 Good afternoon everyone. I hope you have a wonderful Sunday. Hello, am I audible? Can you hear me? Good afternoon. We're going to start in one minute. How are you guys doing? Have you submitted your first submission? Have you done that? I'm doing it. You're going to submit today. Your first or your second submission? My first. I think you are keen to be shown everything. Let's start with today's session. Remember that today we are not going to discuss content. We've done that already. You should have watched the recordings last week. Today we just go on and do activities. So you are expected to do the exercises yourself. And then I will come in to assist where there is a need. We have one hour 30 minutes so we can pace ourselves as well. Okay, so we can start first. Let me stop sharing. So I can share my entire screen. So we're going to do study unit 6 and 7 activities. So the first few activities will relate to study unit 6. And then the last few activities will relate to study unit 7. So I expect you guys to participate fully and engage and do the activities. And then I will come in. So moving on. I hope also you do have your tables. Your statistical tables with you. I've also adjusted the session plan. I know that we were not supposed to have a session sometime in August. But I just adjusted it. We're going to have sessions every Sunday now. Going forward. So this week we are on the 24th of July. We're doing activities next week. We will have question and answer. Bring those questions. If you still have any question that you want to ask while you are busy with your assignment and you are struggling. How can we help you bring those questions so that we can assist you in order for you to submit your assignment. Right. And then in August, then we will start with assignment for preparations, which will be confidence interval and hypothesis testing. Okay. So let's start with this week's session. So we just going to do activities. And this is your first question. Consider the standard normal distribution Z. Which one of the following probabilities is incorrect? So we can just do them one by one. Let's start with A. Let's find out if A is correct or incorrect. So A says the probability that Z is greater than negative 2.8 is equals to one minus the probability of Z greater than 2.8. Is that correct or incorrect? Let's get to it. So we need to validate the statement. So you need to go to the tables because already you are given the Z scores. Anyone? I cannot see. You cannot see the screen. If it's me alone or not. Others? Are you able to see? I can see. If it's my device or not. Okay, probably it's your device. I will suggest that you. Now I can see it. And join again. Let me share. Are you able to see now? Okay. If you have your hand up. Is that a question? Okay. So anyone who wants to try to answer number A? No one. Okay. Okay. I can try. Okay. Since the one on the left, it's greater than we're going to say one minus the. The number we get on the table, which is 0.0026. And then when we minus from one, it's equals to 0.9974. Okay. And then the one on the right, it says one minus the Z greater than 2.8. And then how I do it, I just say one minus. In the open bracket, one minus, because it's greater. And then the value on the table is 0.9974. And then now my answer is equals to 0.9974. And that is correct. Right? Because your left hand side is the same as your right hand side. That's a question. There we go. Okay. And number B? Anyone? You must also ask if you don't understand. And the more we take time for you guys to nominate yourself to answer the questions to move on, the more time we're not going to finish everything because I don't want to do the answers for you. I want you to engage with the questions and see how you answer them. And then if you're making any errors anyway, then I can come in and assist you. So how do we answer number B? We would go find on the normal distribution table, both negative 2.1 and 2.1. And because we have a greater then, we're like minus and then see if the results are the same. Okay. So on the left hand side, what is the probability when you go to the table and look for minus 2.1? You must go to this table, which is table E2 and look for 2.1. It will be 2.10 because it's just 2.1. You will get that it is 0.0179, right? 0.0179. And when you get to this other side, you say 1 minus the value you find on the positive side, which is 0.9. Okay. Because you guys are taking long. 0.9821. And which is equals to? Equals to 0.0179. 0.1. Which makes left hand side equals the right hand side, right? Therefore, that is correct. Then we move to this number C. It says the probability that Z lies between two values minus 2.8 and 0. We know that we need to go and find the probability on the table for Z less than 0. What is that probability? We go find the probability of Z less than 0. And we're going to subtract the probability of Z less than minus 2.8. One of them we do have. Go find the probability of Z equals to 0, which is Z equals to 0.00. What is that on the table? 0.015. Okay. And for the minus 2.8, we did find it. We did find it. It was 0.0026. So subtract one from the other. 0.7474, right? Which is the same as the left hand side, which is the same. So it also, it is correct. So let's do number D. Number D, we're going to find Z of less than 2.1. We did find it. It was 0.9821, right? Because we did find it previously, which is 0.9821. And we need to subtract Z of less than minus 2.8, which is 0.0026. Let me know if you are lost. What is the answer? Okay. The answer is 0.9795. 0.9795, which means it is the same. The last one, Z lies between two values, 0 and 2.1. We did find the probability of Z of less than 0, which was 0.05. We also did find the probability of Z of 2.1. Did we? Yes, we found it at 0.98. So we're going to find, to first get 0.9821. Because we need to first start with the probability of Z of less than 2.1. Minus the probability of Z of less than 0, which is 0.500. And what is that answer? The answer is 0.4821. 0.4821, which is not the correct answer. So we have found which one is our incorrect answer. And that's how we will answer the questions, right? Do you have any question before we move to the second one? More questions? So let's move to question number two. Consider a normal random variable with the mean of 3,000 and the standard deviation of 1,800. Calculate the probability that the random variable is at most 3,800. Choose the correct answer. The key things here is what you are given. So you're given the mean. You are given the standard deviation, which you need to identify as your sigma. You are also asked in the question, what is the probability that a random variable at most, it will be 3,800. What does it most mean in terms of the sign? Is it a less than or is it a greater than? At most, is it a less than or a greater than? It's less than. It will be a less than. So we need to find the probability that X is less than 3,800. Since this is normal distribution questions, you are not given the sample sizes anyway. So you just need to know that this is a normal distribution question. So we need to find the probability that Z is less than the formula is your X value minus the mean divided by the standard deviation. Everything is given in the statement. Let's substitute and calculate the probability of Z less than our X is always the one that is in the question. Which is 3,800 minus the mean of 3,000. Can you please mute? We haven't played music in the background. I won't be able to post this video because of that. It will cut off. Who has their radio on? So please mute yourself. Can I please ask that every time you join the session, if you know that there is music playing in your background to please not unmute yourself. And if you can hear that the music is playing and we are able to hear that, please let the person know that they are on mute so that then they can mute themselves. The problem is when I publish this because I publish them on YouTube, it will tell me that I am using copyrighted music. Do I want to cut it off? And I'm going to say yes, cut it off. And it will be a problem because then the bulk of the things that we discussed, it will be wiped off. So please, please, I'm saying this for the last time because I think I also did address it in the last time. If you know that you have music playing in the background, then unmute yourself. Okay, so we were busy. Let's finish off the standard deviation is 1,800. Calculate that and let me know what is the probability of Z of less than a value. What is the answer? 0.44. If I use my calculator, I can say it's 3,800 minus 3,000 divided by the standard deviation of 1,800, which is equals to 0.44. Then you need to go to the table. On the positive side of the table, we'll look for 0.4 and 4 at the top. What is the answer? 0.6700, which is option number. Any question? I have a quick question because all of our values end in double zero. Could we calculate this faster if we just did it by eliminating those two zero values? Would it still give us the same answer? Yeah, 38 minus 30 divided by 18 will still give you the same answer. When I see 38 minus 30 equals divided by 18 equal and SD 0.044. Right, you will still get the same answer. Okay, this one, I will do it with you as well. The statistics of the STA 1610 module students' exam marks are normally distributed with the mean of 80 and the standard deviation of 20. There are 2,969 students with a score higher than 95. How many students took or wrote the exam? How many students wrote the exam? So there are 2 ways of answering this question. I will show you both ways and you can decide which one you want to use if you get questions like this in the exam or in the test. So the first way, which will be the easy way of calculating it, let's first find out what we are given. We're given the mean. We're given the standard deviation. We are told that we have. I'm going to call this once our Y values of those who scored higher than and higher means greater than 95. And we are asked to find how many students wrote the exam, which means we're looking for N. So our N, you can say our N is made up of those who scored below 95. And those who scored above 95 because all of them would give us 290. Oh, sorry, our N, the 2,969 plus those who scored below should give us the number that we are looking for, which is our N. So the first thing you need to do is to calculate the probability of those ones. So let's find the probability of those who scored more than 95. So we need to go and find the probability of I used those ones as Y. So I'm going to say Y is greater than 95 because Y is those who scored above 95. And that we're going to find by calculating the Z value of those who scored above 95, which means we're going to use the formula because we need to find the proportion of those ones first. So say the probability of Z, you will do the calculation and tell me how much up there. So it's 95 score minus our mean of 80 divided by our standard deviation of 20. Probability of Z greater than what is the answer? What is 95 minus 80 divided by 20? 0.75. It will be 0.005. So 95 minus 80 divided by 20 will give us 0.75. 0.75. So and we know that this because it's greater than we need to say 1 minus the value we're going to find on the table. So on the table, then let's go and find the value on the table. On the table, we're going to find 1 minus 0 comma 7 Z of the probability of Z of he has done 0.75. So 1 minus go to the table on the positive side of the table. Thank you for 0 comma 75. They both meet. And that is the very right, which is 0 comma 7734. Right. So we're going to say 0 comma 7734. And what is the answer? 0.226. 0.226. 0.226. So those are those who's got above is the probability of those who's got above 95. 95 score. Right. We know that if we know that our Y is equals to 0. Those who's got the probability of those who's got 0.226 out of our end. So those who's got above 95, it is 1 over N, which will be your proportion of those who's got above 95. Right. If we know that there are 23% of those ones, we can. Calculate our N because we do know what our Y is. Our Y is 200 and 2,900. So we say 2,969 divided by N is equals to 0.226. Therefore, if we cross multiply, we're going to take N to the other side and take the 0.226 and divide it. Right. Therefore, we'll end up having 2,969 because we're going to just cross multiply and because yeah, there's nothing. We don't have to worry about the one that is below. So we take N, we multiply it to that side and we take our 0.226. We divide it on the other side because we're doing a cross which then it will be 2266. Calculate what is your N? So your N will be 2,969 divided by 0.2266, which is equals to 13102. Approximately 13102. So it means that will be the answer. Alternatively, how can we answer the same question? Like I already said it here, we know what the proportion of those who are above 95. We can go and find the proportion of those who are not above 95 because that will be the inverse of the ones. So we also go and do the same. So we have the probability of Y greater than 95. We know that they are 0.2266. We can also find the probability of those who are less than 95, which they are at those ones, right? The proportion will be 0.734. Quick question, would it be Y less than 95 or Y less than or equal to 95? For normal distribution, it really doesn't matter whether I have it less than or equal there or not because the probability table uses the same values. That is why also when we were doing the utmost, remember, we were supposed to just put less than or equal. So with normal distribution, it really doesn't matter that much. Whether you put the less than equal or less than, you will still get the same answer. Okay great. Yeah, it's not like the others. So now, if I know, I know that the probability of Y greater than 95 plus the probability of Y less than 95 will give me 1, right? So I know those things because the sum of both of them should give me 1. So if I know that, then I know that the following. If I say our, this is the one that I have right here, but I want to do it this way. I want to say, if I have our Y, sorry, over N, and our N should be equals to 1 and the proportion of the Ys, but it will still be the same thing anyway. Remember now, I think I'm just giving you the same thing that we just calculated right now, 0.2266, right? So if I want to know those ones because those are the probabilities that we have. So I can take, sorry, in state of using Y, so remember that I want to use my, this one, which I'm going to call them by X, which they will be 0.7734, right? So then I can also divide, divide, crisscross, I will have, but I don't have my N. So I cannot use that, sorry, my bet, my bet. I cannot use that because I don't have my N, but I have my probability, which is 1, right? And I also have, probably I'm going to confuse you. Let's just leave it as such. Let's use just 1. Let's just use this so that then we don't have to waste more time. More time. I don't want to confuse you even further. Let me just leave it with this only one because what I wanted to do was to use this method of saying, if I know what the probability of Y is and the number of Ys, we can calculate the probability of X and find the number of X, right? But this will be sufficient enough if you use this methodology as well. For now, I don't want to give you the long methodology and then confuse you in the long run. Sorry, can I ask a question? Would they ever maybe ask us how many students did not write the exam? If they do ask you that, then you would have to calculate the, once you have the probability of those, because if they give you those who didn't write the exam, then you can calculate the probability of that. But if they give you this way and they say, how many did not write the exam, you already know, you already know how many have wrote the exam, right? So let's use this one. So if they say here, how many did not write the exam? You follow the same step. You calculate the total number because that will give you your total number. Then you use this N is equals to X plus Y. You know how many wrote the exam? They were 18,100 and two. We need to find those who didn't write the exam, which will be our X. We know those who wrote and got 95. Sorry, but they will not ask you those who didn't write. Let's say they will ask you those who didn't get, not those who didn't write. Sorry, my bad. They cannot ask you those who didn't write because if they didn't give you your N to tell you that there were this many students in statistics. So here they can only ask you those who did not write, but those who did not get greater than 95. If they ask you in this format, then you can say, we know how many of those who got 95, they are two, nine, six, nine. And you can calculate those who got less than 95 by saying, 13, 102 minus two, nine, six, nine. And those will give you the rest of them who got less than, not those who didn't write. You can calculate that. So that will be 13, 102 minus two, nine, six, nine. And that will give you one, one, one, three, three, one, one, three, three. Those are those who received the probability of those who received less than 95. Oh, sorry, not the probability, but the number. Okay, thank you. So they can only ask you those who did not get the score of greater than 95. You are able to calculate them, but they cannot ask you those who did not write the exam. You won't be able to calculate those ones because you don't know, you can only calculate based on the information that they have given you. They've given you those who got higher, and you can use that to get the total number of those who wrote. Right. Okay, so let's move on. So then we save up time. Shagas found that customers travel time from home to a store at one of the stores is normally distributed with the mean of one, one, four minutes and the standard deviation of 72. What is the probability that the customers travel from home to the store is less than 100? Choose the correct answer from the list of options below. So what you are given is the mean, standard deviation, and you are asked for less than 100. So you need to calculate the probability that X is less than 100. I'm going to give you some couple of minutes to do that. Remember the formula is less than X minus the mean, divided by the standard deviation. When you are done, let me know that you are done. T is right. No, I just want to know if you guys are done. And then we will answer the question. I don't want to know which one is correct or incorrect. We still need to do the answers. Are you done? Is anyone still busy? Okay, so with no one saying anything, then I'm going to assume that you guys are done. So that one who answered that number five is the correct one. Let's answer the question. How did you get to that? You said D. D, yes. How did you get to D? Let's do that. Okay, I said that. Oh, you want me to talk? Yes, I want you to talk. Okay, I said 100 minus one. Z less than minus one nine. That will be Z of less than minus zero nine. Five. Seventy-two. Yeah. That will be Z of less than minus 0.19. Then you went to the table, right? On the negative side. Mine is the last one. So was that the answer? No. Somebody says no. The answer is 0.447. Oh, sorry. Which is D. Right. Okay. Moving on to the next question. I found that customers travel time from home to store at one of the store is normally distributed with the mean of 140 and has done a deviation of 72. A research consultant has recommended no more than a certain minutes of a customer's travel time to store. If checkers would like to ensure that 15.15% of customer at year to the recommendation, what is the recommended travel time? We are given the mean. We are given the standard deviation. We are told no more than, what is no more than the recommended time that the customer needs to move from one, from home to the store should not be no more than, what is no more than? Less than or equal to. It will be the probability that X is less than or equal or you can just say less than. It doesn't really occur. What is 15% less than a value? 0.1515. It's equal to 0.1515. So we need to know what is the recommended time. The recommended time is that X amount. So what is the value of A? That is what we need to find, which will tell us what is the recommended time. The same way as we were doing here. So if 100 was A, and this is the probability, what is this A? What is the 100? So we need to go take this value, go find some things so that we can find the value of A. So we first need to go to the table and go find the Z value. That's step number one. Step number two. Once you find the Z value, then you can use X is equals to the mean plus Z sigma formula to calculate that. So let's do that. Let's go find our Z value that corresponds with this amount. So you need to go inside the table, especially on the negative side of the table because the positive side, the values are big, right? So look for inside the table, a value that is 0 comma 1, 5, 1, 5, or a value close to that. If you don't have a table. It's 0, 3. Yeah. Sorry. Minus 1, 0, 3. That. And when you go up, it corresponds to 3. So it is minus 1, 0, 3, right? That's what you've got. So we know what our Z value is minus 1, 0, 3. Then we come here and we substitute into the formula. What was our mean? 1, 1, 4. What is our Z into bracket minus 1, 0, 3. What is our standard deviation? 72. Calculate and let me know how much you get. It's 34.84. I get two numbers. 39.84. 39.84. 39.84 is the answer. And how do we approximate this? What is the approximate amount? Approximately 14 minutes. 14 minutes. And the answer will be C. Is it right? Yes. That's good. Good. Good. Number 6. Which one of the following statement is incorrect with regards to normal probability distribution? And I think this we did in the first. When we were doing normal distribution, I think I'm repeating the question again, but it's fine. Which one of the following statement is incorrect? We will go and do process of elimination until we get to the incorrect statement, right? The smaller the value of the standard deviation, the narrower and the steep, the kev. Correct. That is correct. The mean of a normal distribution can be any numerical value. The mean can be any numerical value. It can be minus one, it can be one, it can be two. Is that correct? Correct. That is correct. The area of the mean, the area to the right of the mean, if we know that a normal distribution is distributed with the mean of zero. If the area to the right, which one is the right? Right is going this way, right? So if this area to the right of the mean, which our normal distribution means, it's at that point. It's 0.5. Then the area to the right to the left will also be 0.5. Is that correct? Correct. That's correct. That is correct because if this side is 0.5, this side will be 0.5 because the area, which is the probability underneath the kev, which is the area underneath the kev, should always be equal to one. And if you split it into half, one divided by two is 0.5. So that is correct. Number D, the z score of a mean of a normal distribution is one. I think that's not correct. We know that the normal distribution is distributed with the mean of zero and the standard deviation of one. So that would be incorrect. And from the empirical rules, we know that a two standard deviation is always equals to 95%. Right? This is from empirical rule and one standard deviation and one standard deviation is equals to 68%. And a three standard deviation is equals to 99. Do you still remember that? From the empirical rules, you just need to remember all those things. Okay. Moving on to number seven, a random sample. Okay. Now, so far, we have dealt with a normal distribution question. The minute you get to questions where they ask you questions like this. A random sample of 120 is drawn from a normally distributed population with the mean of 160. The standard deviation of 50, determine the standard error of the mean and choose the correct answer from below. You just need to know that now you are in study unit. So when keywords like standard error, whereas in the normal distribution, standard errors are not mentioned or the sampling or the sample size is not mentioned where they say the sample of this match is not mentioned. Know that you do the normal distribution. So this is sampling distribution of the mean, which is study unit seven. Now, asking you to calculate the standard error, which is the population standard deviation divides like the square root of the sample size. So sample size, which is our N is 120. The standard deviation is 50. Calculate the standard error. They also gave us the mean, but that is not what we're looking for to use in this question. And then you know the answer. Standard deviation is 50. The square root of N and it's under the 20. So you just go to your calculator and you say 50 divided by the square root of 120. Equal four comma five, six, I don't know how many decimals. So they left it at two decimals at least. So that will be four comma five, six. Or if you don't use a case, you are using your normal calculator. You just say 50 divided by the square root of 120 and you say equal and it should just give you same answer. Which is number B. Is it right? So the question can be determined the standard error or they can say determine the standard deviation sampling is three. They can say standard. You can also say it is the standard deviation three distribution. They will mean one and the same thing. Standard error, standard deviation of sampling distribution of the mean or standard deviation of the sample means of sample means distribution. They will mean one and the same thing. Now consider a normally distributed population with the mean of 190 as standard deviation of 120. A sample size of 50 is drawn from the population. What is the probability that the mean is between 150 and 190? In a minute, you see things like sample size, because if I go back to the other questions, remember, like this question, they didn't give you any sample size anyway here, right? So you need to be able to distinguish between normal distribution and sampling distribution questions, especially now in your assignments and also in the exam. What will happen is because in the assignment, at least we know that the first half of the question comes from normal distribution. The second half will come from the sampling distribution. In the exam, it will also follow the same format. You will get two questions per study unit, so you need to always bear in mind that the first two questions when you get to normal distribution will come from normal distribution. Following normal distribution, then it will be sampling distribution. You need to just know when to distinguish which one. But we will get there when we do the exam preparations. Okay, so let's answer this question. We need to find the probability of the mean between 150 and 190, and we need the correct answer. So what are we given? We're given the mean. We're given the standard deviation. We are given the sample size and our x values. So we need to find the probability that x lies between 150 and 190. We're going to find it by finding the probability that and we need to use the sampling distribution formula. We have sample mean, which is the sample mean, sorry. These are sample means not just x values. So these are sample mean, sample mean. So sample mean minus the population mean divided by our standard error, which is the standard deviation over the square root of n. And then you're going to do the same on the other side, standard deviation over the square root of n, and you substitute the values. Which will be our sample mean, it's 150 minus our population mean and 90 divided by our standard error. If I go back one step up, it's not the same. Okay. I think it is, oh no, it's not. Okay, so yeah, we have standard deviation of 120 divided by the square root of 50, 190 minus 190 divided by 120 divided by the square root of 50. So you just need to give me the answers. On the left hand side, we have 150 minus 190. Those who are calculating this manually, please do it step by step. First find the standard errors before you answer the question or put everything in the bracket. And I will show you just now again how to do that. And I need to also do 120 divided by the square root of 50, which then it's equals to minus, do you also get the same answer? Minus 2.35, no 36, because I need to round it off correctly. That will be 36. And you do the same 190. That will be equals to zero because 90 minus 190 minus 19 should be equals to zero. And we know that we need to find the probability that the Z is less than 0.00 minus the probability that Z is less than minus 2.30. 36. Go to the table. Find 0.00, we did find that in the previous sessions. It's 0.05, right? 0.500. Minus. Minus 2.36. We go to the negative side of the table. We look for minus negative 2.3. We'll move on this other x minus 2.3. And we go to the top of the table and look for 6. They both need 0.091. What is the answer? It's 0.4909. 0.4909. Yes, yes. Which is option number 3. I just want to back it up a little bit and go back to this. 150 minus 190 divided by 120. For those with no cashier calculator. If you want to answer this question as well, using your calculator, you're going to have to use your brackets and the equals side. So let's do the first one, which is 150 minus 190. And you say equal because that is what is at the top. And to do the one at the bottom, I will suggest because it's a division. Say divide by because it will divide the answer instead of say 120 divided by 50. Put that in the bracket and say 120 and put the divide and then do the square root of 50. And close your bracket and say equal. Wait, sorry, because I'm using ratio. It's always going to be that so equal. And then you say equal. And it should give you the same answer as what we got there. Right. If you don't do it that way, then you need to do this one first. You need to say 120 divide by the square root of one of 50. The square root of 50 and get the answer right. The whole answer 16,97056275. Do not cut off and take only two decimals and you won't get the right answer. You must take the whole answer as it is. You can also even get to number six decimals, at least there will be enough. Six decimal by taking 970562. Because if you only say 16,97, let's see what happens. 150 minus 190 because you're going to say that and you say give equal and divide by 16,97. And you say and you say equal and you say that you will still get the answer because I guess because that is the highest values. But sometimes it might not give you that if you might get 35 and then there's a very small number and you won't be rounding off to six. Then it means when you answer the question, yeah, you might find that you don't get the right answer. So you need to take all the decimals, especially those who are calculating things manually and you don't have a cashier calculator. Okay. Let's go to the second last question because I think this is after this. Oh, there is another one. And the other one. So there are two more questions left. So a psychometric test score of new employees is normally distributed with the mean of 80 standard deviation of 20 and random sample of 36 employees were selected. Let X be the test score of new employees. It is further known that the probability that the test score is more than X is 0,8849, determine the value of X such that the probability of the sample mean is more than X of 0,8849. Choose the correct options from below. So this is similar to what we did when we were looking at the normal distribution of that 15.5 remember when we were looking at this, but now because we are using. Now you need to pay attention to the two differences. This is when you are using normal distribution when you are using the sampling distribution because the formula looks like this Z of the sample mean minus the population mean divide by Z of the sample mean minus the population mean divide by the standard error, which is the standard deviation over this square root of n. So it means if you're going to calculate your sample mean, right, your sample mean. You need to multiply your z with the standard error. If we do that, we will have the mean plus z times you put in the bracket, the whole thing, sigma divide by the square root of n. So you will have to do it that way. So number one, we need to go find the z value on the table. And number two, we need to find our sample mean of the mean plus our z standard deviation over the square root of n. So this is just making sample mean the subject of the formula means moving the mean to the other side it becomes positive and multiplying because the standard error is dividing it will multiply with whatever value that is on the left hand side. So that is just simple math. How you manipulate equations. This is different if the value here was less than then we know that this value is the value that they found on the table was that so because the value they found on the table they used one minus we also need to do the same. So you cannot go to the table and find this way and say that is the probability. So we need to say our z value will correspond to one minus 0,884 because we are looking for if I put it this way. This value we found it by using z of greater than a value, which we found that it was 0,8849 and because it's greater than we said one minus the probability of z value that we find on the table and we found that it was 0,8849. So we need to do the same. So in order for us to find this z value we need to say we need to move the probability on the other side so that it becomes positive z of less than a and we move 0,018 to the other side it becomes 0,8849. Right. So the probability of z less than a will be given by what is the answer one minus 0.8849 is equals 0.0.1151 0.1151. So that is the probability we're going to use. So we need to go to the table and look for this probability on the table on the table. Okay, which is minus 1,2 minus 1.2 minus 1.2 which is that way. So we know that our z value of a here. So we know that our z value was minus 1.2 zero. 1.2 zero. We know that. So it means we can go back and substitute into that formula minus. We forgot to put minus the 1.2 zero. So let's go and calculate the x value, which is the value that we are looking for the sample mean value. So let's find it. The mean we're told that is 80. So that is the mean that is the standard deviation. The mean is 80 plus our z minus 1.2. You can say 1.2 zero times the standard deviation of 20 divided by our sample size. Which is our n square root of 36. Calculate 80 plus 2 bracket minus 1.2 close bracket multiply by open bracket 20 divided by the square root of 36 close bracket equal. And the answer is 76. Do you also get 76? Those who are calculating manually work backwards. How do I what do I mean by working backwards? You say 20 divided by the square root of 36. Which is 10 over 3 or you will see it as 3.333333333333. Multiply that with negative 1.2 equal and it will give you negative 4 and multiply that or not multiply add 80 to the answer and it will give you 76. Just work from back to the front. It should work the same way. Remember it's bought muscle. But my says brackets first. So do the brackets first because you forced it to use the brackets. Whatever is inside the bracket first. So the answer is G. And that's how you will answer questions like this. Okay. Second last question I think. No, we still have more. In a sample size of 90 Puma store 72 reported a decline in the number of customer. Calculate the standard error of the proportion of stores that reported a decline in the number of customers and choose this one. You need to also pay attention in sampling distribution. We have two things sampling distribution of the mean and sampling distribution of the proportion and for the proportion. For the proportion. You will be given the population proportion. You will also be given the sample proportion. And if you're not given the sample proportion, then you can calculate your sample proportion by calculating your X observation over. So in this question, they have they're asking you to find the standard era of the proportion of stores that reported a decline now. Yeah, they're telling you that this is a sample. And they're asking you to calculate the standard. So you're not going to use the population standard deviation to calculate the standard error. Because remember the population standard deviation. If you have the populations. The population proportion, which is the mean, you can calculate this by using this. But because in this question, they have given you the sample. So you can use them and you can use X. So when you calculate this standard era, you are calculating the standard era of the sample proportion, which is almost the same as the population proportion. So this is the same thing. This is the same thing. Sorry, it's not of the mean, but of the proportion standardization. So to calculate that we first needs to find our proportion. So our proportion then we calculated P is equals to X over N and our X is 72. Our N is 90. So we also need to bear in mind or make or think outside of the box, especially when you are reading the questions and looking for ways of answering the question, open up your mind for possibilities. Right. So this is just where work labs and only think about one thing, one thing, one thing that we show just open it up and say, but in terms of the information given, can I am I able to use that to do the same to answer the question. And this is one way. So what is 72 divide by 90. You calculated it. 0.8. 0.8. 0.8. Since we do have that you need to point substituted 0.8 times one minus 0.8 divide by our N. 9 to calculate that the square root of 0.8 times one minus 0.8 close bracket divide by 90. And the answer is 0.04 in three decimals. If I look at the options that in three decimals, do you also get the same answer? Yes. Yes. That will be in here. Okay. Let's see. Oh, second last question. And we have five minutes. Consider a population proportion of 0.66 and a sample proportion of 0.75. Given a sample size of 99, calculate the value of the test statistic and choose the correct answer. Now, think of it this way. At some point when we do hypothesis testing, you will be introduced to these things called test statistics and all that. Which is the same as your, let's go, or your Z value. So they mean one and the same thing. So in sampling distribution, we call it the Z value in hypothesis testing, we call it the test statistic. You are given the population proportion, which is your pi. You are given the sample proportion, which is your P. You are also given the sample size, which is your N and you are asked to calculate the value of your Z. Not the probability. Just calculate Z. So P minus divide by the standard N, which is population proportion 1 minus the population proportion to 5 by N. Remember in the previous question, it's the reason why we didn't use this formula is because we were given only the sample. So we calculated the proportion, the standard error of the proportion, but your Z score formula is this. Which this square root of the population proportion times 1 minus the population proportion divide by N is your standard error. So now let's substitute into the formula. Our sample size, our sample proportion P is 0.75 minus your population proportion, which is 0.66 divide by the square root of 0.66 times 1 minus 0.66 divide by our N. You need to give me the answer. We get an answer of 0.04760952286. You get? Oh wait, sorry. I've calculated the denominator separately from the numerator. Sorry, that's the denominator. I need to calculate the numerator and divide. So what is your denominator? I'm going to write it down. You got 0 comma. I got 0.04760952286. I'm just going to do dot dot dot. Yes, that's correct. You can continue and calculate the others. Okay, it's 0.0900. So it means at the top you get 0.09. 0.89. And when you divide by the answer that you got. I'm sorry. When 0.9 divide by 0.47676. That's some odd number. You will get 1 comma 888774, which is this in two decimals. It will be 1 comma 889. So those who are using the case, you're just going to bring it here. So because that one was if you calculate it manually, you get that. Those who are using cashier, you should also get the same. Which is 0.75 minus 0.66 divide by. Put the square root, put the fraction 0.66 times 1 minus 0.66. Close bracket go down 99. Which is equals to I say equal 1 comma 889. Which is the same as what they would have. Right, so that is 1 comma 889. So if I need to double check your values here at the bottom. It was the square root of 0.66. Can just do this action 0.66 times 1 minus 0.66. Close bracket divide by 99. Equal 0 comma 0476095, 09095. It's not what I wrote then 095. So if you take 0.09, because 0.75 minus 0.66 is equals to 0.09. If I divide that by 0.0476095. Where am I having 0.095? I should get 0.189, which is the same as what they have there. So, which is that. And I'm going to leave you with one last question, which I'm not going to do. You can do it as a practice exercise. And I think it's just the same question as the previous last session that we had. So in this question, you just need to calculate standard error, calculate the Z score, calculate the probabilities. And this is between above, which is greater than and more than. Which will be more than will just be above as well. And on that note, are there any questions? By the end of this session, you should be able to do your assignment 3 with ease. Without any problems. Any questions? If there are no questions, then I am going to say thank you for coming through. Do you have a question? No, no, no, I just wanted to say thank you very, very helpful. Thank you very much. I wanted to say that to thank you very much, especially that question of the students. Please now somebody has reached out for what to do because, hey, when I was doing my first attempt, I didn't know what exactly I was doing, but now at least I have an idea. No problem. Steve, I see you have a question. But I guess the same question, you should be able to answer it yourself. After the session that we just went through now. Yes, of course, definitely. Thank you very much. Do you know how many attempts are we going to get with this assignment? How many? How many attempts? How many attempts? Only two, because you have one and then you'll have the last one. So only two attempts. Unless if there are very few students who completed the assignment, maybe on the last week, and looking at the max as well, your lecture might give you the third attempt. So if you did get that attempt with your second assignment, did you? Yes, we did. You did get a third attempt, but I think I'm not sure. I cannot say because he hasn't discussed that with us, but you only get two attempts. And that should be the norm because I guess you need to start preparing for when you write your exams. You only get one attempt to write your exam, right? So you need to start preparing that. And like I always say, take your first attempt to just gauge to see the type of questions and go and revise and look at. Don't panic and start doing your second attempt as well immediately while you still don't know what is happening or what is going on. Then you are using up all your attempts. So do not rush. And because I'm also here to support you and that is the reason why I'm saying your first attempt, you can even take it immediately when the assessment opens up. After you have watched the first video of content and then take it and see the type of questions that are asked. And then the second one, which is your last one, you only take it if you are 100% sure now that you know what you are doing. And that will be after we have done the discussions like this, which is like the third session of us being together, then you can take it. But if after the session of today, you still unsure you still have next week to make sure that you come back with a whole lot of other questions to ask because next week we do question and answer before you take your last attempt. You do your question and answer session, you iron out any uncertainties, any query, anything that you still unsure of. We iron it out so that by the time you do your last attempt, which is your second attempt, you are now clear in terms of study units that you are busy with. Otherwise, enjoy the rest of your day today. See you next week. Thank you. Bye.