 And welcome to your session 17. I must just apologize for last week, Wednesday, or I was so busy that I forgot about our class because I was preoccupied with other things as well. So... No worries. It happens. Yeah, but we are still on track for you to be able to submit your assignment four because now we're working towards your assignment four. So today, we're going to continue with confidence intervals. Like I said, we're going to split it into three sections. So we already covered the two. Today is the other section. And then because today's session will be, the content will be short. We then going to move into doing lots of other exercises just to give you a different perspective in terms of how the questions are posed in the exam as well as if you get those kinds of questions in the assignment as well so that you can assist. And then next week, we will do hypothesis testing. So from Wednesday, the three sessions that will follow up will be doing hypothesis testing. We'll also split it into three sections so that we can do hypothesis testing for the mean when the standard deviation is known and when it's unknown. And then we do for the proportion. And then we come back again before the final submission of your assignment just to do lots of activities relating to confidence interval and also hypothesis testing so that you are ready to submit your assignment four. So let's tack him into today's session. Do you have any comments, question or period before we start? I just have one. Where can I find an update to the T table? I've been looking forward but I'm struggling to find the correct one. Do you have a past exam paper on? Only 2019s, but it doesn't match the one you have. It doesn't match the one that I'm using. Yeah, because you're using 2020s one. No, then it's fine because some exam, in the exam paper, I think they left it at three decimals and you will find that also in some textbooks they left it at three decimals and then others at four decimals. So you can use the one that you have. If it's three decimals, then it's fine. You can still use that. They are the same. It's just that my one is in four decimals and I think I'm using the tutorial letter 101. I'm not sure if in your tutorial letter, let us anyway that are published on my UNISA if you do have the tables in one of the tutorial letters. If not, then it's fine. You can still use the one at the back of your past exam paper or at back of your prescribed book if they do have a tea table. Otherwise, I think there is a tea table in your study guide as well. You just need to check it. I checked the study guide. I didn't find it. There is no table, okay. So, but yeah, you can use the one that you have. Okay, let me see how it goes with this one. I have printed out and take it from there. Okay. Thank you. So as long as it's a three decimals, it's called critical values of teas. Let's go there. As long as it's called critical values of tea, whether it's three decimals or four decimals, it will look exactly the same. It's just that if it's three decimals, then you won't have. So for example, if I'm looking at this value, where it's one decrease of freedom one and 0.10, on your one, it will say 3.078. Yes, yes it does. And mine also starts. Yeah, they would have found it off. Mine also starts at 0.10 and not 0.25 on top. At the top. It starts at 0.10. And you don't have 0.25. It's fine, don't worry about that. Remember, most of the time, like I said, we only use certain cumulative probabilities, which are your one minus alpha. We always concentrate on 10, 95, and that will be that one, and 99. So only those ones. So most of the time, this column, you will never ever even use. Okay, all right, that's fine. You will use one of those. So 0.1, 0.05, 0.03, and 0.01. And sometimes you might also use that one, but it's very rare that you will find where we use 0.05 because then it means you would have been given 0.0, 99% where you have to calculate 99 divided by two. Oh, 0.01 divided by two, which will give you that. But it's very rare that you will use that column and this one. Okay, now then now it should be fine. All right, like I said, yeah, let's see how it goes today in a moment. Yes, okay. All right, thank you. All right, okay. Any other questions, comments? Nothing? Okay, so let's quickly recap on what we did the last week's past just to bring our minds to the same topic that we've learned. We know that the purpose of the three sessions is to understand the basic concepts when it comes to confidence intervals. We also wanted to learn how to construct a confidence interval for the population mean, when the population standard deviation is known, and also when it is unknown. And today, we're going to look at constructing the confidence interval for the population proportion. And I'm going to also include one terminology, which we already have been calculating it and using it, but we've never defined that, which is the margin of error. And so that if you get questions on that, you know what those are, what is a margin of error and how do we calculate it? So the past week, we looked at where the population standard deviation is known, and we know that with that the assumption says the population standard deviation will be given and the population will be normally distributed. And we use this formula, which is the point estimate plus or minus the critical value multiplied by the standard error, which our point estimate is our sample mean and our critical value because the population standard deviation is known. We use the Z table and we find the critical value by using Z alpha divided by two, where alpha is the value we find from the confidence level, which is one minus alpha. And when we divide the value of alpha by two, we find the probability and we go and find that probability on the table and look for the corresponding Z values. And those Z values becomes our critical values, which is our Z alpha divided by two. Multiplying that with the population standard deviation divided by the square root of your standard deviation, which is called also that the standard error. We learned that in the past weeks. Then last Saturday, we learned about the confidence intervals for the mean when the population standard deviation is unknown. And we also learned that when the population standard deviation is unknown, they would give you the sample standard deviation. And your population should also be normally distributed and if it's not normally distributed, the sample size needs to be large enough. And we know that with the critical value for where the population standard deviation is unknown, we go to the T table. And we know when we find that critical value, we use T alpha divided by two and the degrees of freedom. And we know that our degrees of freedom is N minus one. And that will give us the critical value when we go to the table. And the formula to find confidence intervals, it's plus minus or the point estimate plus minus the critical value times the standard error. And you must always, always remember that the plus and plus and minus, the minus gives you the lower limit and the plus gives you the upper limit. So if the question asks you to find the lower limit, you know that you just use the minus side and calculate your lower limit. If it asks for an upper limit, you use the plus side to calculate the upper limit. Otherwise, we know that our confidence intervals are the limits in terms of the lower limit and the upper limit as well. And we learned how to find the critical value on the T table just to refresh our mind. At the top, we look at the upper tail area of the table to look for the probability at the top of the table. And we look for the degrees of freedom on the left side of the table. And where they both meet, that's where you will find your critical values. Different to how you find the critical value with the Z table. Remember with the Z table, when you have a Z table, we use our Z values, which are at the top and the bottom. So the probability we find on the table, we go out and out to find the Z value. Whereas with the student T table, we go in, we don't go out, we go in the table to find the critical value inside the table by using the degrees of freedom and the probability. And that's what we did the last time and we looked at the exercise as well on how we calculate the confidence interval. Today, we're going to continue and look at confidence interval for the proportion. And when we look at the confidence interval for the proportion, we need to remember the following. So, and confidence interval estimate of the proportion can also be calculated by using the sample point estimate, which also will give you some uncertainties and we know, remember from the sampling distribution, we know that if we don't get the sample proportion, they would have given us observation satisfying the sample proportion divided by the sample size. So they will give us the X and the sample size and we can calculate this sample proportion. So with confidence interval, our standard error will, from the sampling distribution, remember, so if I can go back, remember with the other, for example, with the mean, we always use the same sampling error. But when it comes to the proportion, you need to bear in mind that in the sampling distribution, when we were doing sampling distribution, because at that point, we knew what the population proportion is. We were using the standard error as such. When we do confidence intervals, because we do not know the population proportion value, and that is what we're trying to estimate to find where does it fit in or where does it falls between those limits, whether the limits will include the population proportion because we don't know it. Then we use the sample proportion to calculate the standard error. So it means on the confidence interval, when we multiply the critical value with the standard error, our standard error will be the standard error calculated using the sample proportion, not the population proportion. Okay, so similar, we're going to continue with confidence interval. We know that it's point estimate plus or minus the critical value times the standard error. So for the confidence interval, where we calculate the confidence interval for the proportions, the formula will be our point estimate plus or minus our critical value times the standard error. Now, our point estimate will be our sample proportion plus or minus because we're using the proportion always, always, always. For the proportion, we always use the Z values. So it means the critical value of the proportion, we go to the cumulative standardized normal distribution table to find the critical value times the standard error, which is our square root of your sample proportion times one minus the sample proportion divided by N. And that is how we're going to find the confidence interval for the proportion. Let's look at an example. A random sample of 100 people shows that 25 are left-handed from a 95% confidence interval for the true proportion of the left-handers. Sometimes, how will you know what question is this? Because then they talk about confidence interval. When they don't mention anything about the mean, the standard deviation, you must know that you are dealing with proportion. And sometimes also key weights in the question will give you guide in terms of whether you need to calculate the confidence interval for the proportion or confidence interval for the mean. So for this one, it says 95% confidence interval for the true proportion. So it means now we're doing the proportion. So if I'm doing the proportion and I know what my formula looks like, no, sorry, I know that my formula is our P plus or minus Z alpha divided by two times the standard error, which is one minus P, P times one minus P divided by N. If I know that, I need to make sure that in this question, I am given my sample proportion. So reading the question, I can see that I have my N and I have my X. So therefore it means I'm able to calculate my P, which is my X divided by N. Therefore my P will be 25 divided by 100, which is zero comma two five. That is my proportion. And then I can also go and find my critical value because my critical value is Z alpha divided by two. And I know that my zero comma nine five is the same as one minus alpha where alpha will be equals to zero comma zero five because alpha, if I bring it this side, it becomes a positive alpha. And if I take zero comma nine five to the other side, I have one minus zero comma nine five. Therefore it is one minus zero comma nine five gives me zero comma zero five. Then I take this, I need to divide it by two. So alpha divided by two will give me zero comma zero two five zero. If I don't know my critical value by now of nine to five, then I take that value and I go to the Z table, go to the Z table on the negative side because in the negative side, that way I find small probabilities. And there is my zero comma zero two five. I go out, it corresponds to minus one comma nine. And when I go up minus one comma nine and when I go up my last digit is six. So therefore my critical value is minus one comma nine six or one comma nine six in this instance. So therefore I know that my Z alpha divided by two is one comma nine six. So since I know that, then I can start calculating my substituting into the equation. I'm sorry about that. Let me write it some way. Z alpha divided by two, which is one comma nine six. So I just use my formula that I know, substitute into the equation. My P is 25 divided by a hundred. And I know that it's zero comma zero two five plus or minus my critical value. I just did find my critical value of one comma nine six times our P, I've already simplified it, which is zero comma two five. One minus zero comma two five is zero comma seven five divided by a hundred. And I just calculate the standard error and I find that my standard error is zero comma zero four three three multiplied by one comma nine six. And I can split it into two because I'm looking for my lower boundary or my lower interval or limit and my upper limit. And simplify the equation, zero point two five minus one point nine six times zero comma zero four three gives me zero comma one six five one and zero comma three three four nine. And that is how you find the confidence interval for the proportion. And you can write it as such as well by relating it to that population proportion. And any question, if there are no questions, then we can look at the exercise. This is how you interpret. I'm not going to go into interpretation because you do not need to know about how you interpret. Let's look at one of the exercise. According to a recent report by the Census Bureau, 26% of single male household owns stock bonds and mutual funds. Although Census Bureau estimates are based on a very large sample, for convenience assume that this result is based on a random sample of a thousand single male households. The 99% confidence interval for the proportion of all single male households that own stock bonds and mutual fund is. So they're asking us to find a 99% confidence interval for the proportion. So how do I know that I need to be doing for the proportion? It's because in the question they gave me the hint and when I read the statement, I've never saw anything that has to do with the mean or the standard deviation and given a 26%. Therefore, I am also told that this comes from a large sample, so that is my sample proportion. I'm also given my N, so I don't even have to calculate my sample proportion because they have given it to me. All I need is the formula. Sample proportion plus or minus, the critical value alpha divided by two times the standard error. My standard error sample proportion, one minus sample proportion divided by N. Now I need to go find my 99% confidence interval. Finding the 99% confidence interval, I know that it's 0,99 is equals to one minus alpha. So therefore alpha will be equals to 0,01. And I need alpha divided by two. So it's 0,01 divided by two, which will give me, let me quickly open my calculator. It will be zero, zero, zero, zero, two, five. Thank you. It should be zero, zero, zero, zero, five. Yeah, you're right, right. Ease, so the answer is, sorry, sorry, sorry, sorry. The answer is zero, zero, zero, five, right? Ease. Okay, so now, taking that, we need to go to the table and go find, oh, sorry, we need to go to the z table and go find zero, zero, zero, five on the table. Let's go to the z table. Let's look for zero, zero, zero, five inside here. So I am gonna stick with that last one, which is zero, zero, zero, four, because if I round it off correctly, this is way past zero, zero, zero, five. So that would be minus 2.58. So my critical value is 2.58. So my critical value here will be plus or minus 2.58 or we can just leave it as 2.58. You don't even have to worry about it. So now, since I have my critical value, I can just substitute into the formula, substituting. My P is 26%, so therefore it is zero comma 26. Plus or minus, my critical value, I just found it. It's 2.58 times my standard error, which is 0.26 times one minus 0.26. 0.26 divided by my n was thousand. Close bracket, let me go find. Let's find the standard error, calculate that side first. Yeah, I just want to do it on my calculator. You can also do it on your side so that we can see if we get the same answer. Bring it close by. So doing this side first, the square root first, square root of point, I'm going to, since I'm using this calculator, it's got a fraction. It makes life easy to 0.26 times one minus 0.26. Close bracket, go down and put the thousand. And say equal and I get, I'm gonna keep only four decimals at this point, zero comma zero one 39. So yeah, I have 0.26 plus or minus 2.58 times zero point. Now I need to go back to my calculator because I forgot all the digits. Zero one 39, since I'm keeping only four decimals. So now I can just calculate the other side and multiply my answer, multiply by 2.58 equal, 0.0358, I'm gonna keep three decimals, four decimals, 0.3578, let me write it first, 0.035, I'm writing zero. I think it was zero point, yes, I'm writing it right, 0.03558, 0.03558, 0.03558, 0.03558. And that is 0.26, now I can split it into two, which then it will be 0.26 minus 0.0358 and the side will be 0.26 plus 0.0358. Plus 0.0358, and like many of us, we find it easy to do the plus first and we find plus 0.626 equal 0.2958, 0.02958. So I must remember that this is the side 0.2958 and I must do the same on the minus side, do the same, can say it is 0.26 minus 0.0358. Equals 0.2242, 0.2242. Because I rounded off quickly, you will see where my challenge is with this. Because of when I am leaving all the digits and rounding off quickly, you might find that you're missing out on certain digits as well. So the answer is number three, as you can see that the rest of the other question, not even way too close. Just to do it again using the full calculator, those who have this type of a calculator, I can just show you, you can do this whole step, the first step on your calculator, but you need to be very careful when you do that. So 0.26 minus, I'm going to start with the minus first, 2.58 times, I must do my square root of my fraction, which is 0.26 times 1 minus 0.26, close bracket, go down, put 1,000 there. And then use my arrow to go to the end. So I must just make sure that I'm right at the end and I put my closing bracket. And if I want to see the whole equation, I can just scroll, scroll, scroll, scroll and see that I have captured all the answer correctly and then press the equal side, and that is your answer. And if you look at that is edit a one because they rounded off quickly when they were doing the calculations. So you just need to be very careful as well. So let's say we want to do the plus and also check. So I just use my arrow to go and change the minus to a plus, go back one step, put the plus and equal to find my answer. And the answer is 0.2958. So this one, they also didn't round off correctly because it should say 58 and that's how you use your calculator and how you find the confidence intervals manually. Any questions before I give you your own exercise to do without my help and then we can see how we're doing? Any question? No questions. Do I still have people in the room because I might be talking to myself? Oh. We're still here. See, you're still here. Okay, so here is your exercise. I know it's a long sentence. Identify what you have. We're still doing the proportion. I'm going to write the formula here. P plus or minus Z alpha divided by two times the standard error, which is P times one minus P divided by N. Remember to go find your critical value. If you still remember the table that I shared with you the last time of the critical values, you can still use that table even today without going to the table so often. If you have it, you need to have it. Actually, you need to have it close by where it shows you your alpha divided by two values where it says for a 90%, it's one comma 645 for a 95%, it's one comma nine six for a 99%, it's two comma five eight. And I think for an 80%, it's something. So you need to keep that from a table close by so that you don't always constantly go to the table to find the critical value you can just apply. You can use the critical value tables when you use the T table because with the T, you cannot memorize it this way. Okay, so we stop talking, you do the exercise. When you're done, you can post your answer in the chat. Let me open the chat so that I can also see. I thought you already answered the question, okay? So let us know when you are stuck so we can help. I see answers are coming through. Anyone who's lost with assistance, Fiso and Slyly, they are on the same page. They say it's option three and somebody else as well. Like others, you still need more time. Just a minute or so please. Thank you. Thank you. Okay, who wants to do it with me are given a thousand, which is N and you are given our X, which is 230. So we need to find P, which is X divided by N, right? Which is 230 divided by a thousand, which is equals to, our P is, since you guys worked it out. Zero comma two, three. Zero comma two, three. Then we also need to go find our critical value, which is that alpha divided by two. We know that it is that alpha divided by two, our alpha is zero comma zero five. And that alpha divided by two, which will be zero comma zero two five zero, which means our critical value for this will be one comma nine six. One comma nine six. I don't have to go to the table to go find it. So let's substitute into the formula. So we set our P zero comma two, three plus or minus a 95% is one comma nine six times the square root of zero comma two, three times one minus zero comma two, three divide by our N of a thousand. Have you calculated these values before the plus after the plus or minus? Yes, that's zero comma zero two, six, eight, three, five. Yes, slowly, zero comma. Zero, two, six, zero, eight, three. Well, I kept a lot of decimal. I don't know if I should. Then it's five, zero point two, three. Have I wrote it right? That's the most important part. Yes, you're right. Yes. So then we split it. We start with the minus side first and then we go to the plus side. Zero, two, six, zero, eight, three. And what do you get as an answer? Zero point two. Zero, three, nine. And on the other side, we can leave it as three decimals, actually, because we have three decimals. So that will be zero comma two, zero, three, four. Zero comma zero. Okay, so that will be zero comma two, zero, four. Zero comma two, zero, four. And on the other side. Zero comma two, five, six. And the answer is option. Three. Okay, so that's how you answer confidence intervals for the proportion. Now, I want to introduce what we call a margin of error or what we call a sampling error. A sampling error, which is also the same as margin of error, is the amount of positions in the estimate for the proportion that we use. And we either edit or subtract it from the point estimate to form a confidence interval. By now, you should know what I'm talking about because when we have for confidence interval for the mean, we have x bar plus or minus the critical values times the standard error. Oh, x bar plus or minus t alpha divided by two and the critical error. And for the proportion we have, oh, we just did it now. How do I get it wrong? Plus or minus z alpha divided by two times e one minus p divided by n. So in terms of what I just said, is the amount added or subtracted. So therefore, it means this or not cross out, this, this is what we call the margin of error, the sampling error. Because we edit to the point estimate or subtracted from the point estimate in order for us to find that position of estimate. That's all what I needed to share with you. How do we calculate it? You've been calculating it. This is your margin of error. So you've calculated your margin of error there and you find that is zero comma zero, three, six, zero, eight, three. And that's how you calculate the margin of error. And you can do it for all the questions. That is your margin of error. Zero comma zero, three, five, eight. And that's how you will calculate the margin of error. It is your critical value times your point estimate. In terms of the mean, it is just that gives you your margin of error. There is nothing fancy or nothing technical that you need to know about it. You've been doing it. You just need to know that the values after the plus or minus calculates your margin of error. You can also do it for the proportion calculates the margin of error. With that, I'm not gonna do any exercise to ask you calculate the margin of error because we've been doing it. All I just want you to do is go through the activities and answer all the questions. So here, all the activities that are following up are a mixture of today's session, Saturday's session, the past Saturday's session, and the past Wednesday's session, that other Wednesday's session. So we'll go through each and every one of them until our time is up at two o'clock. Or if we finish early, we can go home. Early and enjoy our weekend. This is your first exercise. When you have your answer, you can post it on the chat. If you need help, shout. I'm gonna time all the exercises. You'll have five minutes for each exercise. So this, you have five minutes. If you want to take a break in between, you are more than welcome to go and make yourself coffee. In Cape Town, it's very cold today. It was raining. So I will do a feedback at five past one. You know, when you've killed someone and you see them. Close your eyes. Okay, are we winning? Yes. Do you want more time? Or we can do the activity now? Do it. Okay. What are we given? We're given X and N. We're given X and N. So our N is 200 and our X is 34. So it means we need to calculate P, which is X divided by N, 34 divided by 200. What do you have? Zero comma one seven. One seven. We need to find our critical value. By now you should know what the critical value is when we use proportion. What is our critical value? Z alpha divided by two. One comma nine six. One comma nine six. Okay. So we need to substitute the values into the formula. Okay. Our P is zero comma one seven. Plus or minus our critical value. One comma nine six. Times zero comma one seven. Times one minus zero comma one seven. Divide by our N of 200. Calculating the marginal error. What did you find? Calculating your critical value times your standard error. Zero comma zero. Zero comma zero five two one. Did I catch it right? Zero five zero comma five two one. Five two one, not four. Yes. Okay. So let's split it. Zero comma one seven. Minus zero comma zero five two one. On the upper limit, zero comma one seven. Plus zero comma zero five two one. And what is the answer that we get? Zero comma one one seven nine. And on the upper limit. Zero comma triple two one. Triple two one. Yes. Hey, looks like it's that one. This one is 12. This is 10. This is 12. So it might be that they rounded off too quickly. Somewhere when they were doing some of the calculations. Especially when they calculated the standard error. I'm going to assume. Let's see if I use the standard error of point one seven times one minus point one seven. Point one seven, divide by 200. So if they rounded off quickly, they would have said this is, they kept it at two decimal. Then they would have said this is zero point zero two. So it will be three, zero point zero three. So point zero three times point one nine six and nine six equals. And if I add point one seven, the number. Cause they have two, two, three, three one, which is not even two, two. Okay. I don't know unless the digits here, they were more. I just want to double check how, how do they get to those digits that we can get to them? We do the whole equation. Point one seven minus one point nine six times point one seven open bracket one minus one seven close bracket down to 200. Okay. I don't know, I feel sorry for you guys. I think this pay pass and exams and all that. Cause all this I get them from your past exam papers and all that your past tutorial letters. Okay. So that is one. Let's go to number two using the same information. Now instead of using, instead of using a 95 use a 99. So a 99 would mean our z alpha divide by two would be our alpha year will be zero comma zero one. And that would be zero zero one. And that would be zero zero one. And that would be zero zero one. That would be zero comma zero one. And if we divide that by two, we'll have zero comma zero, zero five. I was to double check my story. Zero comma zero, zero five. I think we did deal do this. We found that it was 2.58 is 2.58. So we know what our critical value is. We just substitute back into, into that formula that we had. So which is P plus or minus. And since we calculated the P, you don't have to go back and calculate it again because it's the same equation or question. So we know that our P was zero comma one seven plus or minus 2.58 times zero point one seven one minus zero point one seven divide by two hundred. And you must be very careful. It says they only looking for their apartment. With this one, I'm just going to give you two minutes because we repeat. Are we winning? We happy? Do you have the answer? Yes. Yes, sir. If we calculate the, I'm not going to go and calculate the margin of error and all that. Let me just use my calculator and see if you guys did it. We all have the same answer. So we only do the plus sign because we're looking for the upper limit. I press equal. Do you all get the same answer? Yes. Which is option number four. So you only do zero point one seven plus the West part is I didn't calculate the margin of error. Do that quickly. Which is just deleting the rest of them. Other questions. Which is zero comma zero one five four. I think it deleted the two before the decimal point as well. Zero point zero six eight five. And that should give you the answer. There we go. So next question looks exercise. Do you know what you need to do on this one? No, this will need to be used with a T table. This will need to use the T table because the population standard deviation is known. It's unknown. We are given S. So you will need to go find the critical value, which is T alpha divided by two and the degrees of freedom, which is N minus one. So they say at 90%. So we know alpha will be zero comma one zero at 90%. Then T will be zero comma one zero divided by two and the degrees of freedom and is hundred minus one. So therefore T will be zero comma zero five and ninety nine. So we need to go find the critical value on the T table. I can just assist with that. And like we discussed the T table, sometimes they use three decimals, sometimes two decimals. So I think it's best to use three decimals for some reason and not for decimals. So but on the hour one, we need to go and look for zero comma zero five, but our degrees of freedom should be 99. So it will be on the second page. We know that we are on this column. So that we don't lose track of where we are. 99, 99, 90, 90, 90. Okay. We're going to lose that. We just highlight it. And we don't lose that. And that is your critical value, which is 1.6604. Like it. 1.6604. So you need to go find choosing and that is your formula. Substitute the values and are we winning? Yes. Okay. Do we have an answer? We've been quiet for some time. So let's work it out. Our mean is 200. Critical value we did find it. It was 1.6604 times our standard error, which is five divided by the square root of 100. 200 plus or minus if I calculate the marginal error. So the sampling error on one side, I get 0.8302. 0, 8302. 0,2. Then I expand this 200 minus 0, 8302. And we do the same 200 plus 0, 8302. And on the lower limit we get 200 minus 0,8302. So 1.199. 1.1698. And on there upper limit it will just be 200. 0.8302. Which is option number two. So now let's do some complex questions. Remember what we did when we were doing the on Saturday past. We looked at one of the question where it was asking whether when we increase the confidence interval does it narrow or expand and when we increase the n or decrease n does it narrow or does it expand? Remember that. We still remember those. So to answer this question you need to remember all those things. You need to remember that. Your confidence intervals when it's big or your confidence level of 90, 95, 99. As to compare it to the others whether the 99 the confidence level becomes or the confidence interval becomes wider or does it become wider? So to do that I think we did one of the exercise where we looked at two scenarios where it has the confidence intervals are the same. So let's take this as an example at 95 we have 0, 11 and 0, 22. At 99 let's write I'm going to write it on that slide 0, 0, 2221. I hope I will remember all that. So we have at 99 the confidence level is 0, 17. We didn't calculate the other side. I'm just going to use the same information that we have here because we didn't calculate the minus on this one. So we find that this side is 0, 2385. Let's quickly do the minus 0, 17.17 minus 0, 0, 6, 8, 5 we get 0, 10, 0.5. So 495 because I'm only going to work with the 2 to demonstrate. For 95 we got 0, 11, 7, 9 0, 11, 7, 9 and on the other side we got 0, 221. Now, if I take these two confidence levels and I draw them so I draw the first one and I say it is 0,10 I'm just going to use the last two digits and this is 0,24 and I draw this and say this is my 99% that is my 99. Looking at my 95 it says it is 0,112 So 0,12 will be somewhere inside and 0,22 will be somewhere inside as well. So therefore it means if I draw this this will be my 95% So since 99 is bigger than 95 therefore if I'm going to have a 90 so 90 will be somewhere in between as well so this will be my 90. So then what it tells me is a 99% the limit will be bigger than a 95 and a 95 limit will be bigger than 90 I hope you remember all this so we're going to take this information that we have without the numbers just to remember that 99 is bigger than 95 and 95 is bigger than is bigger than 90 so I'm going to draw it here so we draw that here so we say this is 99 this is 95 and this is 90 that's what we have so it means when we look at A, B and C we need to visualize it in terms of 99, 95 and 90 to match the confidence with the values so whichever one that has a bigger or a smaller starting point will be our 99 and whichever has the bigger interval so this will be small and big, small, big, small, big so let's go between 96, 196 and 194 and 196 which one is smaller which is this so 94 is smaller then it means 194, 194 will be our 99% because 194 is smaller than 196 and 196 so we haven't determined which 196 we're matching with another one because we know if this is a number line this is 0 and this is 200 94 will be here and 96 will be there so we can already identify those so we can also identify which one between the two so 94 so therefore 196, 06 will be this one 196, 06 will be that one and 196, 70 will be that one so we know when we go in this way we go into the bigger side so therefore without looking at the end, the ones at the end by looking at those ones we can determine which one is 95, 99 and so forth so if we look at the bigger side as well we can see 203 and 205 so this will be 205 and this will be 203 we just need to determine which one is which now we know that 30 will be before before 94 so this will be 30 and this will be 94 so I already have already visualised which one is which so this one I did it already by looking at the smallest the lower limit so already I can see 95 is that one that is 95 and this will be 90 and then I can find which one is the correct info so which match the confidence estimates with the appropriate confidence level A is 95 so this one won't be correct that will be correct this won't be correct that won't be correct that one will be correct so we move to the next one B we only have those two that won't be correct but this is correct so therefore our answer is option 5 so you just need to apply the theory that you know things that you know in terms of the confidence level and confidence intervals any questions if there are no questions you will have another exercise to do pay attention to the question so this is what we have been giving today we have been posting on the chat sorry ma'am just to go back to exercise 6 so what I did is because I didn't want to visualise I just subtracted the two values from each other and then I compared which from big smaller smallest and then thus assumed my 95 which is almost the same but yeah I just wanted to just get your opinion in terms of doing a subtraction of the upper limit minus the lower limit so like for example for I just looked at the value for A then I did 203.94 minus 196.06 and then for B I did 205.18 minus 194.82 and then for C I did 2 3.3 minus 196.07 and then I just compared my answers yes so you can do it that way as well because when you compare we know that when we compare a 99 with a 90 a 99 will have a bigger value and a 90 will have a smaller difference between the intervals because they are too close to one another whereas with a 99 the intervals will be wide or bigger or wider and this will be narrower so also if you compare a 99 and a 95 you will see that the bigger the confidence level the wider the interval and the smaller the confidence level will be so you can you can use any method you want as long as you can remember that if you visualize it you will remember that the 99 will be bigger than 90 in terms of confidence level so in math to skin a cat you can make use of any method there are many ways okay thank you exercise are we winning are we done still calculating are we done okay so we are given x we are given and we know that our z alpha divided by 2 for a 90% is the only exception 1.645 it's 1.645 and we need to calculate our point estimate p plus or minus critical values z alpha divided by 2 times the standard error which is p times 1 minus p divided by n our p x divided by n 70 divided by 100 p equals to 0.7 plus or minus 1.645 times the square root of 0.7 times 1 minus 0.7 divide by 100 sampling error or margin of error 0.7 plus or minus what do we get 0.07 0.0754 0.0754 0.0754 we all get that okay we split it 0.7 minus 0.0754 and on the other side we have 0.7 plus 0.0754 on the lower limit we get 0.6246 and on the upper limit 0.7754 any questions I think we are left with two more questions or one oh no we have more okay any questions if there are no questions I think this one we did it I might be wrong I might be right but if we didn't do it there is your exercise do you know what you need to do here because this is not proportions for the mean are you going to use z or t I suspect z what are we given in this question well we given the standard deviation so so what standard deviation is given sample so if we given the sample standard deviation which is s then it means we are going to find t you must read the questions carefully remember if you are given the sample standard deviation we use t you will need to go and find your t alpha so we using 99 so our alpha will be 0.01 therefore our alpha divided by 2 will be 0.005 and we are given our n so our n minus 1 will be 30 minus 1 which is 29 she's 29 therefore to find the critical value so which is the critical value of 0.005 and 29 we need to go to the table the t table so let's go to the t table we are looking for 30 so we need to go to the next so reduce the table a little bit and the top we are looking for 0.05 we must go do you have a table with 0.05 yes alright so it's only for the 0.75 that you might not have so we need 29 where are we this is 29 so I've already highlighted everything next to 29 so let's move this a little bit bigger so that those with no tables can see so we know that we using the last color so we are going to use the table with me 2,7564 2,7564 let's just double check that 7564 so we can just substitute into the formula our mean and sample standard deviation respectively so this is our mean which is our standard deviation which is s so our mean is 90 plus or minus our critical value was 2.7564 times our standard error which is 18 divided by the square root of our n our standard error which is 18 divided by the square root of our n this is with the calculations 2.7564 times 18 divided by the square root of 30 which is my marginal error is equals to 9.058 we can leave it at 4 decimals since our answers are at 4 decimals so we can leave it at 4 decimals so if I round it correctly it's 5 so we add 1 it's 9.5 so 90 minus 9.058 5 and 90 plus 9.05 5 8.5 and the answer for this is 80.945 80.945 80.945 do you also get the same answer as me you can see yes so let's do the other side which is the plus side 99.058 90 the side we get 99.058 5 which is cannot be that one this is option 2 and this is probably because of the number of decimals that they use when they win the answer so I'll always look at the true last digit after the comma just to give an idea in terms of the correctness of the answers as well because we get the same answer as that so that will conclude today's session and they is this one question you can go ahead and do it yourselves you must be very careful of this one as well when we do it because this one uses the population standard creation but that does not because what they're asking you actually is using a 90 percent of that statement that you have which one of those questions are or statement is correct is calculating correctly so you'll have to calculate the confidence level at 90 at 95 and at 99 and choose whichever one is not correct based on those statement and choose your answer there that is that one next one which is the last one I think we already did this just also calculate the confidence interval of this question and then that's it so we conclude by saying you have learned for the last three weeks or two weeks the basic concepts of confidence intervals how to construct the confidence interval for the population when the population standard deviation is known and when it is unknown you've learned how to construct the confidence interval for the proportion and also what them and how to calculate the margin of error or sampling error and that concludes confidence interval see you on Wednesday when we start with hypothesis testing for the mean when the population standard deviation is known if there is any other question that you want to post now is your time if there are no questions then we can call it a day enjoy the rest of your weekend