 it seems like I didn't, for some reason, it didn't start. Thank you for reminding me again. So I think every day you will be my class monitor. You will keep me in check to see if I do the recording. Okay, so the recording has started. If I can start from here. We're doing sampling distribution. We're going to look at the basic concept and then we calculate the mean and then do the proportion. That's the objective of today. Okay, so if there is any other question or any question, feel free to ask. Otherwise then we can start. Maybe just a question from me. Yes, and it's not specific to this section, but it's specific maybe to the course. In terms of the exams, it seems it's very calculation heavy. So your understanding of the concepts, you mainly reading the series, so you're able to do the calculations. Is that great? Because it doesn't look like you asked a lot of theoretical questions. It's more calculations. Is that right? Yes and no, but there are theoretical questions as well that you will have to learn. Like if you were in this session yesterday, there was one question that we asked after we did the basic concept that asked you a question to say which one is the correct one, where it asked you about the properties of a normal distribution. Those are the things that you need to learn. So like for example, like now you will need to know the basic concept of a sampling distribution. So you still need to learn the content, the concepts, the theory part of it. Not only the calculations, it's a mixture of goals. But yes, 80% of the time they will ask you to do a lot of calculations. Okay, all right. So essentially how do you apply the concepts on a calculation, into a calculation basically, if I can put it that way? Yes and no, but remember there are certain assumptions as well that you need to learn. It's not even about the concept of how you do the calculations. It's also about the specific content. For example, like explaining what a discrete probability is all about. Explaining that it comes from a discrete variable. Explaining that the data needs to be independent. It has two outcomes. All those who don't use them in any calculation, but it's things that you need to know. And you will see as well when we do the sampling distribution, there are things that you will need to learn and know that are not based on the calculation, but based on the basic concept and basic theory of sampling distribution on its own. And then you also learn how to do the calculation in relation to the theory that you have learned as well. So it's two ways. You cannot say, I only concentrate on the questions. There are certain that you will still need to learn. Okay, thank you very much. Thank you. Any other questions? If there are no questions? Yes. All right. On the normal distribution, where are the values in between? Yes. We left z is greater or minus one and z is less than two. So how will I determine which one to subtract from which one? Okay, so when it's in, when is the between? So z will be between minus one and two. Next. Yes. If this is the question, you always start with this one. It will be your probability of z less than two that you will go and find on the table and you're going to subtract the probability for the second one, this one, because this one is the greater than. So we just go and find from the table, we will find the probability of z less than minus one. The second part minus the first part. So you will have to go through the table to go find the value on the table for the second part and then you go and find the value for the first part on the table, then subtract from one another. And that's it. Okay. Okay. So let's learn sampling distribution. So a sampling distribution is a distribution of all possible values that comes from a population selected and you select a sample from there. Instead of only selecting one like we do with the normal distribution where we only worked with one sample, here you select multiple samples from that one population. So for example, if you have UNISA students, you get a pool of all UNISA students into one basket and you go in there and you select 10 students, you put them somewhere. Then you go again and select 10 students and put them somewhere and go and select 10 students and put them somewhere. And those we call them samples. When you have all these samples, you calculate the means for all the samples, the mean for sample one, the mean for sample two, the mean for sample three. Once you have the means for all those samples, you can now calculate what we call the sample mean of the sampling proportion, which is the mean of all the samples. So you will add all the samples that you have divided by three, which will be divided by the three samples that we have. And in that way, we are standardizing your sample values so that they follow a normal distribution. In normal distribution, we only used one population and we selected one sample. In sampling distribution, we use one population. We select multiple samples from that population. And then we use the average of all the means of those samples. That is taking your sample values and transforming them into becoming normally distributed and call that a sampling distribution. So since we talk about measures that we take from the population and we know that those measures are called the paramitas. Those are the measures that we take from the sample. So if I have a population with the sample, with the population, with the four people in it, it's my population. I can define it as such. So there are four people in this population and I want to take a random sample of four of that population. It's just saying I want to repeat the same process again and again and again, four times a game. So if I take my population and I calculate my mean of those populations, I get that in this population, there is an 18-year-old, a 20-year-old, a 22-year-old, and a 24-year-old. I can calculate the mean of that population and I can see that the mean of that population will give me 21. And I can also calculate the standard deviation of that population and that standard deviation will give me 2.2. And if I look at the distribution of that population and I can see that this is not a normally distributed population because it does not follow a belly-shaped calf, it is a uniform distribution. Therefore, I need to take this and transform it and make it a normal distribution. And to do that, it means I must go to this population and select multiple samples so that then I can take the mean of those samples and try and transform them into becoming a normal distribution. And in that process, I am applying a sampling distribution method. And when I do that with the sampling distribution method, I go and I select four samples out of that population that I have. And when I have that, I go and then if I select four samples, therefore it means I will end up having 16 samples. Sorry, I don't select only four. I select 16 samples. So I have 16 samples. Then I take the mean of all the 16 samples. When I take the mean, so all these values at the top are the averages are the means of all the samples that I've selected. So I select all the samples selected and then calculate the averages of that divided by how many samples I have selected. I've selected 16. And when I calculate them, they still give me the mean of the sampling distribution still looks exactly the same as my mean of my population. Remember my mean for the population was 21. So when I take the sampling, the sampling distribution mean and I start calculating that from the sample, then I get 21, which is the same as my population. And when it comes to the standard deviation, when I calculate the standard deviation of a sampling distribution, because then I have all the samples, I calculate the standard deviation of all the samples and I get 1.5. And remember on the other side, the population, the population standard deviation was 2.2. But on this one, the population standard deviation of a sampling distribution is 1.5. What I just said or what I just did is just to explain that when you have a uniform distribution, you can create a normal distribution by selecting more samples from the population and then transforming it into a by doing that by selecting as many samples from the same population, you will eventually try to create a normally distributed function. If you can look, there we had four population sizes and then we have two sample means where we selected 16 samples from the population, 16 values from the population and we created the mean and created the standard deviation. And you can see that the standard deviation for your sampling distribution is different to the standard deviation of your population and the mean of the population and the mean of the sampling distribution are the same. And you can see that the distribution as well changed from being a uniform to being a normal distribution. And this is just to repeat that. So just to say, if you select different samples from the same population, eventually you will yield different standard deviations of those samples. But also we also talk about the standard deviation of your sampling distribution. In other words, it also measures what we call the variability of your means from those samples that you selected. So it will take, it will tell you how variable your samples from one another they are. And usually that sample, the standard deviation of the sampling distribution, we also call it a standard error. And you will hear about the standard error from today until we do the hypothesis we always going to be talking about the standard error. And the standard error, which is the population standard deviation divided by the square root of your sample size, which is also called the sampling, sorry, it is also called the standard deviation of your sampling distribution. And it's also called standard error of the mean. So standard error of the mean is also called the standard deviation of your sampling distribution of the mean. And it is this population standard deviation divided by the square root of your sample. When it comes to the mean, if your population is normal with the mean and the standard deviation, remember if it's normally distributed with the mean of zero and the standard deviation of one, then the sampling distribution is normally distributed with the population mean being equal to the sampling distribution mean and your sampling distribution standard deviation being your population standard deviation divided by the square root of m. Suppose that we have a population that has the mean of eight and the standard deviation of three. Suppose a random sample size of n is equal to 36 is selected. Calculate the mean of the sampling distribution. All what they are asking me to do here is to calculate the mean of a sampling distribution and is denoted by the mean of the population with an x bar because we're talking about the sample means because we're talking about different means that we calculated from here. So in that, remember it is the same as your population mean, therefore it is equals to eight since I know that our population mean is equal. Second question, calculate the standard deviation of a sampling distribution or what they are asking me is to calculate sigma x bar which is my standard error which is given by the population standard deviation divided by the square root of n and my population standard deviation sigma is three and my sample size is 36. So therefore I get three divided by the square root of 36 and that gives me a calculator. Three divided by the square root of 36 gives me 0.5. It's how you answer the question when it comes to this sampling distribution. How will I know that I need to be answering questions on the sampling distribution? Sometimes they will say it follows the sampling distribution. You will have a hint in the question that tells you a sampling distribution. If they don't say that then some way they will talk about the sample means or the standard errors and so forth. Then you will know that you have to be not calculating the normal distribution but you are calculating the sampling distribution. And there is your exercise. For example, on this exercise it talks about the standard error and it also mentions sampling distribution. So you should know what they are asking you to do. So in this instance they are asking you to calculate the standard error which is your population standard deviation. I am writing the formula down because I'm not going to assume that you are going to remember everything that I just said five minutes ago. So I'm just providing you with the formula. So that is your exercise. Do we have an answer? Ma'am, I think it's number two four. Okay, let's wake it out together. I said standard, the deviation is 28. Yes. And then I said the sampling distribution is 49 square root. Square root of 49. And then I did the calculation and I got four. The answer is four. Yes. My question is incomplete. Next, this one. Ignore that one. It's the same thing. What is the answer? Still working it. To exercise three. It should be easy. They are asking you to calculate the mean of the sampling distribution. Ma'am, is it number one, 900? Why is it 900? Because the mean, it equals, what you said at the beginning, I can't remember, but I'm assuming it's 900. The population mean and our population mean is 900. 900. Yes. Thank you. Any question? Not yet. Not yet. So there is no body with the question. So we can move on to doing the probability. So now let's look at calculating the sampling distribution of the mean and how we calculate the probability related to that sampling distribution. By the end of this 30 minutes, you should know how to compute those probabilities related to the sample mean. And like with the normal distribution, when we convert from the standard data to a normal distribution data, we use the Z-scope. So with sampling distribution, the Z-scope for sampling distribution is done. Calculation, which is your sample mean minus your population sample mean divided by the standard error. And we know what the standard error is. It's your population standard deviation divided by the square root of your population. So remember, if they talk about the standard error, they are asking you about everything that is underneath the fraction, the on the fraction, the bottom part. It's called the denominator. And this is your sample mean. It will always be given to you in the question. So it will say, what is the probability of the sample mean being 300? And you know that that is the sample mean. And then in the statement, in the question, they will tell you that this, suppose this population is distributed with the population mean of 95 with the standard deviation of 5. And you know that that is the population mean and that is the population standard deviation. And you will always get those on the statement. And then they will also continue and say with the sample size of some sort. So and you will know that you need to calculate the sample size from there. So we're going to use the Z-scope to go find the values on the table as well. Remembering that when we calculate the probability, the probability of Z less than a value, we go find the value on the table. We're still going to use those concepts. If we have the probability of Z greater than a value, that will be one minus the probability we find on the table. And if we go find the probability of Z lying between two values A and B, that will be given by the probability that we're going to find on the table where Z less than B minus the probability Z less than A on the table. So the same concept we'll continue with that. Here is an example. Suppose the population has the mean of 8 and the standard deviation of 3 with the sample size of n is equals to 26. What is the probability that the sample mean is between 7.8 and 8.2? So I can go and calculate all this in one. So the probability that X lies between 7.8 and 8.2 is equals to and we can calculate that all in one because we know the formula which is the mean minus the population sample mean divided by our standard error which is the square root of n. Since we are going to convert it to the Z, so you can do it in this way. I'm just doing it in a longer way but you can always calculate the Z outside. You don't have to do it the same way as I am doing it. I'm just showing you other method other than what we just did yesterday. Now we can substitute the values. Remember always your X or your sample mean will always be given in the question. What is the probability that the sample mean you are given those in the question? So it means for this one my sample mean of the first one will be 7.8 minus my population mean is 8 divided by my sample my standard error, sorry, standard deviation of 3 divided by the square root of 36 and I do the same on the other side. Our sample mean 8.2 minus the population of 8 divided by the standard error which is 3 square root of 36. Now I can calculate the whole equation. I know that my standard error was because we did calculate this before we found that it was 0.5. Remember that it makes it easy. So to calculate this we can just say 7.8 minus 8 equals minus 0.2 divided by 0.5 equals minus, so I'm running out of space here. I hope you are able to see what I'm writing there minus 0.4 less than and I go and do the same thing on the other side. 8.2 minus 8 equals 0.2 divided by 0.5 equals 0.4. Now I need to go to the table. Then go to the table and go find the values. I'm not going to go to the table now because we showed you all this. So what I just did just now was up to there on the other screen. So I calculated that and I found that it's minus 0.4 which is less than z and z is less than 0.4. Go to the table, go find this probability on the table which is the probability of z less than 0.40 and the probability you get on the table will be equals to 0.6554. Then I go to the table again to go find the probability that z is less than minus 0.4. So I go to the negative side of the table, go look for minus 0.4 on the left and then I go to the top, look for 0 as a last digit on the table and where they meet find that it is 0.34046. 0.6554 minus 0.3446 I get 0.3108 and that is how you will answer the question when it comes to sampling distribution of the mean. A sample of n is equals to 16. Observation is drawn from a population with the population mean of thousand and the population standard deviation of standard deviation of 200, calculate the probability that the sample mean is less than 105. So you need to go and calculate the probability that z is less than the sample mean minus the population sample mean divided by the standard error which is the population standard deviation divided by the square root of n. That's your exercise and when you do that you go to the table go find the value remember also to apply the concept you flag. The probability of z less than a is the value you find on the table. Probability z greater than a minus the probability you find on the table. The probability of between will be the probability of z less than v minus the probability you find on the table for z less than. When you are done let me know so that I can hear how many people have at least attempted the question and finished. Hello I'm Dan. Thank you. And that's how far away. Come on guys are you still busy? Yes one more minute. Thank you. You make sure you can. So just one more minute as well. Thank you. Dan are you done? Those who said to one minute are you done? Yes you're done. Ha you must let me know when you're done. Okay so who wants to go? Okay let me try. I'll try. Yes. I started by calculating the standard error of the mean okay. I said 200 divided by the square root of 16 and then it gave me 50. Hello? Yes. Okay and then now I went and calculated using the formula. I said 1050 minus 1000 divided by the standard error of the mean which is 200 over the square root of 16. Okay so the answer was 50 divided by 50 and then it gave me 1. So you got the answer is 1,00? And then I went on to look for the probability on the table. It gave me 0.8413 which is option 2. Agree everybody? Yes. Yes correct. The next question please let me know if you are done when I ask and if you are busy also communicate with me so that I know how many people are still busy. The screen is still on the previous exercise. Ma'am that question is it based on the heading of the previous one? I'm done. Tan? I'm done. Are we all done? Yes. Yes. Okay but now let's Yes. Who wants to try? I'll try ma'am. So I said 920 which is the sample mean minus 900. So you said 920 minus 900 divided by 100 sorry which is the deviation divided by the square root of 100 which gave me 20 divided by 10 which gave me 2. I then went to the table I looked for 2 um and it gave me 0.9772. Do we all agree? Yes ma'am. Yes. Yes because also this is the less than. Less than. Your next question remember when you're done let me know that you are done. We did this one ma'am. We did this one. No you didn't do this one. I'm done. This is a new question. Oh the sign. Done. And then? I'm done. Done. We did this exercise earlier but we found that Z was greater than 1,00 remember that that's the same question that we did here where the sign on here was less than. We're not going to go and do all this to save time and we know that we got Z of 1,00. So anybody who wants to try to complete it? Although ma'am I went to the table and and it was 0.1 was 0.8413 and because the sign is Z is more you say 1 minus 0.8413 and it gives you 0.1587. Then you get 0.15. 1587. Which is number three? Any question? Okay without any question. So we can take let's take five minutes break or let's come back at 10 past one at 10 past one then we will do proportions. Please stand up from where you are. Stand up from where you are and go around come back exactly at 10 past one.