 forgot when we met on Wednesday. If a question, I don't have all the information, we're just going to skip those questions and I will look at previous year's assignment as well to see if we can find more exercises relating to the same concept. But let's continue with the first question. Which one of the following statement is incorrect with regards to normal probability distribution? So it means you need to know your properties of this normal distribution, right? So we're looking for the incorrect statement. So A, the normal distribution has the mean of zero and the standard deviation of one. Is that correct or incorrect? Can go through each one, state each statement. Correct. That is correct. The area to the right of the mean of a normal distribution is one and the area to the left of the mean of a standard normal distribution is one. Think about it. We just said it, that the mean is zero. Think of the area as the probability. So if you think about the normal probability and in the middle here, it cuts off at the mean of zero and what do we know about the probability which are the area underneath the probability? What is the sum of all probabilities should be equals to? One should be equals to one. So we know that the sum of all probabilities equals to one. Therefore, if I split this graph which has the probabilities underneath here, if I split it into half, therefore, on the left, what will be the percentage on the left and on the right? What will be the percentage of those probabilities? Is it one or is it a for it? That's what this question is asking you. The area to the right of the mean which is the area, this area to the right, they say it's equals to one and the area to the left is equals to one. Can it be true? No way. No, I think it's not 0.5, 0.5, then all equals to one. Yes, so therefore it means the incorrect statement in terms of that is B. If we look at number C, it says the score of the mean of a normal distribution will be zero. We just spoke about it because if it doesn't score with the mean distributed, normally distributed, it will have the mean of zero. And that would be correct. And we know that 95% of the empirical rule it's the same as a two standard deviation because 68% is one standard deviation, 98% is two standard deviation and 99% is three standard deviation. The smaller the value of the standard deviation, the narrower. So we know that the smaller the standard deviation which is the gap between the mean and the outer edge of your normal belly calf. The narrower or the smaller the standard deviation it means the narrower the belly calf will be. And the larger your standard deviation, the flatter your belly calf will be. So those are the things that you need to always remember and moving to question two. I don't know what question two is, looking for the incorrect statement or the correct statement. But we can go through question two and we can do all the statement. Do you still remember how to answer questions like this? So if I go to, we're looking for the correct answer. So for the repeat on what we are looking for. So if we need to choose which one is the correct answer, therefore it means you need to evaluate each and every statement. So with number A, what do we do? How do we answer A? Nobody knows. Remember, because it's the probability of between, you're going to go to the table and look for the probability of the second one, which is the probability of zero comma zero zero minus the probability of the first one, which is that of less than minus 1.90. So it means you go to the table. You will need to go to the table and look for the probability inside the table and subtract each one of them. We're looking for zero. The first one is zero and zero, which is zero comma. I found it. So you're just going to say zero comma five thousand minus and go to one comma nine zero on the negative side and look for one comma nine and then go to the top and look for seven. Sorry, it's one comma nine zero. Why am I looking for seven? One comma nine zero, which is zero comma zero to eight seven. Zero comma zero to eight seven and you subtract one from the other. We might be looking for the incorrect answer, but anyway, that's how you will answer the question. I don't know which one was the question statement was looking for. So is that zero comma four, seven, one, three? The first one is correct. Okay, so the first one is correct. Then do the second one. What is the second one? What is the probability of that one comma six on the table? Did you guys bring the tables? It's zero comma eight, five, five, four. Zero comma eight, five, five, four. And the probability of zero, sorry, z of zero will be zero comma five, zero, zero, zero, because we did find it previously. Sorry, the probability for one point six, isn't it that it's gonna be zero point nine, four, five, two, because it's positive. Zero comma nine, four, five, two. Yes. It's the same as that one. It's zero comma nine, five, nine, nine, four, five, two. Yes, zero comma nine, four, five, two. And the answer will be zero comma four, four, five, two, right? That will be correct. Then do number C. With number C, we do have all the venues here in front of us as well. So it's zero comma nine, four, five, two, minus one point nine, six, one point nine. Minus one point nine, we did find it. It was zero comma zero, two, eight, seven. And the answer is, no, it's not all, it's, you must go double check on the table. And the negative side, it was zero comma zero, two, eight, seven, not like one from the other. What do you get? Zero comma nine, one, five. Zero comma nine, one, two, is it two, five? Six, five, six, five. Six, five. So this is the incorrect one. I'm just gonna make a cross on top of it. How do we answer this? What is the probability of Z greater than minus one point nine? Is it the same as one minus the probability of Z greater than one point nine? We need to validate that. So if we have the probability of a greater than, what do we do? We will say one minus the probability of the value we find on the table, right? So therefore this left-hand side would be one minus zero comma zero, two, eight, seven. And what will be that value? Zero comma nine, seven, one, three. Zero comma nine, one, seven, three. Seven, three. No, zero comma nine, seven, one, three. Zero comma seven, one, three, okay? On the right-hand side, we have the value of one minus the probability of Z greater than. So therefore we need, or I don't even have to write it there. I can write it here. This is the same as one minus. And we know that because it's a greater than, we need to subtract one from the probability of Z. And that's then positive one comma nine. So we need to go to the positive side of one comma nine. So it would be one minus one minus again, if it has to take care of what is inside there. So on the positive side, you go to one comma nine and that is zero comma nine, seven, one, three. Zero comma nine, seven, one, three. And you say one minus zero comma nine, seven, one, three will give you zero comma zero, two, eight, seven and subtract that from one. It will give you the same as zero comma because it's doing the double the thing. So the left-hand side is the same as your right-hand side. But that is how you're going to validate your answers. The last one. So you go to the table and look for minus one comma six on the negative side, one comma six, which is zero comma zero, five, four, eight. So here is zero comma zero, five, four, eight. And on the left-hand side, because it's greater than, you need to go and say one minus the value from the positive side, the positive side of one comma nine, six, you go one comma six. One comma nine, four. It's one comma six, right? It's one comma six is zero comma nine, yeah, nine comma, zero comma nine, four, five, two. Five, two. Five, two. And what is the answer? Is it the same as zero comma five, four, four, eight? It will be. So this is also correct. That's how you're going to validate the question. So actually, this question was looking for which one is incorrect, which is C. Do you understand how we do these things, especially normal probabilities? Is there someone who's still unsure of how to find the probabilities? You must speak now. So let's go to the next question. I just wish they were giving us three hours in exam. But remember, you're only going to get two questions from normal probabilities. You're not going to get more than that. So it's going to be as quick as possible. You just need to know how to work them out. Okay, given that Z is normally distributed or is normal distribution, or given that Z is a standard normal distribution, what is the value of small Z such that the area to the right of Z is zero comma seven, nine, three, seven? That is the probability that Z is greater than, and you must remember that Z is greater than small Z is equals to zero comma seven, nine. So how would they have found this value? You still remember? It's everything that we have done. How do we find the answers that we are looking for of zero comma zero five, four, eight? How do we find this value? Because they say the probability that Z is greater than a value, let's call it the Z, is equals to zero comma zero five, four, eight. That's exactly what we're doing. We need to find this Z because we know that that Z is the positive 1.6 because they would have given us that. So now they didn't give you 1.16, but they want you to go find 1.16. How did we get zero comma zero five, four, eight? That is what this question is asking us to find. This is what they are asking. How do we find this value of zero comma zero five, eight? In this regard, how do we find zero comma seven, three, nine? We would have found it by saying, we would have found it by saying one minus the Z value, right? That small Z value that we have, it would have given us zero comma seven, nine, three, nine. So now let's go find what this Z value is. We cannot go and find that Z value before we can make Z the subject of the formula here. So it will be one minus zero comma seven, nine, three, nine. And that is, what is that? Zero comma two, zero, six, one. Zero comma two, zero. Six. Six, one, right? So now let's go look for this value on the table. Zero comma two, zero, six, one. Inside the table, inside here, so that we can go find the Z values. Zero comma two, it don't be on the positive, it will be on the negative. Zero comma two, six, two, zero, six, two, zero, six, one. There we go. There we go. Zero comma eight, three, two. There we go. Out. It's two at the top and zero comma eight. So the answer is, a small Z would have been minus zero comma eight, two, which is this. Happiness, are we good? If the sign would have been, the probability that Z is less than or equal to small Z of zero comma seven, nine, three, nine, then it would have been a different story because we would have taken this value because we know that they would have found it from the table, right? And we would have said this is, the answer would be zero comma seven, nine. We come inside, we look for zero comma seven, nine, three, one, seven, nine, three, nine. Was it three, nine? And it would have been zero comma eight, two. That's how you would have found it, right? If the sign would have been a less than. Always remember that if we need to find the probability of Z of a value, it is the table value. If we need to find the probability of Z greater than a value, it's always one minus the table value. If we need to find the probability of Z relying between two values, A and B, we always find the table value for B minus the table value for A. Always, always. This is very important, especially now with normal distribution and also sampling distribution. You always need to remember that. Everything you do to answer questions relating to normal distribution and sampling distribution, this should be your base of how you're going to handle the questions. Okay, so let's move to the next question. Okay, so because this one, I don't have the question at the top, we can skip it too. Question five, the Department of Basic Education found that the learners travel time from home to school at one of the remote schools is normally distributed with the mean of 114, which is the mean of 114 and the standard deviation which is our sigma of 72 minutes. What is the probability that the student or the learner will travel from home to school between our X value is between 90 minutes and 150? So we want to calculate the probability that X lies between 90 minutes and 150. So it means now we need to calculate the Z values. So we can from here find the probability that Z is less than and we can start with the, this is the same as A and B, we start with the B, Z of X minus the mean divided by the standard deviation minus and probably in state of using X, I could have just used the actual value which is 150 and you can do the same. The probability that the Z is less than the second value which is 90 minus the mean divided by the standard deviation and substitute the values into the formulas. Z of less than 150 minus 114 divide by the standard deviation of 72 minus the probability that Z was done 90 minus 114 divide by 72. Do the calculation and give me the answers. Negative 0.333. We need two decimal, negative 0.333, right? We only need to keep two decimal. And the next one, but it cannot be negative on this side. No, the negative is the other one. I'm so sorry. Yeah, the other one. I did the other one, yeah, sorry. Negative 0.333 on this side and on the 150. It's 0.5. 0.5, 0. Let go to the table, look for the positive 0.5 table. I got 0.69015. 0.69015. Yes. 0.69015. And then you go to the negative side table and look for 0.333. 0.3707. 2-1. 0, 1, 2, 3. This is the beginning. It's 0.3707. 0.3707, 0.5, 0.7, 0.7. And subtract one from the other. 0.32. I'm going to assume it's 0.3208, right? Yeah, the answer is that. Yes. I also don't know what this other question would have asked in terms of this question. Probably you needed to calculate X because then they are asking you to estimate this. They would have given you the probability. So you would have went to the table to look for the probability and then come and substitute that Z value into the formula, remember, so that you can calculate your X. Your X value for this question. So based on half information, I cannot help with that one at the moment, but we can look for the previous assignment questions if you have questions like this. This is one of those that we actually did a lot of explaining in, I think in the class, we did it twice to explain the concept and then we also discussed it on WhatsApp by now. You should know how to answer this question. So, and there are multiple ways you can answer it. So anyway, let's get to it. The emotional intelligent quotient score of high school learners is normally distributed with the mean of 80 and the standard deviation of 20. If there were 2,969 learners with a score higher than 95, how many students took the test? So we need to know how many, all of them have taken the test. We know how many have scored more than 95, but we don't know how many took the test. So we need to find that. So if N is the number of how many took the test, we know that Y plus X where our Y or our X or we can call it X plus Y, X plus Y, X being those who's got higher than, higher than 95, and Y will be those who's got less than or equals to 95. Right? They should give us, but we are also told how many scored. You know that 2,969 have scored higher than 95. That we know. What else can we do? Let's see if you still remember. So we know all this other information. We also know the mean. So it means we can calculate the probability of those who's got less than 95, right? We can calculate those ones because we don't know them. We don't know how many of them. What is the probability that Lennas scored more than 95? Or we can calculate the probability of those who's got more than 95. So how do we answer this? Let's go. Anyone? No one knows how to answer the question. Okay, let's pick one. Since we know that here we have, we can calculate the probability of higher than 95 because that's what we are given in the statement anyway. So let's calculate the probability of those who's got more than 95. Therefore it means it's one minus the probability of Z less than 95 minus 80 divided by the standard deviation. To it, how many? One minus the probability of Z less than 0.75. 0.75 and therefore let's go find that probability. One minus, go to the positive side to find the probability. 0.75. I got 0.7734. 0.7734. 0.7734. One minus 0.7734. 0.2266. 0.2266. So we know that the probability of those who's got more than 95 is 23%, right? It is 23%. Therefore it means the Y will make up because if I know that the sum of all probability should be just equals 200 plus those who's got less than 95, they will be 77% of those ones, right? So if I need to know how many, what is the proportion of those ones based on the information that we have, I know that 23% makes up 2969. What is my Y? Which will give me my N of 100%. Yeah, I think we're gonna divide, we're gonna say N divided by the value of X to find Y. But you don't know your N, right? So how do I find my N if I don't have a Y? I do have my X and Y, right? I know my X is 292. I know that my, I know these two values. So if, let's call it this way, let's do mathematical function. If X is equals to 23%, what is my Y equals to, right? Yes, something like that. But I know what my X is 2969. And I know that this is 0,23. I know that I'm looking for Y. And I know what this other value is, 0.77. How do I find my Y? You can just cross multiply Y to times that. So this will be, if I need to find my Y, that will be, Y will be equals to 2969, multiply by 0,77, divide by 0,23, right? Because this will multiply by that, but eventually I will have to divide by 0,23. So calculate and tell me what is my Y? 9,939 points. Wait, slowly, 9. 9,939.695. Can round off. Okay. Now, this is my Y and this is my X. So now, in order for me to find N, we said N is X plus Y, right? So if my X is 2969, and my Y is 9,933, because now we're talking about percent. The percent cannot be a decimal, right? Maybe I need to round it off to forge. So add them together. 12,908.695. Okay. I've used the calculator. I didn't punch in 9,940. I just used what was on my calculator. So it's 12,000. Wait, wait, wait, wait, wait. 2,9,6,9. Multiply by 0.77. Divide by 0.23. 12,908. That is the answer that you have. Yes. And it's none of these ones that we have here. There might be something wrong with our calculation, right? What is their answer? They've got dating 102. What is it that we did wrong? So we know. See if I use the whole state of rounding it off. So it's 2,9. 6,9. Multiply by 0.7734. Divide that by 0.2266. Okay. So because we rounded off, we don't have to round it up. So use the full numbers as you see them. Instead of rounding them off, it should give us the same answer. So 2,969. Multiply by 0.7734. Divide by 0.2266 should give you. 10,133. 10,133. So you add 10,133 to that. 0.38304. 8 plus 6,9. It's actually more. The Y, it doesn't give me 10,133. It gives me less than that. It gives you? It gives me 8,011.95. I don't know about the others. Should we do this correctly now? It's giving me 10,133.38. So let's do it step by step and see. I also got 10,000. So, but if someone is not getting it the same way, let's see. 2,969. Multiply that with 0.7734. Equals, and that gives me 2,996.2246. Divide the answer by 0.2266. I got 10,133.38. The number, I mean, the ones that we divided with, so I wrote the wrong ones because I couldn't see very well. But now I've got the right one. Alright, so the answer here is 10,133. 10,133.38. 305, right? So then go and add because we know that n is equals to x plus y. So we have x, we just calculated y. So let's add them together. 2,969. Plus 10,133. That gives you? 13,102.38. So we get the same answer, right? Yes. 2,969. Equals, 13102. There we go. So our mistake here was to round off. So we don't have to round off quickly. But yeah, there you get it. So the answer is this. That's what we can also answer because it's not full question. So let's go to the next question, question 9. Now consider a normally distributed population with the mean of 190 and the standard deviation of 120. A sample of 50 is drawn from the population. What is the probability that the sample mean is 220 at most? So the other thing that I, I think when we start looking at the exam questions, we will have to touch on the key things to know when you, what kind of a question you are answering. Because if you, you know, your study units, study unit 6 and study unit 7, they are almost similar to one another. The only difference is in study unit 7, you start hearing in the question, what is the probability of the sample mean? Whereas in the previous one, they wouldn't mention the probability of a sample mean. They would have just said, what is the probability that the value will be between, right? So now you need to know that the minute in the question they asked you about the sample mean, sample proportion, you know that you're doing sampling distribution, so that you know which formula to use. The previous question we were using set of x minus the mean divided by the standard deviation. The minute you come to study unit, this is in study unit 6, right? In study unit 7, we're going to use the z of x minus, not x, but the sample mean minus the population mean divided by the population standard deviation divided by the square root of n, which is your sampling distribution formula that we use. So you will need to know when are you in study unit 7 and when are you in study unit 6 so that you can use the right formula. So with this question, we are given the mean, we are given the standard deviation, but we are also given the sample size n. So it means we're going to use the z, yes. Are we allowed to have notes? Gosh. Okay. I'm going to ignore that question right now because this is a recorded session. Okay. So, where am I? Okay, so now I don't even know how to address it. Sorry, I may not be rude. Maybe someone can send you a private message on WhatsApp or something like that. Okay, cool. So let's answer the question. So we have z is equals 2. And also we need to read the question carefully remember the sign, right? What is at most? I need to also use the right sign. What is at most? It's greater or equals 2. What is at most? Greater or equals 2. What is at most? So it's less or equals 2. It's less or equals 2. Yes, you need to know these things. Remember we spoke about it on Wednesday? Greater or sorry, at least means there are more or I don't know. So they can be more or that is the number. At most is the opposite of at least. So you've got the two at least and at most. So if it's not at least, then it should be at most. At most will be less than or equal. So we need to find the z score 4 because we calculate in the probability zx, the mean sample mean, because the sample mean is what is given in the question, minus the population mean divided by the standard error, which is the population standard deviation divided by the square root of 5. So we find the probability that z is 2 to 0 minus the mean of 190 divided by the standard deviation of 120 divided by the square root of 50. What is, what is the answer? 1.767. 1.22 decimal. 1.777, right? So now you need to go to the positive side and look for 1.77 and go there. And you tell me what is the answer? 1.77. 0.9616. 0.9616. Easy right? It will be easy. Easy peasy. Easy peasy, lemon squeezy. Okay, because this one also has no question statement at the top, so it's useless for me to try and figure out what that is. So in a sample of 60 school 54 reported the decline in the number of absent, Lena's absent calculate the standard error of the proportion. So remember also that for the proportion we always use the following, right? That's what we use. So because on this question, they didn't give us the population, but they're asking you to calculate the proportion. So we're going to use the sample proportion. And remember the sample proportion uses your x divided by n. And therefore it means for the sample proportion, standard error will be your P1 minus your P divided by n. Square root of that, which is part of the population proportion if they have given us the population proportion. So first let's calculate P. What is our x? What is our n? This will be your n because it's your sample. So your x will be the number of schools. So let's calculate 54 divided by 60. What is your P? 0,9. 0,9. So then let's go calculate the sample standard error, which will be P1 minus P divided by n. And you just substitute the values. 0,9 times 1 minus 0,9 divided by n. 0,039. 0,039. Happy? The previous study has shown only 71% of the schools in Sikukini district. Municipality have reported the decline in the number of LENA's absence since the start of the LENA transport and the school nutrition by the Department of Education. Suppose a sample of 99 schools in Sikukini municipality is drawn at random. What is the probability that at least 80% of the school is reported 80% decline? So that is our P. That is our pi. And that is our n. So it means we need to calculate what is at least. By now you know at least. Raita equals 2. We need to calculate the probability that if Raita equals 2, our P minus pi divided by the standard error, now because we have the population proportion, so we will use the population proportion. So let's substitute. Raita, which is 0.8 minus 0.71 divided by the square root of 0.71 times 1 minus 0.71 divided by 99. And you must remember that this is greater than. The sign here is greater than. Don't forget to go into your piece of less. When you are done. Less than equals to or less than. You can just say less than. The answer that you've got. I've got 1.3. Are they different? Yes, I got. 0. Negative 0.381. I got 1.97. Yeah, 1.97 is and then you go on to the table. And from the table, it's going to be that 1 minus thing. And then it's 0.97. And then you get the answer of 0.9. 0.0. Yes, it's 90.9756. Yeah, that's correct. Okay, let's start first with the proportion. What is the Z value or whatever the Z value that we are calculating? Actually, why am I still keeping P in state of set? This should be standardized set. What is the Z value that you got? 1.97. Okay, so we go to the table, 1 minus the table value. 1.97, so 1.9 and 7 is there. 0.9756, is that what you got? Yes. 0.9756, subtract that from 1. You get 0.244. You need to also pay, yeah, so it's 0.0244. You need to pay attention to the sign so that you know whether you need to subtract from 1. Or is it the value you find on the table and so on. Okay, so that was the last question on this. Let's see if I have more questions. So some of these questions are almost exactly the same as what we just gone through. So this is another way of asking the same question you might have received. Some of you might have received the question that looks like this. Which one of the filming statement is incorrect with regards to normal distribution? Number A, the Z score has the mean of 1. But we know that normal distribution is normally distributed with a mean of 0 and the standard deviation of 1. That is the property. So this would be the incorrect one. And some of this we have already gone through them. Right, so they are exactly the same. So the smaller the value of your standard deviation, the narrower or the steeper. Half will be the mean of a normal distribution can take any value in the negative or positive because we know that it will be... Did I really draw my belly like that? That in the middle, if it's 0, the side will be negative and the side will be a positive. If these are the mean values. So it can take any value of negative or 0 or positive value. The area to the right of the mean, we already touched it. So how it's split into half of 0,5 and 0,5 on the left and the right. Because the sum of all probabilities are equals to 1 and we know that for empirical rule. 95% is one standard deviation. And 99% is 2 standard deviation. And 99% is 3 standard deviation. And 68% is one standard deviation. We know that, right? So you need to always remember all those things. So let's see if there are more questions. And you need to know how to answer questions or evaluate questions like this. Where you are given a statement. You need to make sure that your left is the same as your right to make it correct or incorrect. Because that's what they are asking you to find. And let's see. Questions like this, where they are asking you. What is the area to the right? Area to the right will be greater than or equal to the right of a distribution. If it's this, what will be the value of your z? And this is similar to what we just did previously, right? Because this they would have said it's the same as your probability of small z. Because that is the area to the right. Will be equals to 0,2061. You need to know and remember that in order for you to find this value. At some point you said 1 minus some value so that you can get to 0,26. So in order for you to find the z value, the true z value. You will need to subtract this value from 1 in order for you to get to that. I will share this as well as the other ones on my UNISA as well. You need to know how to find the probability if given. The mean and the standard deviation by applying your depending on the question. So this one is looking for the utmost. You need to know that at most is less than or equal to and then substitute the values. Into your formula to calculate the mean of that. The probability for that value that you are looking for. Let's go on to see the next question. This is the same as what we just did. But yeah, they're asking you for the less than so it means you need to calculate the probability of a less than. So you have to go and say less than your x minus the mean divided by the standard deviation and the value you find on the table. That will be the probability that you are looking for. Moving on. More other questions. Okay, so this is one of those questions that we didn't answer remember I couldn't find. It said approximately. Let's see this one. There was no statement. So here is the question. So we can do this one together. So it says all the other statements. It's normally distributed with the mean of 114, which is our mean and the standard deviation of 72. A consultant has recommended no more than. So you need to know what does no more than mean in order for you to answer the question. What is no more than in a method? Less than or equal to no more than it's less than or equals to no more than is less than or equals to. You need to be able to know those sites and it says the department would like to ensure that 15.15% of the learners are here to the recommendation. So therefore divide 15.15 by 100. So that we get into a decimal and what you need is to find what the Z value is so that we can calculate the probability that Z is no more than less than or equals to. We looking for the Z equals to 0.1515. That's what we're looking. So we need to go to the table. So let's go in and look for 0.1515. 0.1515. Out is 0 minus 1. 0, 1, 2, 3. Probably at the top is 3. So it's minus 1, 0, 3. So we go to the Z here is minus 1, 0, 3. That's what we got and X minus the mean divided by the standard deviation. Everything else it's on the statement. So let's go find our answer 1, 0, 3 is equals to X. Is what we are looking for. Did I really? Oh yes, X is what we are looking for because we don't know because we don't know how many students adhere to this minus our mean is 114. Our standard deviation is 72 and then we can solve for X. So that will be minus 1.03 times 72. And minus becomes positive if we bring minus to the other side 114 equals to X because we multiply. And we move the minus one to the other side. So what is our X? 39.84. 39.84 if we estimate it to a whole number minutes, it will be 40, right? So 40 minutes. 39.84 which is approximately 40 minutes will be your answer. And that's how you will find the answers. So let's see which other question we didn't think it was the previous one. Okay, I know we don't have to go back there. Then that would be the next other questions that we cut off. We did answer this question. So this is the same as what we just did. We're going to go into this. This is another one. So a random sample of 120 are drawn at the normally distributed population with the mean of 160. So we know that n is that the mean of 160 and the standard deviation of 50 determine the standard error. So we need to find the standard error which is your population standard deviation divided by the square root of n. It should be 50 divided by the square root of n is 120. What is the answer? 4.56. 4.56 which is being not going to touch on this. It's the probability of between. We've dealt with the probability of between. Okay, so the others are just as straightforward as the things that we went through right now. So this is also another question. But now you must pay attention to the questions because we are already in the sampling distribution. So this one, if we need to go find the value of X, it's still going to be the same, same, same, same, same, same. But because we're dealing with the, we have the mean, the standard deviation and the random sample. So therefore it means the formula would have changed from just. So let's, let's do it actually because I just want to demonstrate. Let's do this one. So we have a random sample of 36 learners selected. And we are told is more than. So because it's more than we know that it's greater than how do we find this value, the value of Z. So we're not going to concentrate too much on this X because the X is something that will come later. We know that they found the probability of Z greater than a value, which is the Z value that we need and they told us it's 0,8849. So we need to find Z and in order for us to find Z, we need to go find what the probability of 0,888 is. So do the math 0,1151, 0,1151. So we go to the table, look for 0,1151, 1,1,1,1,1,1,1,1,1. It's minus 1.2,0. So our Z, so therefore it means this was Z. Oh, this is one minus the Z of minus. Now what the value is minus 1.2, 1.2. So this would have been the answer that would got us the Z value would have been minus 1.2. So now, since we know what the Z value is, we can then go and find Z value of the sample mean, because that's what we are calculating minus the population mean divided by the standard error, which is the population standard deviation divided by the square root of X. So let's substitute into the formula. Our Z value is minus 1.2 equals the mean that's what we are looking for, because we're looking for this X value minus the population mean 80 divided by the standard deviation of 20 divided by the square root of 36. So let's calculate. Okay, doesn't matter when I show you the answers. So we need to take minus 1.2 multiplied by 20 divided by the square root of 30. And take 80 to the other side, it will be plus 80. It equals 76, not 30. 76 equals to 76, which is option D. So I'm going to put you from here and then we can, and it's up to you because remember it's Sunday, Sunday we go up until six. Then we can look at assignment four. And then on Wednesday, we look at our last assignment, because Wednesday we have only two hours, or one hour, ten minutes. I can't even remember, but yeah, we can take a little bit of a break and then come back. It's up to you. Or we can stop right here.