What is our sample proportion? It's 34 divided by 200, right? Which will be equals to? What is 0.17? Which is 0.17. So our population proportion is 0.2. Our sample proportion is 0.17. So which makes option number for our correct answer. If we need to calculate the probabilities, then we need to standardize our sample proportion. And to standardize the sample proportion, we use the z value. And the z value is your sample proportion minus your population proportion divided by the standard error, which is the population proportion times 1 minus the population proportion divided by n. We must always remember this is the standard error. So the value underneath the square root, or the value with the square root, anything in the square root and the square root is what we call the standard error. So let's get an example. If our population proportion is 0.4, our n is 200, what is the probability that the sample proportion lies between two values? We do the same. We can first calculate our standard error because it's a complex formula at the bottom there. So we can calculate it and get the value of the standard error since our formula is our p minus the population proportion divided by the square root of our population proportion 1 minus the population proportion divided by n. So we calculate everything, the square root, you can see that all this is everything there. So we calculate that first and we get 0.03464. Then now we can calculate our z. So to find the probability that the sample proportion lies between two values, we standardize it by using the z and we use the formula. Our sample proportion, the first one, is 0.4. Our population proportion was 0.4 divided by our standard error which we calculated. The second one, our sample proportion is 0.5. Our population proportion is 0.4 divided by our standard error. And we find that z lies between 0 and 144 and we go to the table to look for the probability that z is less than 1.44 minus the probability that z is less than 0.00. Going to the table, going to the table, we can, that we first need to find 1.44. It is in the positive side, 1.4 and 4 at the end where they both meet, which is 0.9251. And 0.00, going to the table, it's also positive. Also from the positive, we go and look for 0.00 at the top and that is 0.500 and we subtract one from the other and we get 0.4251. And that is how we standardize the values, especially for the proportions. Let's look at another example. From past knowledge Africa check knows that the true proportion of ghosts profile is 0.2. We know that we've calculated that previously, that is our population proportion. Suppose that our sample size n is 200 and we found that our ghost profile x is 34 and we know that our P, it was 34, sorry. It was x divided by, we did calculate this but I just want to repeat it here again. It's 34 divided by 200, which was equals to 0.17. We calculated that. The question is, what is the probability that our sample proportion is greater than or equals to 0.17? Now this tells me that I'm going to be finding the probability of z greater than a value and to find that probability will eventually have to say one minus the value we find on the table. Now let's go and find the probability. So the probability that z is greater than or equals to the sample, we're not talking about sample mean, we're talking about population proportions. So the sample proportion minus the population proportion divided by the square root of our population proportion, one minus population proportion divided by the square root of n and we can then calculate this. P is equals to z, greater than or equals to our P was 0.17 minus our population proportion of 0.2 divided by the square root of 0.2 times one minus 0.2 divided by n of 200. And that will be P of z, greater than or equals to and let's calculate that. You must give me the answer. 0.17 minus 0.2 divided by 0.2. Square root of two times one minus 0.2 plus back it divided by 200. Let me know what you get. I will let you know what I get. I get minus 1.06. Let me know if you also get the same and if you also get the same, therefore it means we get z of less than minus 1.06 because we need to go to the table now. And since the table contains the values of less than, so we're going to take the complement of that value. Let me get on the table. So let's go to the table. Come on. Minus, it's on the negative side. So we go to the negative side on the table and we're looking for negative one. Now I need to minimize this so that I can remember. We're looking for negative 0.6. We're looking for negative 1.06. So negative one is there and we go to the top. We look for six. So in this column, they both meet and that is 0.1446. 0.6, which is option number four. And that's how you find the probabilities. Okay. Lizzie, sorry, are we not supposed to subtract it from one? Yes, you are right. At least we've got someone who is wide awake. Yes. One minus zero comma one, four, six, six. Yes, four, four, six. Yes, you are right. My bet. I mean, I have to finish. Okay. So one minus 0.1446 is equals to zero comma eight, five, five, four, which is option one. And I guess that's how most of us do in the assignment. We see the first answer that we see and we think that that is the final answer. We just need to make sure and double check before we finalize our answers. Yes. So we would have gotten it wrong. Otherwise in your assignment, we would have gotten at least maybe two or one, depending on how generous your lecturer is. But the answer is option number one. Okay. And that concludes what I needed to share with you today. So, but we still have more exercises. Let's see what time we, how many times we have. In summary, we looked at sampling distribution, how to define the basic concepts of sampling distribution and how to calculate the probabilities of a sampling distribution of the means and the sampling distribution of the proportion. All you need to remember are the following. For the mean, the sample mean is the same as your population mean. Your standard error or the standard deviation of the sampling distribution is the same as the population standard deviation divided by the square root of N. And to find the probability, we use Z of means minus the mean divided by the standard error. All right. For the sampling, for the proportion, we know that the mean of their proportion, sample proportion is the same as your population proportion. And your sample proportion, if not given, it is calculated by using the observation satisfying the proportion divided by the sample proportion. However, the standard error, which is denoted by sigma p, you see that it's the means, the proportion. It's given by the square root of your population proportion one minus this population proportion divide by N. And your Z is given by the sample proportion minus the population proportion divided by the standard error, which is the square root of your population proportion minus the population proportion, one minus the population proportion divided by N. It's so exhausting. And calculating the probabilities, always remember the probability of Z. Less than a value, we always go to the table. The probability of Z greater than a value, we always say one minus the table value. And the probability of between, we will say the table value of Z less than B minus the table value of Z less than A. We just go to the table and get all these values. Right, that's as easy as it and straightforward. However, sometimes in your assignment and in your exam, always remember that you need to be able to answer the questions, whether it's calculations or whether it's theory. And let's look at some of the activities or exercises. So let's look at this one question. And this is a theory question. So you do not have to study only how to calculate certain things, how to use the formula and all that you need to also know the properties of these things. Because the questions can be either in a calculation formula format or they can be theory questions. So one of those are this. The standard deviation of the sampling distribution of the means is also called the standard error. It's the standard error. And I have been saying it so many, many, many times. So you just need to always remember the standard deviation of the sampling distribution of the means or the sampling distribution of the proportions, it's called the standard error. Another question, which is more of theory can look like this. They can ask you, which one of the following statement is incorrect? The mean of a sampling distribution of the means is equals to the population mean. Is that correct? We have just done that. The mean of the sampling distribution of the means is the same as the population mean. That's what I'm asking, is that correct? Is this statement correct? So the statement, okay. Yes. The statement is correct because the sampling distribution of means which is denoted by the subscript means is equals to the population mean. The z-score of the sampling distribution of the mean is equals to the difference between the sample mean and the population mean divided by the standard error, which is divided by the square root of the sample size. Huh? Hey, hey, hey. Okay, let's see what this is saying. So this says the z value is the same as the difference between the mean sample mean minus the population mean divided by, because it says divided by the standard error, which is the standard deviation square root, standard deviation of sample mean. And it also says that divided by the square root of N. Is that correct? Well, I know that this should be z of the mean minus the population mean divided by the population standard deviation divided by the square root of N. So is that the same? Let's go back. I can go back and show you. We know that the standard error, which is sigma mean, is the same as the population standard deviation divided by square root of N. So you need to pay close attention. So there they say it's the mean is divided by, the square root divided by that. So that is incorrect. So that will be the question that is incorrect. But we can look at the other question so that you can see how they ask you tricky questions. The standard deviation of a sampling distribution of the mean is equals to the standard deviation of the sampling of the standard deviation of the population divided by the square root. So what they're saying here is, the standard deviation of the sampling distribution of the mean, which is sigma x bar, is given by the standard deviation of the population, which is sigma, divided by the square root of your sample size, which means this is correct. We know that that was correct as well. That's how you define that. Remember, it's the same as that. That's what they are asking you to know. Number four, a sampling distribution, a sampling error is the error resulting from using a sample statistics to estimate a population characteristics. And this is one of the things that I didn't touch, but I can just give it to you that, yes, you will create a sampling error when you do your estimate, when you estimate, or you will create an error as you estimate the population parameter because you're not going to get exactly your sampling. Let's say you calculate in the mean, the sampling or the mean of the samples won't be equal, exactly equal, exact, exact, exact. The resemblance of the population mean, there will be some slight adjustment errors that happens on the bat anyway. In terms of that, that would be correct. So from now on, you know that a sampling error results from calculating a, or when you are using your sample characteristics to estimate your population parameter, it will create an error. So that is correct. Option five, it says, regardless of the shape of the distribution, as the size gets larger, as your sample size gets larger enough, the sampling distribution of the mean is approximately normal. Remember, central limit theorem says the larger your sample size, the bigger your sample size, the more your distribution becomes normally distributed. And that is what they say, yeah. So that is correct based on central limit theorem. So the larger your sample size, your distribution, the more your distribution tends to become normal. And those are the things that you need to know as you go through your module. Okay. Now, I'm not gonna do all these questions. I just want to demonstrate something because one of the questions that I received this week was that, how do I know what is this question asking me to do and I get so confused? And I said, you will notice that the majority of the questions are exactly the same. The only thing that is different is the numbers that they are giving you. So for example, this is one of the questions. We've done similar questions before, but I'm just gonna, we can look at this one. So this is easy to calculate because this says the diameter of a branding ping pong balls is approximately normally distributed with the mean of 1.3. So they give you the population mean and the standard deviation of zero comma, they give you the standard deviation. So it's sigma and the mean. So now we know that we're doing sampling distribution of the means because the minute they give me the mean and the standard deviation, this are the means. And anyway, because this value is more than one, so it cannot be proportions anyway. If a random sample of four, so they give us our N, N of four ping pongs are selected. The mean and the standard deviation of the sampling distribution of the means is, so they are asking you what is the mean and what is the standard deviation? That's all what they are asking you. What is the mean? Easy, right? Answer should be quick. We can spend literally two seconds on this question. I think it's 1.31. 1.31, correct. And what is the standard deviation because they are asking you to find the standard deviation of the sampling distribution, which is sigma x bar, which is the population standard deviation divided by the square root of N. We identify that our standard deviation is zero comma, zero eight. Our N is four square root of four. And what is the answer? Should be easy and quick. Answer is? The answer is zero point zero four, right? Which then it is option number, option number four. Question four, based on the same information that we just used there. So the next question they can ask you, what is the probability that the mean is less than? So the sample mean, so they giving you the sample mean is less than, we know that it's less. And therefore you just need to be able to calculate the probability. So this can be question number one and then question number two that follows, as you can see that they are almost related. The information is there already. If you are able to identify it in the previous one, you can identify it in this one as well. The mean, the standard deviation, and this is our x bar and they're asking you to calculate the probability that the mean, sorry, your x bar is less than 1.28 and you can calculate that. But I don't want to do this exercise right now. I want to come to this point because then now, yeah, I want to demonstrate the differences or the easy way of waking out the questions and using the examples that you have because if you have written down all these exercises and did them on the book, the next time you get a question from the assignment, you can look at what you did from your examples and exercises and related to the question you get from the exam or the assignment. Now, when we did the example previously, remember the 1,050, the question read like that. A sample of N16 observation is drawn from normal population with the population mean of 1,000 and the population standard deviation sigma of 200 calculate the probability that the sample mean is less than 1,050. Most of the values given or the signs or symbols given to you. Then we looked at now before this exercise, which I said I'm not gonna do, but the question we looked at it, which gave you the mean, the standard deviation and I showed you and I showed you that this, you can just write there the mean and that you can write there the standard deviation because they will give it to you in this statement. They will also tell you that the sample and you will know that that is your N. In the question, they will tell you that the sample mean, you will notice there, the sample mean, sample mean is less than and there is less than, there is less than. One person said they are not sure about the sign and the other time someone said less than is the same as greater than, like they wrote less than, the question was asking less than and they did this as the sign. Now, if you forget what less than greater than mean, always remember that less will be left. Less, left current, right? Less, less current, less, left current, which is less than. So it means left current, if you use your elbow, you will see that you do your left current. If it's not less than, it says it's greater than. So great right and great means bigger. So you can also use the sign to say, but if I look at the number line, bigger means going to the bigger side. If I'm writing numbers like one, two, three, going that way is bigger. It's always going to be right, right? It's always going to be right, right? So then you know that it will be right. Current, right, current, which will be right. If you use your elbow, right, is your right hand, your right, which will be greater. That's how you identify these things. But that is not what I wanted to discuss with you. They left the right for now. How, what I want to discuss is the values changes, but not actually the values, but the things that are given. So if you look at this, it talks about the sample from a normal population. There is no actual, like a real, real life example behind it. With the second one, there is a real life example because they talk about the ping-pong. Let that not confuse you. All what you need to look out for is the key weights. The mean, the sample, you see sample, sample. So those are the key weights. The mean, the sample, the standard deviation, those are your key weights. The probability in the question, in your question, the question will tell you the probability of the observation or the age that will be for the normal distribution. In sampling distribution, it will say the sample mean. So you know that that is your x-bar. Sampled mean, that would be your x-bar. Now, if I look at these two questions, question number four and question number five. Question number five is the one that we'll follow. You can see there, it's Africa check that you would do. If you look at these two questions, you will clearly see that almost the question is the same as the first one. So in state of just starting a sample of, they added suppose Africa check and the sample, right? And then they talk about the mean, be 900. You can see there is the mean of 900 and the standard deviation and the standard deviation of 200, as you can see that almost exactly, exactly the same. It's just, they just changed there and there the weights. Next type, they can just remove Africa check and put their shop right, conducted further research using a sample of 100. You cannot get confused with that. It's just the name that they swapped, but everything stays the same. Sample, the mean, the standard deviation will be given. Then you get the probability. They can write it in weight format or they can write it in a mathematical format. You need to know all this. So this less than or equal is the same as less than in terms of how we work with these questions. So these three questions, you are going to do exactly the same thing. Is the probability of a less than? Less, the value you find on the table will be that value. And that is one thing that I can show you how to identify the question because you, all the questions are the same. This might come from past the tutorial letter. This might come from a past exam paper. That might come from a past exam paper as well. So you can see that the format is almost exactly the same, questioning is just that they just change the context of how they asking the question. That's it. So question, exercise five, you need to find the probability of the sample mean less than 920. This one says at least, we know that at least is greater than or equal, which then it means it's the same thing you went to find the probability that the sample mean is greater than 95. You can write it that way because then when you use what you know that z of greater than a value, it will be one minus the value you find on the table doesn't really matter whether you put an equal sign or not equal sign in this regard will not change what you're going to be doing. And this is the same. Remember, we did an exercise that looks exactly like this. What is the sample proportion? What is the standard error? You just need to know that the sample proportion if they didn't give it to you, you will just need to go and calculate x over n. They give you the sample, they give you your x, they give you your population proportion because they tell you that the proportion of success and we learned about proportion of success instead of unit five, it's path. So the sample proportion, the standard error, we know that the standard error is standard deviation of p, which is the same as the square root of your population proportion, one minus population proportion divided by n. Remember that, and then you answer the question. Finding the probability, this is proportion. Now, here is the other catch. How do I know that I'm dealing with proportion than I'm dealing with the mean? So going back one step back the back, population mean standard deviation, right? And in this question, it says the population mean and the standard deviation of 90 and 18, respectively. So 90 goes with the mean and 18 goes with the standard deviation respectively, right? But the mean standard deviation is different to when they tell you that proportion. So the key weight will always be proportion. Proportion will tell you which one you are dealing with, whether it's the mean or the proportion. And proportion will always have decimals and those decimals will be zero comma something. It will never be more than one and it can never be less than one, less than zero. Less than zero or more than one. It's always going to be between zero and one. And also they will tell you the sample proportion, calculate the sample proportion and it's at least which is greater than or equal, which is the same. And this is between sample proportion and you can be asked to calculate the proportions here as well. So there you have two things, the yes and the no's. So you need to be able to know which one you're calculating because if they say, you need to calculate the, and I can see, yeah, the difference because they didn't say which one proportion you are calculating. Is it the proportion of yes or the proportion of no's? You can validate that. This is one of the previous exam exercise from one of the past exam papers, but I didn't check, validate the questions and the answers. It might also have some errors here, but if not, then it should be fine. And this is the same. What is the standard deviation and the mean? We did this. You know what the mean is. The mean is, and what is the standard deviation of the sampling distribution? So the questions are almost exactly the same, same, same, even in the exam you will find almost exactly the same questions. So yeah, remember the N is very important whether you're doing the Z score normal distribution and especially now when you have two question, two chapters that looks exactly the same. How you will identify study unit six and study unit seven is based on the N. Study unit six, they don't give you N. They don't give you that. So you will need to know that the minute there is a sample size or there is an N reference to the question or statement, then you know that this is sampling distribution. And you should be able to calculate the probability of between the probability of above and the probability of more than, more than. So all of them, they are greater than. Standard error, always remember that. And looking at the information given, if it's the mean, therefore it means they are asking you to find the standard error, which is the population standard deviation divided by N. And find the probability of between and finding the probability of at least, which is greater than, that's it. I wish you all the best. You have the notes. You can go and do all these activities. If you're struggling to answer any of them, there is the WhatsApp group. You can always get in touch with us. Show us how you calculated it. And what is your answer? And then we should be able to assist you in staring you into the right direction. Other than that, are there any questions, comments? Yes, Desmond, do you have a question? No, no, no, no. Oh, I used the hand that I raised long time. Yeah, that was then. I don't know why it doesn't release it. Okay. Right. So many things to remember. Yes, so many things. Yes, especially this section, like the sampling distribution, because we do two things in there. The sampling distribution of the mean and the sampling distribution of the proportion. So it becomes very difficult to remember any of these things. So you just need to practice and practice. But the more you practice, the more you go through the activities and work out, you will see that it's actually easy. Okay, thank you, Lizzie. Thank you very much for your help. No problem. If there is no other comment or question, then enjoy the rest of your Sunday. I will see you next week Sunday. Okay, thank you, thank you, Lizzie. Bye. Bye-bye.