 Okay. So the recording has started. Welcome to your sessions. I think it's session 14. Where we're going to do activities linked to study units 6 and 7, which are required for you to do your assignment 3, which is due I think on the 21st or something like that. On the 30th, if I'm not mistaken, on the 31st or 30th somewhere last week, next week. So we're going to continue. I'm not going to do any recapping. The only recap that I will do is based on how you read the table by using the probabilities. You just need to always remember that and also remember the formulas because normal distribution and sampling distribution have almost close to lookalike formulas. But we use the same table as well. So do you have any question, comment, query, suggestion, consent? Now is your time. You have let's use these two more minutes that are while we wait for other people to join to raise any consent question or query. Anything? Are you happy? Are you winning? Am I muted? Are you guys hearing me? Can hear you, ma'am. You can hear me. Oh, you just decided not to talk to me. Now, yeah, yeah. On my side, I'm seeing flames because I have lots of assignments. So I'm just waiting to finish this class and then I'm going to take my other chance of assignment too. And yeah, I'll try and catch up. It's long videos. I normally have to take long breaks, but yeah, it's a lot of pressure on me. I'm trying to cut down also on the assignment on the other modules because I had five for each semester. So now it's basically 10 because of what happened. Oh, yeah, because of the super semester thing. So then it means you're doing your first semester and second semester in one year. Yes, and now it's a whole lot of work. Yeah. Okay. Are you guys happy? Are you fine? Are you coping? Talk to me. Hi. Yes. I had a problem with the first hand from the previous session. I just can't seem to get the right answer. I don't know what is it that I'm doing wrong or is it fatigued? I don't know because I tried to redo it over and over, but I still can't get an answer from the given options. I get it totally. Is it for the assignment? No, it's not for the assignment. One of the activities that you are busy with. Yeah, it's from session 13 that we take. But when you are doing it and you are stuck, why can't you ask? Because today we're going to be doing exercises. I will just ask them now so we can all do it together so I can see where I'm actually drawing. Yeah. But you know that your learning is your own responsibility. You cannot wait for others. Yes, I do. You know that. I said, anytime you work through something and you're struggling, you can always post on my UNISA or on WhatsApp and ask for help. Because I think you guys, you wait until we have a class and you hope and pray that things will be unpacked and you will understand. That's not how you meds work or stats work. If you don't practice on a weekly basis, you're going to forget. And every week we're going to introduce new things, new concepts, new calculations, new formula that you also need to remember when you go to write the exam. And if you don't practice, if you don't ask questions when you get stuck, you're never going to move from where you are because you will be stuck there. So I will encourage you to constantly ask questions, constantly post. When you're doing your activities and you are not sure about the answer, just post. Don't care about what other people are seeing what you're doing. It doesn't matter. It's not about them. It's about you. Post and ask for help. Let's help you so that you can move on from where you are to the next place. So yeah, take responsibility for your own learning. That's what I'm trying to get at. Okay. Me too on the other end, I'm struggling because it's all the work pressure and I mean, probably I'm basically working 24-7. It's difficult getting to, and I know like you're saying it's our only responsibility. It's just that time is not on my side right now. I'm struggling to catch up with all the activities. On a Saturday, the two hours I get for the classes, it's basically the only free time I get. Other than that, it's constantly work, work, work. And that's difficult. But yeah, I'm trying. I'm trying my side. Okay. Yeah. Yeah, but you just need to also make sure that you find time during your busy schedule to go and do a revision of the work. Yeah, but with COVID, it's difficult because the casino is open and with the increasing cases, the contact tracing and all that, it makes it really difficult. But I'll find my time. Okay. All right. So let's continue with what we're supposed to be doing. So we're going to spend a little bit of time on the normal distribution and then you move to the next section. So before you start answering the question, you must always remember those things that you need to know. Oh, sorry, my mouse is not, my pen is not active. You must remember that the table contains the probability of less than. We use the value on the table for the probability of Z greater than a value. We use one minus the value we find on the table. For the probability of between, we use the probability of Z B, which is the second one minus the probability of the first one. You remember all those things. Do you still remember how to use the table? Then because this is normal distribution, you also, when you calculate your Z, you remember to use the formula Z is equals to X minus the mean divided by the standard deviation. With that, try and answer this question without my help. Since we're doing revision, you have, because it's Z score, you have three minutes. If you have any question, you can ask and remember to use the chat. Z score for a raw score of 19. Any hint ma'am? Roscoe, please. I see two responses. One says option three is the answer. Do you all agree? That's what I also get. Yes. Okay, let's see if option three is the answer. So we know the formula that we need to use. Z X minus the mean divided by the standard deviation. So what is our X? 19 minus the mean 25 and standard deviation five. So 19 minus 25 divided by five. It's minus two. Minus 1.2. Minus 1.2. So then it means option three is the correct one. Which one of the following statement is correct one? This one you can shout it out. Option three. Option three. The distribution that has the mean of zero and the standard deviation of one is called the standard normal distribution. Calculate the value of X. That's what they need to calculate. What is X? Using that formula, calculate the value of X. You have three minutes. When you get the answer, you can post it on the chat and we can come back after three minutes to check if the answer we all agree. Are we winning? Yes. Are we done? I think five minutes. That's enough. It was more than enough. Okay, so let's answer this. We are giving most of the information. Our Z. Our Z is minus two. Our X is what we're looking for. Our mean is 30 divided by our standard deviation is five. So we need to multiply five by minus two. We are left with X minus 30 and minus two times five. It's minus 10 and we move 30 over to this side. It becomes plus 30. 30 minus 10 or minus 10 plus 30 is the same as 20, which is option five. And three people agree from the chat. Next question. This has a dot there. That is a dot. If the score is given as Z of 1.96 and the distribution of X is normally distributed with the mean of 60 and the standard deviation of six, then find the value of X. So you still need to do two. Z is equals to X minus mean divided by the standard deviation. Find the value of the X value, which is the rose core. Are we winning? Four people say it's option one. Is there anyone who's still working need extra more time? Okay. Silence means everything is fine. So what is our Z value? I think I said today I'm not going to be talking. You guys need to be talking. I'll be your transcriber. Z is 1.96 equals X minus 60 minus 60 is divided by six. Okay. Then you multiply, you take over the six to the other side and multiply by 1.96. So you say 1.96 multiply by six plus 60. Is that what you are saying? Yes. X and the answer is 71.76. The answer is 71.76, which is option one. Okay. Just give me a sec. Okay. Which is option one? Okay. So moving to the next exercise. Given the information, which one of the following statement is incorrect? You, because this might take you a while. Remember that if it's the probability of Z less than a value, it will be the value you see on the table. If it is the probability of Z greater than a value, it will be 1 minus the value you see on the table. If it's in between, it will be the probability of Z less than b, which is the second part, minus the probability of Z less than a, which is the first part. And those will be the values you find on the table. So go through this. I'm going to give you at least five minutes on this one to answer option number one up until option five. And then we'll come back. Remember to use the probability table because this one says you need to find the probability of every value. And don't keep quiet if you are lost. You can ask a question in between. Are we winning? Yes. Do you still want extra more time? There are two options on the chat. I'm not sure if it's for the same question. One is option three and the other one is option four. So it means we're getting two different answers. Are you guys still busy or can we do the feedback? Are you done? Talk to me. So none of you wants to talk to me except one person who said they were done. Okay. So what is the probability that Z is less than one minus 1.52? It's 0.0643. So this is correct. Yes. Agree, everybody? Nobody says anything. One is correct. Okay. What is the probability that Z is less than 1.48? 0.9306. Are we all in agreement? Yes, in agreement. Number three, the probability that Z is greater than minus 0.43. Remember, you will go and find the value on the table and subtract it from one. Is this correct? Yes. It's okay. So this is correct. Find the probability of Z greater than 0.74. This one, you need to say one minus the value you find on the table. Is this correct? No. And the person with the TV mute themselves? Oh, they can mute the TV. Thank you. So this is the incorrect one because this is the value you find on the table. You will need to go and say one minus this value. It should give you the probability of that. So then it means also this is correct. The probability of between. So the only incorrect one is option number four. Anyone who does not know or understand how we got there, speak now. I got lost. Where did you get lost? I was trying to calculate the first option. Yes. So the Z value is minus 1.52, right? Yes. And since the formula is X minus the mean. Okay. I will stop you there. Okay. This is already the Z value. So it means you already, they already calculated Z of X minus the mean divided by the standard deviation. They already calculated this for number one. The answer they got was minus 1.52. So this is that formula. They already calculated it. You only use the Z value. If they ask you what is the probability that X equals to whatever the value is, then you use that. So as long as you see a Z value, you just know that they already calculated it. You use the value of Z to go to the table. You just use these values to go to the table. Okay. Let's see now. All right. Thank you. Others, whoever is the lost, speak now. Just on the last one on number five, that is the probability of B minus the probability of A. Yes. So on the last one, you were going to say the probability. You were going to go find the probability of Z 2.34 on the table and you're going to subtract the probability of Z less than 2.10 on the table. So you were going to first go find that probability and write it down and minus go find the probability and write it down and that should give you 0.083. Okay. Thank you. Right. Next, a psychologist has been studying eye fatigue using a particular measure, which she administers to students after they have worked for an hour writing on a computer. On this measure, she has found that the distribution follows a normal curve using a normal probability table. What is the probability of students showing a Z score below 1.5? How do we write it in a formula format? So it will be the probability, what are we looking for? The probability of Z, what is below? Less than. Oh, this below? It's less than. Less than 1.5. Similar to what we did previously, here they give you the Z score. So you just need to go and find the probability. Should be easy, quicker. We have an answer. Option three. Option three is our answer. When we go to the table, we go look for 1.5 on the side, 1.5 on the side, then we go to 0.00 at the top where they meet. It should be 0.9332. And that is the probability we're looking for. Any question? Anyone who is lost? Speak now. If you don't know how to read the table or how to find the values, ask. The shaded area under the curve follows this normal distribution curve. It's equals to, we want to know what is this shaded area? What is the probability of this shaded area? Remember the shaded area represents the probabilities. What is this probability? You are given 0 and 1.5. So it means we're looking for the probability that Z lies between two values, 0 and I can call it 0.00 and 1.25. So you have five minutes. I see a hand. I understand. Do you want to say something? Sorry, I don't have access to the chat, but I wanted to give you an answer. If you want to give an answer, you just unmute and say it if you don't have access to the chat. Okay. I got option number four. For the last question. For exercise seven. Okay. How do we answer this? We say we find the probability of Z less than 1.25 minus the probability of Z less than 0.00. What is the probability of Z less than 1.25? 0.8944. And what is the probability of Z 000? 0.500. 0.500. And the answer here should be 0.3944. Which is option number four? Happy? Are we all happy? Can we move on? Yes we can. Okay. Moving to the next question. Sorry, I clicked on the wrong. Okay. Given Z follows a normal distribution with the mean of 0 and the variance of 1. What is the probability that Z is greater than minus 1.44? You have three minutes and I guess people who are quick and know what they're doing they already have the answers. Minus 1.44. Pay attention to the sign. Pay attention to the sign. Pay attention to the sign. I'm talking to you specifically Garabo. The sign is greater than. Are we done? Do you still need more time? We could listen. We good. Okay. So let's do the answer. So since we're looking for the probability of Z greater than minus 1.44 we need to go find 1 minus the probability of Z less than minus 1.44. What is the probability of Z less than minus 1.44? 0.0749. And that gives us 0.9251 which is option three. Moving on to the next unless if there's anyone who wants to comment ask a question or none. X is normally distributed with the mean of 50 the variance of 40. You know the mean of 50 the variance of 25 the probability of B greater than the probability of X greater than B is 0.017 and the probability of X between A and B is 0.10. What is the value of A? So what do we need to do here? We need to find Z first. We need to find we don't need to find Z. Do you find the probability of X is greater than B? They did give you that. We need to find the probability of Z less than B which will be 1 minus 0.017. Yes which will be 1 minus 0.017 which is how much? 0.9830. 0.9830. Yes so if I have that then I need to go find the Z value for B isn't it? That's step number one. Is that done? Step number two we need to go find the Z value for B. We need to go and take this value that we have there and go find that Z value. So go find the Z value and tell me what is it? One question. Yes. There at number one is it probability of Z or X that's less than B? This is the probability of X is greater than B which is the same thing as because for you to find this the probability of X greater than B you will convert your X value to a Z value. I'm not going to use a B. I'm going to use a value of A which is the same as 0.017 mean one and the same thing. Because remember we take the B value we go find we substitute it into this question. It's not going to be where A is. A represent X minus the mean divided by the standard deviation. We do this because we take it from the X question. We go find the probability of Z less than to go find the probability in a way we are finding the same thing. Okay got it. Yeah. So now did you go find the probability of Z on the table and then locate the Z values? So it will be on the positive side of the table. I got 2.012. 9830. It's 2.122. 2.12. So our Z corresponds to 2.12. Do you know how we got that? Do you want me to show you how we got it? Are you happy to find it yourselves? You go inside the table on the positive side of the table. Go look inside the table for 0.9830. Then you go out to your left and go out to your top of the table and read those values. The first digit before the comma and the second digit after the comma on the left and the last digit at the top. Combine both. They should give you 2.12. Found it? Happy? Talk to me. Happy? Yes, happy. Happiness. Okay so now we have found the value. We actually haven't found the value of B. We found the value of Z for the B value. We still need to go find B, B is X. So we need, actually we didn't even have to go find the value of the Z value because we can take, we know if this is the case. If that is the case, we know that the probability of Z lies between the value of A and B in this instance will give us 0 comma 110 and we know that this is the same as the probability of Z. Minus Z. Z less than B minus the probability of Z less than A is equals to 0 comma 110. We didn't have to do number two. So what I did is I took that 0.9830 and substituted it into this one. So you substituted here minus the probability of Z less than A is equals to 0.110 and then we need to make this the subject of the formula because it's negative. I'm going to put it to the left anyway. So I'll take probability of Z less than A and put it to this side. Move it over and move 0 comma 11 to this side. So you'll have 0 comma 9830 minus because it is positive on this side. We bring it this side. It becomes negative 0 comma 110. And what do we get? 0.8730. So I can write it here again. The probability of Z less than A is 0 comma. The answer that we get here is 0 comma 8730. Are we all happy? So we have the probability of Z less than A. The next step is for us to go find the Z value that corresponds to this A value there. Remember this is not the A that we're talking about here. I could have put this as a question mark. The Z value here is a question mark or used another letter. So this A here is not this A. This is an A observation Roscoe. This is A that represents the Z score. So now let's go find this. Go to the positive side of the table and look for 0 comma 8730 and go find the Z values. It's 1.14. So you will find it. It will be 0 comma 8729. We'll use the 2.9. 8729. We'll use the Z values that correspond to that. Which is 1 comma 1, 4. So we go here, we go find that our Z, our A. So that will be Z less than 1 comma. Am I doing it right? 1 comma 1, 4. Please. And that will be 1 comma 1, 4. And since we have that, then we can calculate Z is equals to X minus the mean divided by the standard deviation. So our X here is the A that we are looking for. The real A that we are looking for that we need to find here. So substitute the value and solve for A. So our Z, we did find it. It was 1.14. Our A is what we're looking for. Let me make my sigma smaller. Minus our mean. They gave us the mean is 50 divided by our standard deviation. They gave us the variance. So we need to calculate the standard deviation, which is the square root of 25, which is equals to 5. Our A will be equals to. So if we do it all at once, it will be 1 comma 1, 4 times 5 plus 50 is equals to A. I ran out of space. Just do the calculation. What is the A? 55.7 is 55.7. 55.7. And that's how you will find the value of A. Any question, any query, any comment. We should be at the end of normal distribution questions. Given that Z is standardized, normal variable, the variance of Z will be equals to, is it greater than 2? Is it always greater than 1? Is it equals to 1? Is it equal to 0? Cannot be calculated. What is the answer? Option 2, actually option 3. Option 3 is the correct answer because normal distribution is distributed with the mean of 0 and the standard deviation of 1. The variance will also be 1 because the square of 1 is 1. Okay, let me see how many slides still have. Still have a couple for normal distribution. Okay, suppose that X is normally distributed with the mean of 100 and the standard deviation of 20. The probability X greater than 145 is. So they're asking you to find the probability that X is greater than 145. So you need to go find the probability that Z is greater than X minus the mean divided by the standard deviation. You have three minutes. Okay, we have the answer. Are we winning? Yes. Do you want extra time? What is the answer? So what is our X? X is what is in the question. Our mean, they gave us its 100. Standard deviation? 20. So 145 minus 100 is 45 divided by 20. You get? 2.25. 2.25. 2.25 because it's greater than we'd say 1 minus the value we're going to find on the table. Isn't it? Let's do it. And when you go to the table? The table value is 0.9878. 1 minus 0.9878 and that is equals to? Option number 2, 0.0102. Any question? No question. Find the value of A. So this we do the same as what we did previously. So because this is the Z of less than A, you must go find the Z value and want to have the Z value. You must come and substitute into the formula and calculate the value of A again. They say A. And you are given the variance. So you can find the standard deviation, which is the square root of the variance, which is equals to 4. This one, you have 2 minutes. Are we winning? Since it's 1515, you must go to the negative side of the table. Because the negative side of the table contains the smaller values of probabilities. Okay, are we done? What is your Z value? Hello. Negative 1.03. Minus 1.03 equals our A is what we're looking for. Our mean is state. Our standard deviation is equals to 4. Minus 1.03 times 4 plus 30 is equals to A. Therefore our A is how much? 25.88. Anyone who's still lost on how we do this, how we go and find the probability on the table and then locate the Z value. Are you finding this information or doing this on your own? And there is no answer. Okay, so let's do a sampling distribution. Questions in the last hour that is left. Other than that, if we have extra time at the end, we can come back to question 18 because it's the last one on the normal distribution that we didn't do. Okay, sampling distribution, the same thing that we've learned with normal distribution, we're going to still learn. Remember, yeah, the formula is Z sample me minus the population me divided by the standard error, which is your population standard deviation divided by the square root of n. Or if it's for the proportion, let's go to the next slide. If it's for the proportion, it will be the Z of the sample proportion minus the population proportion divided by the standard error, which is the population proportion one minus the population proportion divided by n. And if p is not given, they would give you x divided by n in order for you to be able to calculate p. Propositions, it's decimals or percentages. To know that you're doing the mean, sampling distribution for the mean, you will be given the population standard deviation. You will be given the mean. You will be given the sample mean in the question. For the proportion, so this is for the mean. For the proportions, we always know that they will give us the p, the sample proportion, or the population proportion. And also there will be n, which is your sample size. But you just need to know, to identify to know whether are you doing the proportion or are you doing the mean. So for proportion, you will see things like decimals, like zero point decimals, or you will see percentages. And that will tell you that you are answering questions on proportions. Okay, so with that being said, there is your first question. So what is the probability that a sample proportion of children with ASD in a special needs school is at least 0.7? In a formula, mathematical formula, what does at least mean? Does it mean this or does it mean that in terms of probabilities? Is it greater than or is it less than? In fact, greater than. It will be greater than. So then it means they're asking you to find the probability that p, because they're talking about sample proportion, p is greater than 0.7. Therefore, you need to go find the probability that z is greater than p minus population proportion divided by the standard error, which is the population proportion 1 minus population proportion divided by n. So do the calculation you have. Let me give you five minutes. Are we done? Yes. Yeah. Okay, so let's do this. Sorry, my bad. What is our p always in the question? 0.7 minus our population proportion? 0.74. Divide by the standard error, which is 0.74 times 1 minus 0.74. Divide our n, it's 100. 0.7 minus 0.74 is 0.04. Minus, minus 0.04. Minus 0.04. And the standard error? 0.0. Did I write it right? 0.0439. What is the answer? Minus? 0. Are we all in agreement? Yes. Yeah, so therefore it's 1 minus the probability 0 comma minus 0 comma 9 1. So when we go to the table 1 minus 0 comma 1 9, when we go to the table what do we get? 0 comma 1 8 1 4. 0 comma 1 8 1 4. And the answer is 0 comma 8 1 8 6. 0 comma 1 8 1 8. And that's how you answer the question. Any questions? If none, I'm going to give you three minutes just to look at these questions and then we will do look at them individually and then we will do a recap just now. Okay, are we are we done looking at the questions? So let's let's do this. Let's do this. So this is also proportion questions. Okay, so number one, is it correct? Yes. So you need to you need to calculate P by finding X divided by N because they gave you the number of respondents. So your proportion will be your if it's they are both split yes and no. Let's say let's take the yes. Your sample proportion will be 32 divided by 64 and that will give you 0 comma 5 which is correct. Correct. Number two, calculate the standard error. So you needed to go and find the standard error which is 1 minus divided by N. So our population proportion is 0 comma 7 5 times 1 minus 0 comma 7 5 divided by our N of 64. And is it correct or incorrect if you calculate it? Incorrect. Is it incorrect? Yes. What do you get as a standard error? 0 comma 0 5 4 1. 0 comma 0 5 4 1. Yes. Okay, so then it means that is the incorrect one. Which makes that is the correct one. The sample proportion of 0 comma 5 is the same as a half. It's correct. The probability of the proportion less than 0. So yeah, they're asking you to find the probability. So now we need to find the probability of less than Z less than P minus the probability divided by the standard error. I am going to put it like that because we did calculate the standard error. P Z less than our standard, our P is 0 comma 5 minus 0 comma 7 5 divided by our standard error. 0 comma 6 5 P. P is 0 comma 6 5. I'm taking the first, which is 0 comma 0 5 4 1. So what is our Z value? 0 comma 6 5 minus 0 comma 7 5 will give us 0 comma 1 0 minus 0 comma 1 0 and divide by that. By 0 comma 0 4 5 what do you get? Minus 1 comma 8 4 8 4. 1 comma 8 4 8 4. So it's the same as 1 comma 8 5 because we need to keep. Then you go to the table on the negative side. You go look for 1 minus 1 comma 8 5. So and the answer is 0 comma 0 3 2 2 2. Which is correct. And that's how you will answer the question. Let's move to the next one. Find the probability of between. We already or we didn't calculate that. So you need to go find P minus K. So calculate the probability of between. So we we actually did calculate number A which we found that it was minus 0 comma 9 1. So do the other part. Calculate the second part. When you have the answer you can shout it out so I can write it. Excuse me ma'am. Shouldn't the sample size A be 100? Sorry. Yes we should be any 100. I got 2.28. I also get 2.28. Okay so that it's 2 2.28. So now you must go to the table and go find Z less than. 2.28 2.28 and subtract the value you find on Z less than minus 0.91. So I get 0.8073 of the same one. Okay. So what is the value for 2.28? What is the 0.9884? 87. 87 and 4 minus 0.91. 0.1814. And when you subtract one from the other you get 0.8073 it's not the. So it's not one of these answers so it means from where I got this question they got all their answers wrong there. So you must write it correctly. So the answer that should be here is 0.8078. 8073. That is our answer that we need to get. This is for the mean. We were dealing with the proportion all along. So now we are given the standard deviation, the mean, and they want you to find what is the mean of a sampling distribution and what is the standard deviation of a sampling distribution. Remember the standard deviation of a sampling distribution it's sigma which is the population standard deviation divided by the square root of n. Remember the mean of a population mean is the same as the mean of a sampling distribution. We have the answer. Option four. Okay. So what is the mean of a population standard deviation? The same 1.31 and the standard deviation 0.08 and the question is missing the sample n so our n is 0.04 divided by 2 0.08 divided by 2 is 0.02 so which makes it 0.04. It's 0.04 because 8 divided by 2 is 4. I'm sorry I divided by 4 instead of 2. I kept on saying 2 but I divided by 4. So the answer here is option number 4. Now calculate the probabilities. So based on the information that you got there so you have already 0.04. Now we need to calculate because we're using the same information. Calculate the probability that the mean is less than 1.28. So therefore we need to find the probability that z is less than the sample mean minus the population mean divided by the standard error. So for some of this it's easy because we already calculated the standard error. Our mean is in the question 1.28 minus our mean of 1.31 divided by the standard error we calculated it. I'm not going to ask you to calculate it again. What is the z value? Negative 0.75. Negative 0.75 so go to the negative side of the table and look for negative 0.75. It's 0.2266 which is option number 1. Next what is the standard error? It's 0.05. Okay so we first need to solve it. Standard deviation is 10. It's 10. And the square root of n our n is 200. I made a mistake. 10 divided by the square root of 200. Remember this is the sample size which is our n. This is our standard deviation. This is our mean and this is our population. Our population size capital letter n. 0.707. 0.707 which is option 2. Consider the population mean of 70 standard deviation of 6 and a sample size of 36. What is the probability of between? You don't have to do it the same way as I've been doing. You can calculate these things outside separate so let's do that. The first one which is z of 70.5 minus the mean of 70 because we know that our equation is z of sample mean minus population mean divided by the standard error which is the population standard deviation divided by the square root of n. So we can just do that. Divide by the standard deviation of 6 divided by the square root of n and it's 36 and you do the same z 71.5 minus 70 divided by 6 divided by the square root of 36 and you get your answer. I want to use this slide to do the question. Once we get the value of our z. Because 70.5 minus 70 is 0.5 6 divided by the square root of 36 is 6 divided by 6 which is equals to 1 and that will give us that and at the top we have 71.5 minus 70 which is 1.5 divided by 1 which is 1.5. So therefore the site it will be between 0.5 and 1.5. So we need to find to go to the table and find the probability that z less than 1.5 minus the probability of z less than 0.5. For 1.5 what is the value on the table 0.9332 or 0.5332 and for 0.5 0.6915 minus 0.6915 and if we do the calculations 0.2417 I'm going to assume that that was supposed to be the right answer from the question. Okay and I think the time ran out of on us so there is question 21 22 23 so only three questions three questions that are left which I think you can do on your own as well. So this first one is asking you to calculate the probability of a sample mean at least so you will go and find the probability of x bar greater than 67.5 which you will use the formula z is equals to x mean minus the population mean divided by the standard error you will use that formula to answer the question. Next one which is the here you're going to validate each statement the first one is to calculate the standard error which is the standard error the second one they want you if x is that what is the z equals to so this one they just want you to calculate z x bar minus mean divided by square root of n and validate if that is the correct answer the next one they want you to find the probability of between using the same the second one or the fourth one they want you to calculate the probability of above which will mean greater than and for more than also the same more than 132 you also do calculate that and find out which one of the statement is incorrect the last question they are asking you to calculate the standard error so they should be quick and easy for you to do and that would have concluded today's session any comment any query any question sorry ma'am I have a question and my question was the difference between population sample proportion and sample proportion would it be that if they say population sample proportion it is the proportion relative to the entire sample and then if they say sample proportion it is the proportion between the sample that we had taken from the population it will be the the population proportion will be the proportion from the population it's your proportion of from the original population whatever they because remember let's say for example on your population you have a thousand a thousand people in your population and you want to calculate the proportion that let's use those who who tested positive for covid and those who didn't test positive who tested negative so here you will calculate let's say uh only 100 tested positive and the 900 tested negative so to calculate the proportion of positive it will be 100 divided by a thousand and that will give you the population proportion a sample proportion from the same population we select a sample of 600 people so this is our sample and that is the population from this sample we also get people who tested negative i just give me a sec okay so so there are people who tested negative let's say from yeah we say only 50 from the sample that we selected tested negative therefore those sorry what am i saying negative and positive so 50 tested positive and 550 tested negative so yeah we will calculate this the sample proportion of positive by saying 50 divided by 600 and that is where the difference is the difference between population proportion and sample proportion so a sample proportion will be from the same population how much is but when you calculate the proportion remember it will be the sample proportion of observation or outcome that satisfies that from the sample you don't use the population but you use the sample oh okay okay i get it i get it yeah okay thank you yes i just also want to correct this one question because i think i gave you the wrong hint on this question this question is a proportion question and this is where also you need to pay very careful attention when you read the question this is a proportion question so the standard error here they're looking for is the standard error for the proportion which is the square root of your population proportion one minus population proportion divided by the standard error okay any other questions that you have if there are no questions thank you for coming through and enjoy the rest of your