 Good morning. Good morning. Good afternoon. Good afternoon, ma'am. How are you guys doing? Yeah, we are cool. We are great. Come on. We are cool, ma'am. We are great. Sorry, I thought that was a beauty. Yes, welcome to your session today. Today we will be doing sampling distribution. If you have any question or query, now is the time to ask any question or comment. What we previously did regarding the module, anything? Yeah, assignment two. Oh, Jesus. What's wrong with assignment two? Are you struggling? Yeah, yeah, I'm struggling. I really struggle. But I'm going to do my second attempt tonight. Did you at least watch the recordings and did you do the activities and practice? I've tried, but after tonight I will communicate with you after my second attempt. Okay. Thank you so much. After the second attempt, why can't you communicate with us now during the day when you are still preparing to do your second attempt? So that when you go and do your second attempt, you are sure of what you're going to be doing there. In terms of the activities that we did previously, if you are unable to answer those activities on your own, then it means you need to ask for help so that we can help you understand how to answer those activities. Because if you don't know how to answer the activity, you won't know how to answer the assignment questions as well. So practice now with the activities. Ask questions if you still start. Let's help you. And then you go into your assignment. Okay. Thank you, Liz. I think on assignment two I was not prepared enough and then the time was running out for me. Now that's why I say let me just go through the other activities. And then after my second attempt, I will shout for help. Okay. Of course, assignment one was number one. Assignment two I didn't practice much. Okay. Thank you, Liz. I think I had the same problem in assignment two. When it comes to the theory questions, that got caught me a bit off guard. There were some sections in assignment two where I could fly through because I know how to do the calculations and then it goes back to theory questions and I'm completely lost again. It's also just a matter of watching the videos again. That's why I asked you guys now what videos all covered assignment two. So I was really watching all those videos and doing the activities so that I can do my second attempt. Okay. And then also I get confused with the table sometimes. I just need to focus on that table. Okay. Yeah, especially with the binomial tables and you need to be very careful when you use your binomial table as well in terms of what probability of success they have given you. Whether you must use the left hand side or you must use the right hand side based on the probability that they gave. Yeah, but I believe in if you practice a lot and do more activities. You might master the chapters as well. Do not rush to do the assignment without understanding the concept as well as knowing how to do the calculations. Okay, so let's do sampling distribution. So the last time we met, we discussed normal distribution and the same concepts we learned in the normal distribution. We're going to continue with them today. With normal distribution, we're only looking at one population with sampling distribution. We look at multiple samples that comes from that one sample population as well. We're going to be learning how to use the same table, the normal standardized cumulative standardized normal distribution table that you used in the normal distribution. We're going to learn how to still do the calculation for the Z score to standardize the means and then using the Z score value to go find the probability on the table. So we're still just going to do the same thing that you have learned. So let's get to it. So by the end of the session today, you should learn the concept of sampling distribution. You should also be able to calculate the probabilities relating to the sample mean or calculate the probability relating to the sample proportion. And the sample proportion is another avenue that we are introducing. You've learned how to calculate the Z score using the standard, the normal Z score, which will look almost exactly the same as when we calculate for the sample mean. But we're going to introduce the sample proportion Z score formula as well today. So when we talk about sampling distribution, like I said, with normal distribution, we only look at one population and we estimate or we approximate the observations that we get from there in order to standardize them into a normal distribution. With sampling distribution, it's a distribution of all possible values of a sample statistics given for a given sample size selected from a population. So like I said, in sampling distribution, we're going to use samples selected from that one sample. So in state of using one population, we're going to use multiple samples but selected from the same population. So like example is the one that you see on the slide. We've got three different samples that we selected from this population. And when we calculate the average of those distribution, you can see that the distribution might also be, they might look almost exactly the same but they might be different in terms of how we selected the people that are presented in the sample. And you can see that the mean distribution or the distribution of all the three samples, you can see that they are not actually normally distributed. So with sampling distribution, we're going to take all these sample means and we're going to make sure that we convert them into a normal distribution. So we will take all the samples in average or sample means and standardize them in order for them to be in or to convert them to be normally distributed. So when we talk about the sampling distribution measures like parameters for the sampling distribution. So from a normal distribution, we know that we have a population and we can calculate the mean of that population and we can calculate the standard deviation of that population. And if we do that, for example, if this is our population and it has the values 18, 20, 24, we can see that our distribution will be a uniform distribution. This is not a normally distributed distribution, it's a uniform distribution. So we can take this uniform distribution and convert it to a standardized normal distribution by re-sampling from this same population multiple samples and then use those multiple samples to standardize the distribution. And when we do that, then it means we are doing what we call a sampling distribution. And when we select multiple means from those samples and we calculate the means of those samples, we get that the population mean of that sampling distribution or the sample sizes, all of them. The average mean of those gives us 21. And when we calculate the standard deviation of those, we find that it will give us 1.58. If you look at, if we only had one population, you will notice that we had the mean of 21 and we had the standard deviation of 2.2 from this data set. So if we selected four other samples out of this, so re-sampling the values from this population, we get the sampling mean or the mean of the sampling distribution of 21, which is the same as the mean of the population, and we find the standard deviation of the sampling distribution to be 1.5, which is 1.6, which is less than the population standard deviation. You will notice that we divide each one of them by 16 because we selected four samples. So they were four times four times, sorry, four plus four plus four plus four is 16. So that would have been our sample sizes, even though we selected only two samples, but in each sample they were four observations from the normal distribution. So it makes it 16. So in a nutshell, what we're saying is when we have a population with the size of four and we calculate the mean of that population, we get 21 as the mean and the standard deviation of 2.2. If we re-sample and we take two samples from this, from this population of the four individual values from that sample, from that population as it is, but we sample it twice, we calculate the mean of those samples, we find that the mean is 21 and we find that the standard deviation is 1.58. As you can see, we were able to transform a uniform distribution and make it a normal distribution. With the sampling distribution of the mean, if we compare it with the population, we can see that clearly, the mean of a population is the same as the mean of the sampling distribution, but the standard deviation of a population, standard deviation, is not the same as the standard deviation of a sampling distribution and the standard deviation of the sampling distribution is also called the standard error and we're going to learn about that just now. So what we have learned so far is different sample of the same size from the same population will yield the same sample, oh sorry, it will yield the different sample means and also we've learned that a measure of variability in the mean from the sample to the sample is given by the standard deviation of the sampling distribution or what we call the standard error of the mean. So standard error is the same, standard error is also the same as the standard deviation of sampling distribution and it's also denoted by sigma x bar because we're talking about the sample mean so it will be your sigma x bar which is your population standard deviation divided by the square root of your sample. And that is what we call the standard error or the standard deviation of a sampling distribution or the population standard deviation divided by the square root of n and you will need to know this and remember this because in the exam or in your assignment they might be asking questions regarding theory questions and you need to know how to answer that question as well. And the standard error is the same as the standard deviation of a sampling distribution is the same as your population standard deviation divided by the square root of your n. Another thing you need to note is your standard error will decrease when the sample size increases so the bigger the sample size the smaller your standard error will be because we are dividing the population standard deviation with the sample size so if I divide 2 by 8 the value will be less if I divide 2 by 20 it will even be way less so you need to remember that that when you increase the value of your sample size the value of your standard deviation or standard deviation of the sampling distribution of the mean or the standard error will be small let's wrap up what we've learned so far we've learned that the mean of your sampling distribution for the mean is the same as your population mean we also learned that the standard error which is our standard deviation of our sampling distribution of mean is the same or it equals to the population standard deviation divided by the square root of the sample size let's do an example to see if we learnt anything from that suppose a population has the mean of 8 and the standard deviation of 3 suppose a random sample size of n is equals to 36 is selected calculate the mean of a sampling distribution what they are asking you to find is the mean of a sampling distribution what is the mean of the sampling distribution based on the values that we have that is your question am I alone here hello you are not alone you are not alone Lazy okay here is the question what is the mean of a sampling distribution anyone 8 it's equals to 8 because the mean of a sampling distribution is the same as the mean of the population is the same as the mean of a population which is equals to 8 what is the standard deviation of a sampling distribution or what is the standard error 3 3 the standard error will be your population standard deviation divided by the square root of n what is your population standard deviation is 3 what is your sample size square root of 36 therefore this will be 3 divided by 6 which is equals to you can use your calculator and calculate it which is 1 over 2 which is 0,5 and that is your standard error questions if there are no questions then we move on similar to what we do when we calculate the normal distribution we need to standardize the values or the x values or the unit in terms of using the z score or the z value calculation with the sampling distribution we also use the z value to find the standardized value in order for us to be able to go find the probability on the table in the normal distribution our z score was our x observation minus the mean divide by the population standard deviation and that is what we use in chapter let's say study unit 6 in study unit 6 in study unit 7 we use the sample means instead of using the x we're going to use the mean so we're going to say it is the mean sample mean minus we're also not going to refer to the population mean but we're going to refer to it as the mean of the sampling distribution because we're using the means divided by the standard error which we know for sure it is your standard deviation divided by the square root of n so this formula is to standardize our sample means in order for us to be able to use the standardized units to go find the probability on the table and the formula we can write it as your mean bar minus your mean because we know that the population mean is the same as the mean of a sampling distribution so we can just write x bar minus mu divided by the standard deviation divided by the square root of n let's look at the next example suppose a population has the mean of 8 and the standard deviation of 3 suppose a random sample size of n is equal to 36 is selected what is the probability that the sample mean is between 7.8 and 8.2 same we know we want to calculate the probability and here we have the sample mean but it lies between remember we don't have to worry about the greater than an equal or less than an equal because we know we're using the cumulative standardized normal distribution so this will be 7.8 and this side will be 8.2 and that is what we need to be calculating so to calculate this probability we need to standardize the mean distribution and standardizing the mean distribution means we're going to use the formula our mean bar minus the mean divided by the standard error which is the population standard deviation divided by the square root of n change to the z because we are standardizing the mean bar minus the mean divided by standard deviation divided by the square root of n substituting the values in terms of what we are given we are given the mean we are given the n we are given the standard deviation and we are given the sample mean in terms of the question so our sample mean for the first one will be 7.8 minus our population mean of 8 divided by our standard error which is 3 divided by the square root of 36 same this side our sample mean is 8.2 minus our mean of 8 divided by our standard error which is 3 divided by the square root of 36 so now we can calculate the side we know the bottom part because we did calculate the sample standard sampling distribution standard deviation and we did calculate it we said it sorry we said it was 3 divided by the square root of 36 and we found that it was 0.5 remember when we were doing the activity found that it was 0.5 so therefore this will be the probability of 7.8 minus 8 will be 0.2 divided by 0.5 you can just double check my values and minus miss list minus 0.2 yes minus 0.2 and then this 8.2 minus 8 is 0.2 divided by 0.5 and therefore the probability that we need to find will be 0.2 minus 0.2 divided by 0.5 what do you get do the calculation for me minus 0.4 0.4 and I'm going to put 0 there because we always need to leave our values into 2 decimals greater than Z and therefore this side will be 0.40 as well 0.5 so now we have the probability of between and since this is the probability of between we need to say this will be the probability of the why am I dividing by 0.5 again here we say we will find the probability on the table for 0.40 minus the probability we find on the table for Z less than minus 0. for Z minus 0.40 so we're going to go to the table let's go to the table let's keep everything that we have go to the table we're looking for 2 values the first value we're looking for it's on the positive side of the table we're looking for that 0.40 at the top we find that that is 0. 0.554 Iron out of base so I'm going to write it here 0.5 4 and then we're going to subtract go and find minus 0.4 on the table we go to the negative side of the table and go look for 0 minus 0 and I know that this column is 0 at the top there for that we'll see so roll up and you will see that that is 0.0 and then you come down to that column and we get 0.3446 0.3446 minus 0.3446 and if you do the calculation what do you get? Miss Liz can I ask a question before you continue? Yes. Why are we using 0 and where does the minus for both of them? Why do we subtract instead of adding what is the reason for it? Let me take it back remember when we use the normal distribution table probability of Z less than a value it will be the value we find on the table remember that? Yes I do. And when we have the probability of Z greater than a value we say 1 minus the value we find on the table. Oh okay yeah thank you so much for that. Then for the probability of between if it lies between A we say we take the probability of the second value which is Z less than B and we subtract the probability of the first value which will be Z less than A and it's the same so the value of table value of B minus the table value of A. That's what we're doing because it was in between we take the table value of B which this side will be B this is A and this is B. So we take the table value of B table value of B which is the probability we're going to find for B subtract the value we will find for the table value for A which is minus 0.4 What do you get when you calculate? Okay since you guys you don't want to talk to me 0.3108 0.3108 and you can see that we took the population distribution that we had we converted into a sampling distribution and then we went and found the probability of that normal distribution which now is standardized Miss Liz going back to the table can you please go to the table why do we use the first column 0? Is it always going to be 0 if it's in B? No. You asked the question why we using so the question the z's go actually with 0 comma for 0 remember the first the one before comma and the one after comma we always find it oh yes the second one the last digit we find it at the top so at the top that is why you always need to leave your answer at 2 decimals oops oops oops that is why you need to always leave your answer into 2 decimals so that the last digit the last digit will tell you in which column you need to be at the first two digits the one before the comma and the one after the comma will tell you which row you need to be at okay Thank you so much Miss Liz No worries let's do an exercise this is your exercise to do a sample of N16 observation is drawn from a normal population with the mean of a thousand and a population standard deviation of 200 calculate the probability that the sample mean is less than at 1050 remember the formula for z z is the sample mean minus the population mean divided by the standard error the sample mean you are given in the question the standard deviation the N and the mean are in the statement you have 5 minutes and your time starts now remember when you have your answer you can post it on the chat are we winning? Yes Miss Liz I'm unable to type on the chat can I share my answer? Yeah when we get there when we get to are we done? anyone still need more time? okay violence means you are all done so we need to take that probability of the mean less than 10,000 150 and standardize it I'm just going to use the same formula and you guys need to tell me what you did so give me the values what is our mean sample mean is 1050 1000 and our population mean is 1000 divided by our standard error it's 200 200 for the population for the deviation 10 by the speed of 16 10 divided by the speed of 16 1050 minus 1000 gives us 50 divided by our standard error is how much? it's 50 divided by 50 that is 200 divided by speed of 16 is 200 divided by 4 that is 50 and then 50 divided by 50 so that will be 50 divided by 50 so the standard error is 50 and we need to go find the probability that such is less than 50 divided by 50 is 100 so we are looking for the probability of a less than 100 so we need to go to the positive side so we go to the positive side of the table and we need to go to where it is 1 because we are looking for 100 0 so 1 from 0 and 0 at the top and the answer is 0,8 0,8 0,8 0,8 0,8 0,8 0,8 0,8 0,8 so I didn't go to the table that I'm not looking for the net. The minute that ask you to find the probability you must always make sure that you go to the table so you need to always feel you I'm not sure how you did it Lady where you got 0,538 but I hope you know where Can you please show me the table again? The table is the cumulative standardized normal distribution. You need to look for 1 comma 0. Oh, I see you went to 0 comma 1. You need to go to 1. Remember, the digit before the comma is 1. So you need to go 1. And then the digit after comma is 0. You need to go to 0, 1 comma 0, and 0 at the top, the last digit 0, and that is that. So I can see you used that. Yeah, that's exactly what I did. I used the wrong value before the comma. Yes, so you need to pay attention to your values. Right, otherwise you do understand how to use the table. Just pay attention to the decimals and the values. All right, let's continue. Now let's look at how we do for the proportion. So far we've learned the sampling distribution for the mean. How do I know which one I'm working with? As long as they mention the mean, the standard deviation, you must know that you were working with sampling distribution sampling distribution of the mean. And to know that is if they give you things like the mean, the standard deviation, you should know that. Because you can also find the sampling distribution for the population proportions. And with the population proportion, we use the pi, which we used in the binomial distribution as our probability of success. Here we use it and we call it the proportion. Population proportion. For sample proportion, we use or it's denoted by a pi. Remember always for the population, we always use the Greek letters. For the sample, we always use the English letters that we always know, like the small letters like p's and the x and the n. With the proportion or parameters, we use the Greek letters like mu, sigma, pi, which is one of those. So for example, sometimes when we calculate the sampling distribution of our proportion, they might not give us the sample proportion, but they might give us the observation satisfying that sampling proportion or the sample size. And when they give us those observations that satisfy that sample, we can calculate the sample proportion because we're going to take the sample or the observation and divide them by the sample mean in order for us to calculate the sampling or the sample proportion. And always remember, your probability should always be between 0 and 1. The same thing that goes with the sample proportion. It's always going to be between 0 and 1. And with this, we always assume that when we do the sampling distribution as well as with the mean, we always assume that the sampling that has been done is with replacement or without replacement. With replacement, it means when we take out one, we create the first sample and then we take them back. And then we go and re-sample again, and then we take them back. And then we re-sample again, and we take them back. When we do without replacement, when we take out the first sample, we don't have to replace everything that we took out. So we are left with whatever we are left with. And then we go back and we select another sample from what we left with. So it's different. So with replacement, it means if I have all these dots, if I take this dot out and that dot out, and I create a sample, I must take them back. I must take them back and then recreate another sample, and that can be out, and that can be out. And then I must take them back and create another sample. Without replacement, if I have those dots, if I take out this one and I take out that one, I've taken them out. I don't have to replace them again. So they are gone from the sample. So it means they are gone for the next sample. Then when I create another sample, I just select that one and then select that one. Then they will be gone out. I cannot re-select them again. But with this one, I must put them back because they can be eligible for being selected again and again and again and again and again because I'm replacing them again. And that is without or with replacement. And that's what it means. It means the samples that we create, they can either be without or with replacement. Okay, similar to what we did with the sample mean, sampling distribution as well, we can approximate it to a normal distribution when our sample size multiplying by the population proportion should be greater than five. And our sample size times one minus the population proportion should also be greater than five. Our sampling distribution is distributed with the mean of the sampling distribution of the proportion, which is equals to our population proportion of sampling distribution, which is the same as the pi, and our standard error, which is our standard deviation of our sampling distribution of our proportion. It's given by the square root of your population proportion times one minus the population proportion divided by n. For sampling distribution, the mean of a sampling distribution is the same as the pi, which is the population proportion. The mean of a sampling distribution is the same as the population proportion, which is the pi. The mean of a sampling distribution is the same as the population proportion. The standard deviation of a sampling distribution, which is also called the standard error of the sampling distribution is given by the square root of your population proportion times one minus the population proportion divided by the sample size. Let's look at an example. From the past knowledge, Africa check knows that the true proportion of ghost profiles on Facebook is 0.2. There we are given our population proportion. Suppose that we take a sample of 200, which is our end Facebook profile, and we found that only 34, which is our X because this is from our sample of 200. We found that our only 34 of the ghost profiles. What is the value of your population proportion? The population proportion is given in the question. Our population proportion is 0.2. What is the value of our sample proportion? Remember, if we are not given the sample proportion, but we are given the observations that satisfies that sample divided by n will give us the sample proportion. What is the sample proportion? Our observations, they are 34. Our sample size is 200. Therefore, divided by 200 is how much? 0.17. 0.17. Population proportion, yes, it's correct. No, it's not correct. Not correct, not correct. This is the value for the sample proportion. This is the value for the population proportion. Population proportion is 0.2, correct. Sample proportion is 0.17, correct. That's not correct, that's not correct. Option number four, the answer you were looking for. Any questions? If there are no questions, then let's look at how we calculate the Z-score and go find the probability on the table. With the proportions to standardize the value of our sample proportion, we use the Z-value formula, which is our sample proportion minus the population proportion divided by the standard error, and we know our standard error is the square root of our population proportion times one minus the population proportion divided by n. This is for the proportions. If the population proportion is 0.4, our n is 200. What is the probability that our sample proportion lies between 0.4 and 0.45? We can go and find this probability. I'm going to use the Z-score without using the probability of between-between, we can substitute them later in the formula because I'm going to run out of space if I do that. I'm going to first calculate the Z for the first one. We know that we use P minus the population proportion divided by the standard error, which is the population proportion, one minus the population proportion divided by n. I calculate for the first one, our sample proportion is 0.40 because it's given in the question. Our population proportion was given in the statement, which is 0.4 divided by the square root of 0.4 times one minus 0.4 divided by our n, 200 equals 0.40 minus 0.40 divided by the other value, which I'm not going to even bother doing because I know 0.4 minus 0.4, any value divided by 0 will just be 0. I'm just going to leave it as 0.00. Then we do the second one, we'll do the second one, Z equals our P minus the population proportion divided by the square root of our population proportion, one minus the population proportion divided by n. 0.45 is our P because it's given in the question, minus 0.4 divided by our standard error, which is 0.4 times one minus 0.4 divided by 200. Now I need your help because on the first one, I could have went and calculated the standard error, but I was so lazy because I know that the first one is 0. Any number divided by 0 will just be 0. On this one, we will need to calculate it together. 0.45 minus 0.4, the answer here will be 0.055. Please go and calculate for me, the standard error. This will be 0.4 times 0.6 divided by 200. If you quickly do that calculation for me, and then take the square root of the answer. 0.034. 0.034. Then divide 0.05 with 0.034. Is 0.034 the answer for the whole thing or are you? No, no, it's the only number that's about 0.1. Okay. So do the 0.05 divided by 0.034? It's 1.470. You guys, I'm going to rely on my slide now. 0.034, yes. So that is 1.470. So we leave it to two decimals, so it's just going to be 1.47. Now, let's calculate the probability. So we know what we need to be calculating here. So we need to find the probability of Z. Z lies between two values. It lies between 0.00 and 1.47. So we need to go find the probability that Z is less than 1.47 on the positive side, and we also need to go find the probability that Z is less than 0.00 on the positive side. So let's go to the table and I keep going to the table. We're looking on the positive side for two values, 1.47 and 0.00. 0.500. We need 1.47, 1.4, and we need to go find 7, which is 0.9292. 0.9292 minus 0.500, and that gives us 0.4292 and that is how you go find the probability on the table. So, Lizzie, can you please go back to the table, find that value again and just want to see something. 1.47, 1.4 on the side, and at the top we go look for 7. Okay. Thank you so much. And for the 0, 0 and 0. Thank you, Lizzie. Okay. So we calculated the standard error and then we calculated the probability and we did find that it is so you guys, you told me it was 0.147, it's 1.44. So depending on how many decimals you kept, so always keep it to four decimals. Don't round off quickly because you see when you round off quickly, you're adding up values as well. So please pay attention to how you round off. So do not round off, what does the golden rule? The golden rule for doing math is while you're still in the problem mode, you do not round off, you only round off when you get to the final answer. So we rounded off the standard error to 0.34. Actually it should have been 0.35. So you said it's 0.34, so you also didn't round it correctly. So because we are rounding off too quickly, we're adding up there the digits at the end and that is why the answer we got there was 0.47 instead of 1.44. This is what is going to happen in your exam or your assignment while you are busy. Always, if you want to shorten the value you see on your calculator, keep it to four decimals or five decimals, there should be enough to cover the loss of the last digits every time, or the incrementals of the last digits. That concludes what we needed to learn about the sampling distribution. In the next couple of minutes that are the next couple of hours that we have, we're going to do a lot of other exercises that talks to the sampling distribution as well. Then I send you guys an announcement via my UNISA. So next week, Wednesday, there's no class because it's a public holiday. The following weekend, which is Saturday, next week, Saturday, we are going to repeat chapter six and chapter seven activities. So that we can be well acquainted with the two chapters before you do your assignment three, because it's due on the date yet. We still have enough time so that you can do a lot of practice before you submit your assignment three. So we will discuss the activities on Saturday. Do you have any query before we start with a lot of exercise? But before your queries and questions, let's answer this question. Then I think after this, we can start with the questions. Just to familiarize ourselves again with the proportion. Now, remember with the mean, with the sampling distribution of the mean. Remember that the key thing that you need to remember, when you see or when you read the question is, things like the mean, the standard deviation. Say with the sampling distribution for the proportion, the key thing here will be, they will tell you proportion, they might give you decimal or they might give you percentages. So you need to know those things. They will not mention the mean or the standard deviation. So you need to automatically know that this is a sampling distribution of the mean, sorry, of the proportion. For example, yeah, from the past, knowledge Africa check shows that the true proportion of ghost profiles on Facebook is 0.2. The sample size is 200 and they found that only thing for our ghost profile. If the population proportion is 0.2, the sample proportion is 0.17, what is the probability of that? The sample proportion being greater than or at least 0.7. So they're asking you to find the probability that such is greater than the sample mean minus the mean divided by the standard. No, they are not asking you that. Why? Because this is for the sample mean. They are asking you to find the sample proportion minus the population proportion divided by the standard error, which is your population proportion 1 minus the population proportion divided by n. You need to know the formula as well, which one to use for which question, whether for the mean or for the proportion. That is your exercise. You have five minutes. We did calculate the proportion. Yes, and yet they've given it to you. So find the probability that it is greater than 0.17. Your five minutes starts now. Remember those who are done, you can post your answer on the chat. Okay, how are we doing? Are we done? Okay, so let's work it out. What is the probability? The proportion will be? I got option 1, 0.8554. Yeah, but let's work it out. Our p is 0.17 minus our population proportion. We were given 0.2 divided by our standard error, which is the square root of our population proportion, which is 0.2 times 1 minus 0.2 divided by our n. 200, sorry, 0.17 minus 0.2 minus 0.03, our standard error, 0.028384, 0.028284. Did I write it correctly? It's 0.2, there's no 0 after the other 0, after the call. What? Is it 0.028284? No, 0.028284. Did you guys calculate this? Adidas, do you agree with Laleigh? Yes, it's 0.028284, correct? I agree, then we can move. So what is the answer? Minus 0.03 divided by 0.028234 is? 1.0638. Minus 1.0638. Am I right? I think I will check something. So, Lizzie, why don't you check? Those who got the answer, someone gave me the answer and they said it's 0.8554. What was your z value? 1 was 1.06066. OK, just need to round it off, but yeah. What happened to the minus? It's minus 1.06067. Is it 1 rounded off to 2 decimal? Give me only 2 decimal. Yes, that's right. 0? Minus 1.06. OK, so if we go to the table, we must remember that. Why did I tend the sign? The sign must be greater than. Yeah, I was actually about to ask, but OK. The sign must be greater than, so we must go find the value on the table and say 1 minus the value we're going to find on the table under the z of less than minus 1.06. That will be the value. So let's go to the table. We're going to the negative side and we're looking for minus. 1.06, that's what you told me. 1.0, and I must go to the top to look for where 6 is. 6 is in this column. So where they both meet, 0.1446. Therefore, the answer here we find will be 1 minus 0. I forgot the number. 446, 1, 446. 1, 446. And when you calculate this, what do you get? Is that the answer? Yes, 0.8554. And that's how you will find the probability of the sample proportion greater than 0.16. OK, let's fit in one more exercise. Any questions? Nope. OK, we will do a couple of exercises. We have 30 minutes. The standard deviation of a sampling distribution of the mean is called sampling mean. Is it called residual? Is it called standard error? Is it called standard normal? Is it called sum of deviation? What is the standard deviation of something? Isn't it the standard error missus? It is indeed the standard error. Let's go through its statement and choose the one that is incorrect. The mean of a sampling distribution of the mean equal to the population mean. The mean of a population distribution is the same as the population mean. That's correct. The set score of a sampling distribution of the mean is equal to the difference between the sample mean and the population mean divided by the standard error divided by the square root of the sample size. If I rewrite this, it says our z is the sample mean minus the population mean divided by the standard error divided by the square root of n. Is that correct? Yes, that's correct. Is that correct? Because standard error, remember what the standard error is, is the population divided by the square root of n. So are you telling me that this, what they are saying is the mean minus the mean divided by population divided by the square root of n divided by the square root of n. That's what they are saying. Is that correct? Because the standard error, remember the standard error is the value underneath the, is the population standard deviation divided by the square root of n, which is our standard error for the z formula. Take number two, it says the z value is the mean minus, the sample mean minus the mean divided by the standard error divided by the square root of n, which means they are saying is the sample mean minus the population mean divided by the population standard deviation divided by the square root of n divided by the square root of n. That's what they are saying. That is incorrect. That is incorrect. The next one, it says the standard deviation of a sampling mean is equals to the standard deviation of the standard of the population divided by the square root of n. So what they are saying is the standard deviation of a sampling distribution of mean is the same as the population standard deviation divided by the square root of n. Is that correct? Correct. That is correct. Number four, a sampling error is the error resulting from using a sample characteristics to estimate the population characteristic. Is that correct? I'm not sure. From chapter one, what did we learn? There are two types of statistics. There is one, where we use descriptive and where we use the inferential. And we also learned that we use statistics to estimate the parameter, isn't it? A sampling error is one of the values we use because this comes from a sampling distribution. Remember that it's all the samples, samples, samples, samples. The standard error from there will be the error resulting from our statistics, which are our sample characteristics to estimate their population parameter, which is our population parameter, whichever one we use, whether we use the population proportion or we use the population mean. So this is also correct. So this goes back to, if you still remember what you've learned in your chapter one, you still can apply that concept here. Number five, regardless of the shape of the distribution, the sample size gets larger enough. The sampling distribution of the mean is approximately normally distributed. So regardless, if we have a uniform distribution, if we increase and we create a lot of sample sizes, we can approximate that to a normal distribution. Is that correct? We've learned this just a couple of minutes ago. Yeah, yes, yes, that's correct. That's correct. The only thing that is incorrect is number two. That is your exercise. Suppose Africa check conducted best research using a sample size, which means our N is 100. The number of times an AI algorithm is successful and detecting fake news is normally distributed with the mean of 900 and the standard deviation of 100. Let X be or X bar be the number of times the algorithm is successful detecting the fake news. What is the mean of the sampling distribution of the mean? I think the answer's in the question 900. The answer is 900 because we know that the mean of a sampling distribution is the same as the mean of the population which is the same as 900. Number four, you can see what is the standard error. They're asking you to find the standard error and I'm just gonna give you the formula for the standard error. What is the standard error of the mean? It's 100, it's the same as standard deviation. Nope, it's not the same as the standard deviation. You have the formula there, did you calculate? Remember that your standard error of the mean is your population standard deviation divided by the square root of N. Is it number four? Okay, I've got two answers. Let's calculate. What is our standard deviation? 10, 10. 100, 100. Right, square root of 100. 100, 100. So this will be 100 divided by 10. 10. 100 divided by 100. So which is equals to? 10. Which is option? You must use your calculators guys. You must calculate. When they ask you the standard error, you need to know that it is your population standard deviation divided by the square root of N. And you must substitute the correct values and do the right calculations. Okay, now calculate the probability that the mean is less than 920. So you need to find the probability that z is less than your mean minus the population mean divided by the standard error. You've calculated the standard error of this question. It's here. So you can just substitute the values. So we can just say the probability that we're looking for z less than our sample mean is always asked in the question, which is 920 minus our population mean, which is 900, divided by our standard error. We've calculated it. I'm not going to substitute the values again. I'm just going to put the 10 and do the calculations, which is 20 divided by 10. Which will be z alpha, what is 20 divided by 10? Two. It will be 2 comma 00. And then you go to the table and go find 2 comma 00. Go to the table. Go to the positive side of the table. And we look for two. And what is the answer? 0.9772. And it means it's none of this. And the answer here is, that's what happens with the copy and paste. The probability is 0.9772. And that should be your last, last, last, last exercise that you do. Actually, let's not do that one. Let's do this exercise seven. In a random sample of 100 people, 25 are classified as a meeting classified as meeting a characteristics of interest. That is success. Suppose the proportion of success is known to be 0.3. What are you given? Let's first identify what you are given. What is 100? N. 100 is N. What is 25? X. 25 is X. What is 0.3? Pi. Population proportion. So based on what you know that you are given, we know that this is the population proportion question because it's also guided by those key words that you can use. There is no mean, there is no standard deviation. The question says what is the sample proportion? Sample proportion X divided by N. You just need to calculate that. The next one it says what is the standard error? Your standard error for the proportions. The square root of your population proportion 1 minus the population proportion divided by N. What is the number? It's option 4. It's option 4. So let's do this. Sample proportion X is 25. Divide by our N. 25 divided by 100. 0.25. 0.25. Sampling the standard error. Standard error is 0.3. Our population proportion is 0.3 times 1 minus 0.3 divided by 100. And the answer we get is 0.0458. 0.0458. Happy? Yes. Okay. So you have additional exercise that you can do on your own. Remember we didn't cover exercise 6. So you must do exercise 6. Exercise 8. Exercise 9. Exercise 10. 11, 12, 13, 14, 15, 16, 17. On your own. And then next week Saturday we can also do some of those activities as well in class. I will add them to the pack of activities that we will do next week. The first hour we will dedicate it to chapter 6. And then the next hour we will dedicate it to chapter 7. Just to recap on what we did today. You have learned the sampling distribution. You have learned how to calculate the probability of a sampling distribution for D. When you are given the mean standard deviation and the sample size. And you are also able to calculate the probability of a sampling distribution of a proportion. When you are given the population proportion and the sample size and the sample proportion. And in case where you are not given the sample proportion. You are given the observations that satisfy that sample proportion and you can calculate the sample proportion. And we've learned how to look for the probabilities on the table. With that it concludes today's session. Any question? How are you feeling? Any comment that you want to share with the rest of the class? Honestly I keep forgetting the square root and it's giving me wrong answers. Guys please don't do what I'm doing. Thank you. That's a good advice. Just to comment. I think initially when you had mentioned that we are changing uniform distribution to standardize it. Initially it looked like perhaps we cook in the books. But I think now I'm getting the idea that actually what we're doing is we are looking closer into the data that's presented to us. And then give and then change it into a form that's easier to understand. Is that a correct way of thinking? Not necessarily. We converted to a form that we can make conclusions about the information. Because you cannot interpret a standardized value. You can interpret it but I mean like when we talk about the Z score of minus 1.5 or let's say that. Then we're talking about how the difference. We're talking about how far apart the values are one another in terms of the Z score. We're talking about the average of those values. So because we're standardizing it, it's not in the original value. But it is in a standardized form. So yeah. So we're not also actually cooking the books. We're converting them in order for us to be able to make conclusions about that information. Because usually you cannot, when you have the sample values, remember you selected them from a population. And whatever calculations that you do and whatever conclusion that you want to reach it. If your data is not normally distribution, you cannot infer your results back to the population. We cannot make a generalized statement about the population. So only when your data is normally distributed. You are able to generalize your conclusion back to the population where you selected the information from. And what we are doing. We're trying to make sure that whatever the conclusion that you will reach using the values that you have. You can generalize it back to the original population. Okay. All right. Thank you ma'am. No worries. Okay. And that concludes today's session. You can have a lovely weekend. I will stop the recording and see you on Saturday.