 Hello, I am sorry for that. It took me more than 10 minutes to come back. So let's talk about the proportions. Are we all back? Yes, we are. Yes. Thanks. So now let's talk about the proportions. So the same concept that we applied, we're still going to apply is just, but now we're going to be using a different formula. But this we're going to still use the table, go to the Z table, look for the probabilities from there, and nothing different with that. So when we talk about the proportions, proportion is like your percentage. We did the proportion when we were doing the probabilities probably. So remember, probability is always going to be in a decimal, and a proportion in this instance will be in a percentage format that gives you the proportion of the values. So when we talk about the population proportion, which is your population parameter, we use the pi. Remember, all the population parameters, we use the Greek letters. All the sample statistics, we use the normal standard letters that we are comfortable with, like the letters of alphabet. For example, for a sample proportion, we use a P, and we're going to use the sample proportion to estimate the values. Remember, somebody on the WhatsApp group asked, what do they mean by a point estimate? Point estimate is a value that you use from the sample, or a measure that you use from the sample to estimate the value for the population that you selected the data from. And in this instance, our point estimate will be a sample proportion. In the previous session, our point estimate was our sample mean. So with this one, our sample proportion, sometimes they will give it to you, or they might give you the observations that comes from the sample and the sample size, you will need to calculate the proportion by just dividing the observation from the sample divided by the total sample. And always, like we've learned with the probability, the sum of all probabilities should be equals to one, so therefore it means the proportion, the sample proportion should always lie between zero and one. Let me repeat this. In case when you are answering the question, they didn't give you the sample proportion in the question. They will give you the observation satisfying that sample, which are the outcomes, divide by, and they will also give you your sample size. So you will take those outcomes and you divide them by the sample size. And that will give you the proportion. So if they say 500 people are left-handed and out of those, they selected those people from 800 people who they interviewed. Your sample size is 800. The number of those who are left-handed are 500. To calculate the proportion of people who are left-handed, we say 500 divided by 800. And that's how you will calculate the proportion. We're also going to learn how to calculate the probabilities, but we also need to know the properties of a sampling distribution for the proportion. Is that the mean of a sampling distribution of the proportion is the same as your population proportion. And your standard era of your population, your standard era of your sampling distribution of the proportion is equivalent to the square root of your population proportion times one minus the population proportion divided by n. And to calculate the probability, we're going to use the Z score. And the Z score for the sampling distribution of the proportion, we use this formula. Where your Z is given by your sample proportion, remember your sample proportion, your P, if you do not know your P, they will give you your X values. You will divide by the sample size. You say your sample proportion minus the population proportion divided by the standard era, which is the square root of your population proportion times one minus the population proportion divided by n. Let's look at an example. The population proportion is 0.4 and n is 200. What is the probability that the sample proportion lies between the two values? So to answer this question, we use the formula, but before, because if we go back to the formula, remember that this looks so complex if you look at it. So the easy way is to calculate the sample, the standard era, separate. And then come back and substitute into the formula. And that is what I am doing there. I'm sorry. What I do is to calculate the standard era first. So because the standard era is the square root of the population proportion times one minus the population proportion divided by n. So our population proportion is 0.4. You substitute our n is 200 and we find that our standard era is 0.3464. Please make sure that when you do this, you write all the decimals. Only round off when you go to the final answer, when we run off to two decimals for the Z-band. One, we're still busy solving the problem, write all the decimals down and divide by all the decimals. Because then if you round off quickly, you are chopping away some of the valuable decimals. You might not find the correct value because your number is rounding up too quickly. So be very careful when you work out. Okay. To calculate now the probability, so we do the formula. Remember it's your sample proportion minus your population proportion divided by the standard era. We calculated the standard era. Our sample proportion is 0.40, it's in the question and our population proportion was pi 0.4. Your Z less than 0.45, we calculate substituting into the formula, the sample proportion of 0.45 minus the population proportion of 0.4 divided by the standard era, which we calculated earlier. And once you have solved the fractions, you will find that for the first one, you get zero and for the second one, you get 1.4. And you go to the table, you first find the probability of 1.44 on the table. So you go to the Z table, you look for 1.4 on the right and 0.04 at the top where they meet, it will give you 0.9251. Then you do the same with the probability of less than zero, which will be 0.00, you got, it will be your first column. You go to the top, for 0.00, 0.0 and you go to the top, you look for 0.00 and it will give you 0.500. And you subtract one from the other and the answer you will get will be 0.4251. Any question before I give you an exercise? If there are no questions, given the information from the paragraph, what is the value of your population proportion and what is the value of your sample proportion? When you're done, just let me know so that I can see how many people are done. We can recap on this. Thank you. Are we done? It should be quick and easy. Dan. They didn't give you the sample proportion, they will give you observation satisfying that sample proportion and your sample size. You just need to calculate it. Dan. There. One minute, please. And? Done. Who wants to try to answer it? It should be easy from the statement. From the statement, the population proportion is 0.2. Yes, it's 0.2. And your sample proportion? The sample proportion is going to be equals x over n. The x over n is going to be 34 over 200. Because they told you that from the sample of 200, they found that only 34 are ghost profiles. Then we are able to calculate. Then it gives us 0.17. And it gives 0.17. And then it means the correct option is number four. Based on the same information, what is the probability that the sample proportion is greater than 0.17? Remember, our z is our sample proportion minus our population proportion divided by the standard error, which is the square root of our population proportion multiplied by 1 minus the population proportion divided by n. OK, do we have an answer? You guys are so quiet. You don't even tell me when you are done or you're still busy. Talk to me. Are you done? I'm done, I'm done. We're going to wait here until two o'clock with you not saying a word. We're done, done, done, and I'm hearing nothing. OK, so who wants to try this one? Who wants to answer the question? Nobody, since no one wants to volunteer themselves, first I will do the proportion, which is 0.2 times 1 minus 0.2 divided by our n, which is 200. And that gives us 1 minus 0.2 equals 0.8 times 0.2 is equals to 0.6 divided by 200, used as 0.008. At the square root of the answer, we get 0.028284. I'm going to write all the values 4271, because I'm still in the problem mode. I need to write all the values. So our sample proportion, it's given in the question, is 0.17 minus our population proportion of 0.2 divided by our standard error, which is 0.028284271. And when you calculate that, you will get 0.7 minus 0.2 equals minus 0.03 divided by 0.0282. 0.0284271, you get, I hope you also got the same answer as me. And the answer I get is equals to minus 1.06. If you go to the table, you must go to the table and go find the value on the table. But remember, this is greater than. So the value you find on the table, you will have to subtract it from. This will be the probability that z is greater than minus. Maybe I'm talking to myself and I'm mute. I'm not mute. So you guys decided to just be quiet. No, we are following. 0.06. You should not be following. Why are you following? Because you should be doing, this is your exercise. I'm doing it because you are quiet. And I'm asking, did you also get minus 0.6 as the answer? I might be conflicting into all wrong. Yes, I got 1.06. And now go to the table and go look for this probability and then subtract that value you find on the table from 1. 0.06. What is the value on the table? 0.1. What is the value on the table? It's 0.1446. 0.1446. And when you calculate this, you get 0.8. 0.5. There should not be any shortcut when you answer the questions so that you are able to come back to the recording and remember how you got there. If you use shortcuts, you won't know the steps. You won't know how to get it done by yourself or somebody who is lost. I don't know how to get there. So you're trying to also show them how to answer the questions. OK, so I have exercises. We have 15 minutes. The sample size is? 100. I'll show it. It should be easy and quick. Now you must remember the two things. If they talk about the mean, they will always introduce the mean. For the proportion, they will talk about proportions. So you need to be very careful when you answer the question also in the exam as well. If you see for sampling distribution and they refer to the mean, you must know that they do in the sampling distribution for the mean. If they talk about the proportion, they do in the sampling distribution for the proportion. For example, we just finished doing the proportion. Now I give you an exercise. Here they talk about the mean. You need to know that this question is asking the sampling distribution of the mean, which is the sample standard deviation given by square root of. Are you done? It should be quick and easy. Ma'am, is it number one, 900? Because the sample distribution equals? Nope. Is it number one? Because I give you the formula. No, it's number 310. So how do we calculate it? What is the standard deviation? Standard deviation is 100. And the square root of N, our N is? Is 100 also. And 10. The answer will be 100 divided by the square root of 100. 100 divided by 10. This one also, we did this one, so you don't have to do it. The last question. Also this one, because they're talking about standard deviation and the mean, you need to use that is equals to the sample mean minus the population. Sampled mean divided by the standard error. And you must also remember the weights. It says at least. What does it mean? And if you read the statement, we start with ASD. It says adults with ASD normally distributed with the population mean and the standard deviation. And they give you the values and say respectively. So then it means the first one is your population mean. The second one is your standard deviation. Are we done? Done. Done. Those who are done, let's work it out. What is your sample mean? Sample mean is? About 95. The population mean? 100. Sorry, 90. 90. The standard deviation. 80. 80. 18. Square root 30. Square root of N and our N is? 30. 30. And when you wake it out, what do you get? It's five over the answer. I got five over. Five over. You can give it the way you have it. 3.286. Over 3.286. And the answer is? 3.286. And I got. And then I got 1.521. Yes, since we leave it to two decimals. Therefore, it means we can calculate that. The probability that Z. It's greater than or less is greater than 1.152. 52. Yes. It's given by one minus the value you find on the table. What is the value you find on the table? It's 0.9357. And when you solve that, the answer you get is? 0.0643, which is number two. I actually had the last question for the proportion. But you can do it on your own. It's also not complete because I'm missing some of the information. I don't have my population proportion in this question. But I took this from one of your assignment, I think. So you will find it. You can work it out. I didn't copy the whole question. Then it means. So by the end of today, you should be able to submit your assignment to entirely. In totality. Do you have any questions? If there are no questions, then we can just recap. And just give me one sec. If there are no questions, then we can recap. I know there's no questions. Yeah, cool. So by the end of today's session, we know that we have learned what the sampling distribution properties are, that the population mean is the same as the mean of the sampling distribution. And we also learned that the sampling distribution standard deviation is also called the standard error. And it is your population standard deviation divided by the square root of n. We also learned how to find the probability by using the z, the z value, which is our z distribution value for the sampling distribution for the mean we use z is equals to the sample mean minus the population sampled mean divided by the standard error, which is the population standard deviation divided by the square root of n. We also went on to learn how to find the probability and also understand the basic terminologies and concepts of the proportion, the sampling distribution of the proportion where we said the mean of the sampling distribution of proportion is the same as your population mean. And we also learned that if you are not given the sampling or the sample proportion, you will be given the observation satisfying that sample and also the sample size, then you can calculate your sample proportion by taking the x, which are your observations divided by your sample size, which will be your n. And we also learned that with sampling distribution of proportion, the standard error or the standard deviation of the sampling distribution, which is also known as the standard error, it is given by the square root of your population proportion times 1 minus the population proportion divided by n. And to calculate the z value in order for us to find the probability, we use the z value and the z value for the proportion e, your sample proportion p minus your population proportion pi.