 The purpose of today's session is just simple. Very simple, simple, simple. We just here to show you how to use your calculator, regardless of whether it's a case or a shop calculator, to know how to do certain calculations on your calculator. So we're going to, because we already touched on this, you have learned about the sampling distribution, so I'm not going to go into too much detail. So this is an exercise. It says autism, South Africa knows the true population proportion, population proportion of children with ASD in special needs school is 0.4. Assume a sample size of 100 children with ASD is selected. What is the probability that the sample proportion of children with ASD in special needs school is at least 0.7? So we can from this question already deduce that. Let's use the pencil. I'm going to use it. That this is a sampling distribution of proportions, right? So if it's sampling distribution of proportion, and the question is asking us to find the probability, therefore it means we will have to find the Z score of I, not I, P, the sample proportion minus the population proportion divided by the standard error, which is population proportion times one minus population proportion divided by N. So what we need to do is go and identify the things that we are given here. So we know that they told that the population proportion is 0.74. Therefore, we do have our pi, which would be 0.74. We do have our N, which is the sample size of 100. We are told in the question to find the probability that the sample proportion, which is P, small P, is at least, which is greater than 0.7. Because of that at least, so we will put the greater than. And the sample proportion, which is also stated in the question, is the small P. So now we need to substitute into this formula so that we can find the Z value so that we can then go find our probability, which then I can put here, the probability, right? That is what we are looking for. So let's go find that's probability. So to find this probability, our Z will be equals to, let's substitute the values, P is what is given in the question, which is 0.7 minus our population proportion, which is 0.74, divided by the square root of our population proportion, which is 0.74 times 1 minus 0.74. So we need to solve this. So what I've realized is most people, especially those with a sharp calculator, you cannot try and answer all these things at once. So with a cashier calculator, it's easy. I'll start with the cashier. Those with a cashier calculator, and especially a cashier calculator that has this fraction button, it makes things easier for you because I'm just going to put this here next to the question. You can write the entire exercise after the equal sign on your calculator and press equal, or press answer. It will give you the answer that you are looking for. By using the fraction button and putting in the values, which is 0.7 minus 0.74, and going down, you use the arrows. So to move to the bottom one. And because this is the square root, I'm going to put the square root. And within the square root, oh, sorry, I forgot to finish off this. So this is divided by N, N is 100. So you can see that underneath the square root, it's also a fraction. So we're going to say inside the way it's flicking, you put the fraction button, and then you say 0.74 times into bracket 0.74. You can put 0. or you can say 0.74. The calculator will also remember that. Oh, sorry, I made a mistake there. 1 minus 0.74 and close bracket. And then to go down to the bottom one, we just use the arrow again and you press the 100. Is it 100? 100. Why did I put 10? It's 100. Then you put 100. And then you say equal. And that will give you your Z value. And the Z value here is 0 comma. It's minus 0.91. So we need to only keep two decimal. So therefore this will be P of Z. Of 0 comma or negative 0.91. Now the other thing. Negative 0.91. The other thing that you need to pay attention to is the sign. We said greater than the right. So we should have actually changed the sign from equal to greater than. And once we have that, we then can say 1 minus because on the normal table or the Z normal table, we're going to only find the value of Z of less than minus 0.91. And then we need to go to the Z table. One thing is I didn't open. Let's open one of the table statistical tables. So we need to go to negative 0.91. So there is negative 0.9. And then one will be at the top. So it's the second column. So that will be 0.1814. So if I go back here, 0.18. None of them. 0.1814. The bet choice to choose. So there is no 0.1814. But 1.848. Probably that would have been. Oh, sorry. No, it's not. No, I must not look at the question because that is 1 minus 0.1814. All right. My bet. I keep on coming to the answers as quickly as possible. Then now we need to take our answer that we got there. Subdirected from 1.0.1814. And it should give us. And once you have an answer that looks like this as a fraction, you just press the SD, the S with an arrow. And that will change your answer. And the answer will be 0.1814. 0.8186, which is option number three. So that is, if you're using. Acacia. So if you're using a shop calculator, we're going to do the same. So probably you are using either the financial calculator or you're using a normal shop. E L for calculator, whether it's a 531 or some calculator, it will look almost similar to that. So let's redo this. So you will be doing the same thing. You will be finding the probability that Z is greater than UP minus your probability. Divide by the standard error, which is your. Propositions. One minus the population proportion divide by N. And you just substitute the values. So. That it's greater than. Our P is what is given in the question minus. Zero comma seven four. Divide by the square root of zero comma seven four times one minus zero comma seven four. Divide by hundred. Now, because on your calculator, which looks like this does not have the fraction. This calculator does not have fractions. It means you'll have to calculate things step by step. So. In order for us to do that, we're going to have to find Z of greater than and we first solve what is at the top and we go and calculate what is at the top. It's a point seven minus point seven four, which gives us zero comma zero. Zero comma zero four minus. So at the top will have minus zero comma zero four. Then you go divide by and then you go to the bottom where it is the square root, right? We go there at the square root. The other thing as well. Yeah, you will have to do each step by step. As well. So how do we do that step by step? I already know what the values are because my calculator looks like this. The height. My things. So either it becomes bigger. Or it becomes smaller. Can I take away this way? The same way so we can be able to drag the calculator. So I can drag even the calculator to the side anyway. So since we know what values are, this are the values that we're working for. So looking at now. So we go into take. What I will encourage you to do is first do the bracket. So it will be one minus zero point seven four and press equal. That is the thing and that inside the bracket. Multiply that with zero comma seven four. And then you just say multiply by point seven four and press equal. It's very important that you do the equal sign after doing every action. So equal and then we need to divide by hundred. Then you say divide by hundred and then say equal and that will give you everything underneath the square root. Now in terms of if we want to find the answer of the square root. Then we just press the square root button there. And just press square root and then press equal and it will take the answer and put it in the square root. And the answer will be. Zero comma zero four. Now the other thing, make sure that you write all these numbers. So at least maybe you can stop at number six or number seven depending. So it's zero comma zero four three eight six three eight six three. I'm just going to stop right there. And then once you have that you are not done. We need to solve it. So you go back. I'm going to clear my calculator. And the minus. This is the negative. So this answer had a negative in front. The plus or minus here at the bottom is your negative. So you just press that. We'll give you your negative back point. Zero four. Divide by point. Zero four. Three eight six three eight six three. I stopped right there. And then press equal. And there is still my answer. Negative zero comma nine one. Right. Negative zero comma nine one is still my answer. And I can come back here. We also still need to do one minus. The probability of Z less than negative zero comma nine one. And we know that it was equals to one minus. Zero comma one eight. One four, which is equals to zero comma eight. One eight six from the previous one. That's how you use your calculator. Those who have a case your calculator, it's easy with the fraction. Remember with the fraction. Those with a sharp calculator, you don't have the fraction. So do things step by step by step. Let's do the last one. And then we are done. So I have another activity. Sorry, my back. If I work on a presentation slide, I hope it will work. So with this one exercise, it says given a normal distribution with the mean of hundred and the standard deviation of 25, if n is equals to 25, which one of the following statement is incorrect? They have given us the mean and the standard deviation. So therefore, yeah, we're using the sampling distribution of the mean. We are also given the end. And we need to find the incorrect statement. So it means we need to validate each and every statement until we find the one that is not correct. So the first one says the standard era is equals to five. So in terms of sampling distribution of the means, we know that the standard era is the standard deviation sample means or it's the standard deviation, which is given by the population standard deviation divided by the square root of n. So you also just do the same. Your standard deviation is 25. Your n square root of n and it's 25. You do the same. Let's go to the calculator. We start with the case. It's easy with case. You go to the fraction and you say 25 and use your arrow to go down and you put the square root of 25 and then put 25 and you press equal sign. And that gives you a five because the square root of 25 is five. 25 divided by five will be five. So that means that is correct. So let's look at the case, the shop one. So which shop you will say 25 divide by and then you will use the square root. So you'll just press the square root button and press 25 and say equal and that will give you the same thing. Other than that, you can say the square root of 25 first and say equal and that will give you five and then you come here and you say this is 25. You can use the same line and you can just say you found the square root of 25, which is 25 divide by five, which is equals to five. You can still do that. So you can find the answer and then go and calculate manual or you can do it easier. Number two, if the sample mean is 110, what is the Z score? And they telling us that the Z score is equals to 1.6. So we need to validate that. So in terms of sampling distribution of the means, we know that our Z formula is your sample mean minus the population mean divide by the standard error, which is standard deviation over the square root of your N. Now, this is easy because if I calculated the standard error there and I got it right, I don't have to go and calculate it again here because I can just substitute the answer that I have. I can substitute the answer here of the standard deviation divide by the square root of N because I did find it that it's five. Told that the sample mean is 110. So you just put the 110 and go minus. Our population mean is given in the statement it's 102 divide by the standard error, which I don't have to repeat 25 divide by the square root of 25. I calculated it there. So I just substitute the answer. And you just go and find going with my casual, which is easier. You go 100 and 10 minus 102. Use my error, go down with the five and say press equal and I get eight over five because I know that I'm working in the statistics. We work with decimals. Then I press the SD change to decimals and I get my Z score of one comma six. Same thing on your shop calculator. Sorry on your shop calculator, you will do the same thing. So we have 110 minus 102, which is easy. You start first by saying one, one, zero minus 102. And you need to press equal sign before you do the division. Don't do division because the minute if let me show you if you are there and you press 102 and you say divide by and then you press five. You press five. The five will only divide the 102. It will not divide the 110. So you need to always press the equal sign after you do one operation. So like 110 minus 102 and press equal and then press divide because that's another operation that I'm doing by five and then press equal and the answer will be 1.6. So you need to know how to use your calculator, right? So because if I set 110 minus 102 divide by five, you will see that you will get this kind of answers and you will say they are none of the above because your lecture likes to use none of the above and you will be selecting that so many times because you didn't use your calculator correctly. So that's how you use your calculator. So let's do a couple of them because now I am in the mood and we still have a little bit of time, 15 more minutes. The probability that the sample mean is between 90 and 110. So now we need to calculate the z-score for two values and subtract one from the other and so on. So let's do that. So let's first find the probability. Let's first do the z-score of 90. I'm going to do the z-score of 90 here. z-score x-bar minus the mean divide by the standard error and substituting the values because we're still using the same question. The standard error was calculated earlier, makes it easier to just say 90 because my mean for the first one is 90. My population mean is 102 divide by five because my entire standard error, which I can also substitute it instead of writing it that way, I can write it this way. And this gives us, and we go to our calculator. Let's see, I'm going to do 90, you're going to do 110. 90 minus 102 equals minus 12 divide by five and change it minus 2.4 minus 2.40. I'm going to put a zero at the end. And same, you will get the same when you're using this calculator, the sharp calculator, which will be 90 minus 102 equals divide by five equal and that gives you minus 2.4 and you can add the zero at the end. So now go ahead and do z of 110 minus the mean over the standard error. And we know that this is 110 minus 102 divide by five. Calculate and let me know what is the answer. Is it 1.6? 1.6, let's see, 110 minus 102 down a row five equals, let's change, 1.6. And we didn't even have to go and calculate it, right? You could have told me, oh, no, we do have because we did calculate it here. And we found that it was 1.6. So you just need to also pay attention when you answer these questions because some of these questions, they might be related to the things that you already calculated at the beginning. So now we do have the z scores of the between, but that is not what we were asked. We are asked to find the probability. I'm going to write the probability. So remember we find the probability of the sample mean lies between 90 and 110. And that we did calculate the z scores and we found that z lies between minus 2.40. And 1.60. So you can add the zero at the end. And then now what we learn or what we know about this is because we need to go find the probability of z less than 1.60 on the probability table. And we need to subtract the probability of z less than minus 2.40 from the table. And then we go to the table. We're looking for 1.6. It's on the positive side of the table. And we are looking for 1.60 at the top. And that is the answer for 1, which is 94.52. So you will have 0.9. Is it 94.94. 0.94520. 0.9452 minus. We need to go and find on the negative side. We need to go find minus 2.4. On the negative side minus 2.4. And at the top 0. So the answer is 0.0082. 0.0082. And then you subtract 1 from the other. So we had 0.9452 minus 0.0082, which is equals 0.937. 0.937. And we can also double check with the other calculator. Just for interest sake. 0.9452 minus 0.0082, which is equals 2 and change 0.937, which makes that correct. Because we can also add 0 at the end, because they left it at 4 decimal. So that is correct. And then you can do the others to find out whether they are correct or not. And then you can choose which one is incorrect. So let's go and do number 4. We need to find z of x-bar minus the mean divided by the standard error. Because our standard error is the same. So our mean x-bar is 104.5 minus the other one minus the population mean of 102 divided by the standard error of 5. Then you can calculate. Those who are using Casio, it's easy now. I'm not going to bother again with the Casio. Let's do one. We'll use this one. 104.5 minus 102 equals. Remember to press the equal sign. Divide by 5 equal. And the answer is 0.5. The answer is 0.5. And we can also double check here by using the fraction. 104.5 minus 102. And go down and press 5 and say equal. And the answer is 0.5. Go to the table. We look for 0.50. So it's on the positive side of the table. Z score. We're looking for 0.5, not 5, right? 0.5, which is 0.6915. Now I use the equal sign there. So because this is above, so we need to find the probability that Z is above 0.50, which is the same as 1 minus the probability of Z less than 0.50, which is what I found on the table, which was 1 minus 0. Let's go back there to get the actual number. 6915. That will be 0.6915. And that answer will be equals 2. So 1 minus 0.6915. Equal change. 0.3085. Lessing. 0.3085. Yes. Sam, why did you use the 0.5 in this calculation? Because I got it there. Yes, because we needed to calculate first the Z. Oh, okay. We calculated the Z value and we found that it's 0.50. And since the question was above, then it means it's greater than 0.50. 0.5. And then because it's greater than, we need to find the probability on the less than table. So therefore we have to find the complement of the greater than, which will be 1 minus the less than. Okay. Thank you. It's no problem. So since all of them are correct, I don't even have to bother and do this one. So this will be the incorrect one. But in a way, you can go and check if it's correct. Z of X bar minus the mean divided by the standard error, which will be 132 minus 102 divided by 5. And if we double check that, we left with 3 minutes. 132 minus 105 equals divide by 5 equal 6. It's equals to 6. Come to the table. It's the same as 1. I don't know why they say it doesn't exist. Oh, because they said it's more than. Also pay attention more than. So therefore we go find the probability that Z is greater than 6.00. But it does exist on the table, which is 1 minus the probability of Z less than 6.00. So we go to that will be 1 minus 0.99. 0.9999999997. Let's see. We do have that. So I don't know why your lecture would have placed it there to say it doesn't exist. Because at this table, it does exist. There is a more than a 6 and more than a 6. And then that is the incorrect one because it says it does not exist. And we know that it does not. It does exist. So the question here was does not exist, but we know that it does exist because we can calculate it. So that is the incorrect answer. And that conflicts what I wanted to show you. And thank you for coming. Even though I didn't allow for a lot of participation, but I guess if you still don't know how to use your calculator after this, then I really don't know how to how I can help you. The very the most important thing is on your calculator is to know the following. How to when to press the sign, the buttons and how to use your calculator efficiently like make sure that you do things step by step, especially those who do not have a case your calculator. You step manually step by step, not skip, do not be in a hurry. Take your time when you calculate because like I do, because I'm rushing to finish showing you something and I do a lot of mistakes, but with you take your time because your, your past and will fail depend on it. Make sure that you pay attention. Those with a case your calculator. Things are a little bit much easier for you to do any calculation with your case your because as long as you've got the fraction calculate them function on your calculator. Use it to your advantage. Instead of calculating things manually step by step, you can save a whole lot of time, especially with the calculations in statistics. There's one for things like the Z score because from now on we're still going to be using the Z score will be using the Z score in the hypothesis testing as well. So it will save you a whole lot of time. Other than that, thank you for coming and enjoy your evening. Bye. Unless if there isn't another question or comment. Sorry. My bad. Are they any questions? No questions. No comments. If there are no questions and no comments, then feel free to go and have a lovely evening. Bye. Thank you. Thank you. Thank you. Bye. Bye.