 Waiting for the recording to start. Okay. So welcome to your session 15 of your STA 1610 tutorial, online tutorial. Today, we're going to discuss study unit 8, which will enable you to do assignment 4. We're going to study unit 8 covers confidence interval. I think on WhatsApp, I did share that I'm going to split study unit 8 into four sessions so that we cover every component or every section of study unit 8 in different days. So, and I also said, you need to also remember the learnings or the knowledge you learned in study unit 6 and study unit 7, which is normal distribution and sampling distribution. But more so the sampling distribution because some of most of the concepts that we've learned in sampling distribution, we are still going to continue to use those concepts today. We are still going to use the Z cumulative standardized normal distribution table, that you know that it's got the positive and the negative side to it. We will continue using that table when we go find other babies. Let's get to it. By the end of today's session, you should be learning one of those objectives that are there. So, at least two of them. We will learn the basic concepts of confidence interval, how to build the confidence interval. We're going to learn all those building blocks. Because once we've learned them, then it will be easy to do the following. It will be easy for us to construct a confidence interval for the population mean. Today, we're only going to cover when the population standard deviation is known. You will see what I mean by saying when the population standard deviation is known. Next, on Saturday, we will cover the population standard deviation when it is unknown because then we introduce a new table, a t-table or a t-test table. Then on Wednesday, the following, we're going to learn how to construct confidence interval for the population proportion. Here, we're going to use the same table that we're going to be using today and the ones that we've been using all along, which is the cumulative standardized normal distribution. But for today only, we learn the basic concepts and we learn how to construct a confidence interval for the population mean when the population standard deviation is known. So, what is confidence interval? Confidence interval is one of those inferential statistics. You remember, at the beginning, when we started with stats, we said statistics has two branches, the descriptive statistics, where we describe the data, and we also have inferential statistics, where we make conclusion about the population based on the sample. So, confidence interval is one of those that we use. So, why am I not mentioning the normal distribution then and also the sampling distribution? Those form part and base of what do you need to know how to do confidence interval? So, we still, let's say, those are introductions to what's doing inferential statistics so that you know the concepts and then you can apply them. The sampling distribution, we use it in the hypothesis more often because in the hypothesis testing, we use the Z score, which is what we call the test statistic. But now you would have learned how to calculate it, you know how to use it. In hypothesis testing, we will use that in testing the hypothesis of a population or of the proportion. Population mean or the proportion. Confidence interval also is one of those inferential statistics that you can use to estimate the value of your population mean using your sample statistic, which can be the proportion of the mean and we're going to learn that. Today, we're only going to learn about the mean. So, with point estimate, it is just only a single number, which a point estimate can be the mean or the proportion. In a confidence interval, it provides additional information about the variability of that estimate. So, what it means is, if I have a confidence interval, so with confidence interval, it means there should be, it's an interval and with interval, there is the lower boundary and there is the upper boundary. So when I have those lower boundary and upper boundary, I need to make sure that within those boundaries, my estimates fall within. So we're going to create or construct a confidence interval, meaning we're going to know what our minimum value is and what is our highest value and use those values to determine whether my point estimate falls within those and make a conclusion. In your module, you do not need to know how to make conclusions about the point estimate. The only thing you need to know is the concepts and then you need to know how to calculate the confidence interval. You do not need to know how to interpret it. It's not a prerequisite for you to know how to interpret it, but I will show you how to interpret it as well. So, like I said, point estimate, we use them to estimate or to make a conclusion about the population. So we know what the population parameters are, either the mean or the proportion. So if it's the mean, it's the mu. If it's proportion is the pi. We can estimate this population parameters by using the values that we get or the measures we get from the sample, which are your statistic. For example, we can use the mean of a sample, which is your sample statistic. We can use that to estimate the population. We can use the sample proportion, which is your sample statistic. We can use that to estimate the value of your population proportion. And the mean and the sample proportion, the sample mean and the sample proportion are what we call the point estimate. And you will learn, every time we do confidence interval, I will constantly be saying point estimate, you need to know that a point estimate will be the value of your sample mean or proportion. Not the population mean or not the population proportion, but the sample. Okay, I hope we are on the same page. When we construct confidence interval, we actually want to know how much uncertainty are associated with the point estimate for the population parameter. And like I said, we use the point estimate to estimate your population parameter as well. And those estimate that we're using, when we create those upper boundary and lower boundary, which are what we call the intervals, it's what we call a confidence interval. So the minute we calculate those, by using the point estimate, we are creating what we call intervals. We can call them confidence interval and I will tell you why I'm calling them confidence interval, because once we make, when we make a decision, we need to be confident about the decision that we are making and I will demonstrate that later on. And when we talk about this point estimate, this or the interval estimates, the lower boundary and the upper boundaries, those intervals, they give a range of values and those range of values, they need to take into consideration the variation in the sample statistics that exists from sample to sample because different samples can come with, you can calculate or it can have different sample point estimate or sample means or sample proportion. Different sample can have different sample means, different proportions. Also, when you build the confidence interval, you need to base it on one sample. You cannot do on multiple samples like we did, like what the sampling distribution does because sampling distribution takes multiple means and then I calculate the sample distribution. So yeah, we only base it on one sample, not on many samples. It also gives you information about the closeness to the unknown population parameter. Cause remember, we don't know what the population parameter is, but we can estimate that population parameter by using the point estimate. And when we calculate the confidence intervals, we always calculate them by using the confidence level, which is a percentage, which is a probability. I'm going, don't get confused. A confidence level, which gives you the probability, which is a percentage. It also allows us to say, when we conclude, we are 99% confident that the true population mean is between those intervals or we are 95% confident that the true mean lies between those two confidence intervals. But we should never, ever, ever say we are 100% because we should also allow for margin of errors that can exist. So with confidence interval, it can never be 100%. So either 80%, 90%, 75%, 99%, but never 100%. And we'll learn how to use those confidence interval or how to get those confidence intervals or confidence levels. So let's look at an example at the high level. Do not worry, do not panic when I'm showing you this. I'm just showing you what we're going to be doing. I'm not saying concentrate too much on this. So let's look at the serial field example. If we have a population with the mean of 368 and the standard deviation of 15, if you take a sample size of n equals to 25, you will know that. Okay, remember here I'm talking about the population. So if I have the population, then if I am calculating the confidence interval based on the population, then I will have my point estimate for the population which will be 368 plus or minus my confidence level, my confidence level which is that 95% confidence level, we use that 95% confidence level to find that critical value which I will tell you just now about it. But for now, taking the confidence level and multiplying that with the standard error. You remember the standard error? Standard error, your population standard deviation divided by the square root of n that is if we do it for the mean, remember that. And we find that once we calculate this equation, we find that the confidence interval is between 362.12 and 373.88. Therefore it means if I look at my population mean of 368, remember I took a confidence level of 95%. I can say it contains 95% of the sample mean that we have. So this is provided if my sample mean is equivalent to my population mean in this instance, there are both 368 because I use the population mean instead of the sample mean. So if in this instance is because we know what the population mean is, but most of the time and in your module as well, you will learn that you will not be given the population mean. So if your population mean is unknown, then we're going to use the sample mean. And that is what your module requires you to do. You are going to be given the sample mean, not the population mean. So we're going to use the sample mean to estimate what the population mean is. So if my population, not my population, but if my sample is 362, so it means I'm going to replace this 368 by 362 and calculate my interval. And I will find that my interval by substituting that 362, which is my point estimate plus or minus, and we will do this properly, I assure you. I will tell you how to also do the plus or minus. Plus or minus 1.96, which is my confidence level of 95% times the standard error. I find that my estimate lies between 356 and 368. Since my population mean of 368 lies in between that we can make the statement and say the population mean is included in the confidence. And we are 95% confident that the population mean, the confidence interval contains the population mean. Okay, so what we've learned so far is in practice, you only want to take one sample, like we did with one sample there. We're using only one sample by taking one sample. Number two, we also know that we are not giving the population mean, so you don't know if the interval actually contains that population mean. You will know that either, whether it's 99% confidence or it's 95% with our example, you know that 95% confidence interval that was formed contains your population mean because the population mean lies between that. How do we then build this confidence interval? So to build a confidence interval as well, we build it in order for us to do estimation to estimate what the population parameter will be. The estimation process we know just to refresh our mind, when the population is huge, we take a sample. So if we have the population, we select the sample, we take the sample, measure or point estimate, and if we calculate the interval, we find that the interval is 40 and 60 and we found that the mean was 50. At 95% confidence interval or confidence level, then in conclusion, we can say you are 95% confident that the mean is between 40 and 60 based on the information that you have calculated using your point estimate, which is your sample statistics that come from, the sample. So now, how do we then do this calculation? How do we calculate confidence interval? So to calculate confidence interval, we use this formula and the formula is your point estimate, which can be the mean or the proportion. In this instance, we use in the mean. So point estimate the mean plus or minus. Remember, your confidence interval has the lower boundary and the upper boundary. So this plus or minus creates lower boundary and upper boundary. So plus or minus, minus creates your lower boundary plus creates your upper boundary. Plus or minus your critical value, your critical value in this instance is your Z value if we're using the Z table. The critical value is going to be our Z value if we are using the Z table. When we're using other table, I will also explain to you what your critical value is. So how do we then find this critical value? We find the critical value by using the confidence level, which I just said the confidence level is your probability. It's one minus the probability value, but it's the probability that you find in the table. And then you go and find the Z value on the table. I will show you how to do that. So we find the critical value, which is our... For now, just know that your critical value for the papers of today, your critical value is your Z value. Sorry. For the papers of today, your critical value will be your Z value, right? We're going to call it the Z value. The critical value is our Z value. Times the standard error. So since we're doing for the mean, it's going to be the population standard deviation divide by the square root of M. So this formula, we say in a way, let me write it here. If I expand it, it will be your point estimate minus your critical value times your standard error and you put the bracket like that, because this side is your lower entry. And we open the bracket, put the comma, and we put point estimate plus your critical value times your standard error, close bracket. So it means when you do confidence interval, you are always going to expand your table or your formula to be a lower boundary and an upper boundary. If they ask you to only calculate the lower boundary, you're going to use point estimate minus the critical value times the standard error. If they ask you to give only the upper boundary, point estimate plus the critical value times the standard error. Do not make a mistake of swapping them around, putting the plus before the minus. Always know that you need to do the minus first because minus tells you it's a smaller value. The minus first, then plus first. Or if you want to remember always when you get the answer, whether you do the plus first. If you do the plus first and you get the answer for where you were doing the addition and you find that the answer is 56 and then you go and do the minus and you find the answer is 30. Let's say for argument sake. Do not write like this, 56 comma 30. It's going to be wrong because 30 is small. Always remember that lower, upper. So it means you're going to write 30 first and then 56 second. So that it starts from 30, it goes to 56. Pay attention to the values when you do your answer or when you write your answers. I have seen people do the plus first and put the plus answer and then do the negative and then put the negative answer. Or sometimes you might find that your plus and your negative, one becomes bigger than the other. That is, I don't know where you will get that. But if you get a, where you do a negative and you get the value that is bigger but when you write your answer because the chances are both of these two answers 56 and 30 and 56 might exist as one of the option. And if you wrote your answer like this, the chances are you will choose 56 and 30 as your answer instead of 30 and 56. So pay attention to that. Okay. So point estimate, which is our parameter. Oh, sorry. Our statistic, which will be our mean. Okay. And we'll look at an example just now. The critical value, which will be based on the confidence level, which is our Z value that we're going to use or we're going to get by using the confidence level, which is our probability on the table to go find the Z value. And those Z value are our critical value. The standard error, which is your standard deviation divided by the square root of N for the mean. So far, any question? Any comments before I move? Any comments? Any questions? Are you still happy? Are you still here? We are here. Processing. Yeah, it will get better with time just now. When we do an exercise, you will understand everything when we put it together. Okay. So we kept on saying critical value is our Z value and I kept on saying we're going to find it by using the confidence level. So what is the confidence level? Confidence level, we use it to calculate the confidence interval and it will also assist us because the confidence interval helps us with the estimation process as we know that we use it to find the Z value. The confidence level, you will be given as a percentage. Now, I need to be very, very careful here. It is always going to be a percentage and that percentage usually is 98, oh, let's start at 85. Let's start at 75. It can be 75, 80, 90, 95, 99. I will show you a table just now. That is the confidence interval. It's always in 100% format. So with the confidence interval, which is one minus alpha, the confidence interval, so this 95% that I'm referring to, it is one minus alpha where our alpha, it is what we call level of significance. I hope you're going to remember all this. A confidence interval, if for example I'm using 95%, 95% confidence level, it is the same as one minus alpha. And one minus alpha, it is your 95% confidence level. Where alpha, it is what we call level of significance. So if I know that my 95% confidence interval or confidence level is, I need to convert this to a decimal, so I always work with 0,95 and I know that that 0,95 is one minus alpha, so I can make it equal to alpha and then calculate the level of significance which is my alpha value and my alpha which is now this alpha is what we call probability. On the table, if I'm using the Z, so since we're going to use the Z table, on the Z table, if I'm using the alpha 0,05 it is my probability on that table and we're going to learn how to do, how to find the critical value using this level of confidence. Okay, so let's recap on what we just did or said. The others are just theory that you need to go back and read, I'm not going to repeat that. What I'm going to repeat is this. Remember, when we find the confidence interval, we're going to use the point estimate plus or minus the critical value times the standard error and we know that our critical value, we're going to find it by using the confidence level which the confidence level we now know is 1 minus alpha and we'll use that 1 minus alpha to find the value of alpha which is also called level of significance. And it's also called the probability on the table when we're using the Z table. So it means our alpha is our probability and it's also called level of significance and we use that probability or level of significance to go find the critical value on the table so that we can come and substitute it onto the formula. Okay, at a high level let's go back. We know that we're doing confidence interval and for today we're only doing confidence interval for the population. So it means confidence level and the general formula we're going to apply. When we do the population is known, sorry, population means when the population standard deviation is known when the population standard deviation is unknown and when the population is known. When we need to do confidence interval for the proportion. So for today we're only going to concentrate on this part and apply confidence level and the general formula for today. So how do we then do that? When we find the confidence interval there are several assumptions that also needs to be met in order to know which confidence interval you need to be constructing. The first assumption that needs to be met is that which is the key to everything that we do. The population standard deviation has to be given. They have to give you sigma or they have to tell you that the population standard deviation is this. I hope I am clear. You will be given sigma which is that symbol you will be given sigma if not sigma in a symbol they will tell you that the population standard deviation is this much. Then you will know that the population standard deviation is known or it is given. The population needs to be normally distributed and if the population is not large enough then we need to use the large sample. Those two assumptions for the purpose of your module you do not have to worry too much. The only thing that you need to worry a lot about but you need to know those assumptions in case they ask questions on confidence interval it's one of those concepts where they can ask you questions theoretical questions about the population standard deviation. So you just need to know the assumptions as well. But the only one that is important for now that will guide us in terms of whether are we using the T table or are we using the Z table it is the population standard deviation. If it is given or it is known to us from the question from the statement that they give you you know that we do a confidence interval for the mean where the population standard deviation is known. And if so then we can calculate the confidence interval by using the point estimate remember P E point estimate plus or minus our critical value times the standard error and in terms of the mean when the population standard deviation is known our point estimate is our sample statistic which is the mean plus or minus the critical value as you can see there it says Z alpha divided by 2 which will be our critical value by using Z alpha divided by 2 we divide alpha by 2 because there are two sides on the confidence interval there is the lower boundary and the upper boundary so we need to split that confidence interval so it will be Z alpha divided by 2 times the standard error which is the population standard deviation divided by the square root of N so before we do everything let's find the critical value finding the critical value for example it's easy remember we have two sides so I'm going to not use this because I think it's more complex for now so we're going to go black screen black screen okay now if they tell us remember we're looking for the critical value so if they tell us we need to find at 95% confidence interval or 95% confidence level in your module they don't talk about confidence level they will always say 95% confidence interval you must know that they are referring to the confidence level when they say at 95% confidence interval at 90% confidence interval at 99% confidence interval you need to know that they are referring to the confidence level so if they say at 95% confidence interval all that you need to know or what you do is to convert a 95% to a decimal which is 95% and what we know as well is that 0,95 is the same as 1 minus alpha remember that it is your confidence level then we need to solve for alpha so since alpha is negative on this side we can bring it this side it will be positive and we can bring 0,95 to the other side then it will be 1 minus 0,95 and our alpha will be equals to 0,05 that is our level of significance since I have the level of significance I still need to go find the critical value and I know that my critical value is alpha divided by 2 so if my critical value is alpha divided by 2 I know what my alpha is so therefore my z will be 0,05 divide by 2 which tells me my z will be equals to 0,025 and I am going to put a 0 here at the end now I need to take this I need to take this information and go find my critical value on the table I will go out and leave this slide show and stop sharing and share my entire screen just give me a sec I've got so many things open on my screen please bear with me as I close all of the things that are open and this slide show not any of those things that I wrote so we need to go to the table let's bring up remember this is what we worked on we need to go find the critical value for z 0,025 0 so we need to go to the accumulative standardized normal distribution table that we have been using for the past weeks she's table E2 and remember we need to go to the negative side because on the negative side that way we find the smaller probabilities don't even bother to go to the positive side so we come here in the negative side table we go onto this table we look inside here because we are looking for z 0,025 so we are only interested in those values there at the bottom of z so inside this table we need to look for a value that looks like that 0,0 1.96 then it means I must go down and 1, minus 1,96 so not this one it's that one 0,250 so we need to go and we get 0,025 0,250 we find minus 1,9 we go to the top and we find 6 now at this point I'm going to say ignore what we've learned in the previous sessions where we stick to the rule and keep the minus and then go find the plus and then this and then that, no so I'm going to say once you have the answer there you can just say it's plus or minus this is our critical value we don't even have to worry about the plus or minus because remember our equation which was the mean plus or minus the critical value which is z alpha divided by 2 times the standard of the square root of n divided by so this whole thing it is this whole thing because it is all this the plus or minus are taken care of by that plus or minus on the equation so we're going to ignore the negative sign in front and only keep the actual value and that's how you're going to read the table so you're just going to say it is your critical value is 1,96 and then we use that to substitute into the formula so it means once we find the confidence level and we find our level of significance then we calculate or we calculate our P value or our probability by dividing the level of significance by 2 which is still going to give you the probability so all this are still probability but I don't want to call it probability as yet I'm only going to call the last bit so that you don't get confused you can only get confused when we do hypothesis testing because then with hypothesis testing I'm going to say probability as well for now we use the confidence level to find the level of significance and we use the level of significance to go find the probability and we use the probability to find the critical value which is equals to 1,96 and once we have the critical value then we can substitute into the formula now I want you to find let's go back to our presentation I want you I'm going to go black screen again I want you to find a 99% confidence interval I want you to find the critical value for a 99% confidence that is your exercise let's do it step by step actually find for me the level of significance and once we have the level of significance then we can move to the next step what is your level of significance see if there are any comments it's 0,99 is equal to 1 minus alpha let's do that 0,99 is equals to 1 minus alpha then our alpha will be 1 minus 0,99 ok ok ok and we get 0,01 0,01 that is your alpha so now I want you to go find your z alpha divided by 2 0,05 so it will be 0,001 divided by 2 which will give you 0,0 how many zeros? 0,002,5 0,005 2 zeros 0,002 0,005 so now I want you to take this value and go find the value on the table let's go to the table remember we're only going to be on the negative side so we're looking for z of 0,005 inside the table double 0,5 it seems it lies in between two values should we divide or take the average of the two z values ok so it lies between these two values because if I take this one is one more and if I use this one is one less so in terms of this in terms of 99% we take both of the values no, we don't take both of the values we take one so you're always going to take the value that it is not more than so let's see this is less so if I round off this one will be more accurate than the other one because this is more than so therefore for 99% it will be 2,558 and that's what we use in statistics going to use 2,58 ok so the other value that has more exceptions as well so let's go back to the presentation so we know that this one is 2,58 our convenience level ok the other value that I want you to find is 90% can somebody mute in the meantime sorry that's me 20% let's go find the level of significance 0.1 ok so it is 0,90 is equals to 1 minus alpha alpha will be 1 minus 0,90 which will be equals to 0,10 then let's go find the critical value which is z alpha divided by 2 z our alpha is 0,10 divided by 2 when you answer the questions as well please do not do shortcuts follow all the steps that I'm showing you you will never get it wrong ok so what is that 0,05 0,05 so now if we go to the table if we come here onto the table and we're looking for z of 0,05 you will notice that it also like similar to the one that we had before but this one is a little bit different because this is 5 up it's not even one up one less or one more in statistics what they did in state of selecting one they took both so for 0,05 we're going to use 1,6 and we're going to take 4,5 so for 90% the critical value is 1,645 for 95% our alpha is 0,05 and our critical value is 1,96 for 99% our alpha is 0,01 and our critical value is 2,58 for a 90% our alpha is 0,10 and our critical value is 1,645 and there are other critical values we're going to go and learn more about them just now so what I've just explained how to find the critical values we use them to define our boundaries so you can see that the minus remember we had our point estimate or let's use the actual point estimate which is the mean the equation is the mean plus or minus the critical value which is alpha divided by 2 times the standard error which is the population standard deviation times the square root of n so because of the plus or minus minus will be in our boundary the plus will be in the upper boundary so we're creating this boundary this area here it's 95% which is the bigger area it's the 95% where the smaller areas which are this outside this our alpha divided by 2 those are the areas that we are looking for so we're saying if the value of the mean falls outside of this boundary we can conclude if it falls within we can conclude that it falls within this if it falls outside then we can not use the estimate okay so for a 95% confidence interval our critical value is 1,96 our alpha divided by 2 is 0,02 and these are some of the confidence intervals we covered 90, you know how to find that we covered 95 and 99 so you can also find 98 which will find that the critical value will be 2,33 for 80 is 1,28 the majority of the critical values that they are used in most of the questions in your study guide or in your module 95% is used most often followed by 90% the others likely highly and likely that they will use them and also 99 sometimes they do use it so those are the most commonly used with 95 being the most used 95% confidence because when we do confidence intervals you will see with lots of exercises they will keep on asking 95% confidence, 95% when we do hypothesis testing most of the time they will ask 95%, 95% so they always constantly use a 95% confidence interval okay so let's now do an example and see how we can apply all this that we just learned a sample circuit from a large normal population has the mean resistance of 2.2 we know from the past testing that the population standard deviation is 0.35 remember the assumptions and I said you can ignore the first one where they said the population needs to be or the sample needs to be large enough in this instance the sample is not that large enough but anyway the two the one that is most important I said population standard deviation if it's known so since they have given us so it means population standard deviation is known therefore it means when the population standard deviation is known we use point estimate plus or minus we're going to find the critical value using the z table which will be z alpha divided by 2 times the standard error so going back to our question we identify what we are given on to this question we are given n which is equals to 11 we are also given the mean from the population is large large normal population so one of the other assumption large population is also there we are given the mean from a sample remember this is a sample second from a large population has the mean so it means this mean comes from this 11 sample so it means we are also given the mean of 2.2 we know from the past testing that the populations standard deviation because they have given our populations standard deviation and it is 0.35 so we have the mean we have the populations standard deviation we have the sample we need to find the critical value 95% confidence interval remember at 95% confidence interval what is the true mean resistance of the population we need to estimate if the population mean falls within that estimate within the interval so we need to go find 95% confidence interval so we know that at 95% confidence interval it is 0.95 is equals to 1 minus alpha therefore we know that alpha will be equals to 0.05 and to find the critical value we take z alpha divided by 2 which is z 0.05 divided by 2 which is equals to z of 0.0250 just to recap so that people don't get confused so we go to the z table and go find our critical value we come to the table we look for 0.0250 which is there we go outside the table and we find 1.96 9 sorry 1.9 and we go up we find that it corresponds to 1.96 then we come back to our question and we say our critical value is 1.96 so this is critical critical value so since we have the critical value we need to take our equation which we have the x bar plus or minus the critical value times the standard error so x bar plus or minus our critical value which is z alpha divided by 2 times the standard error which is the standard deviation divided by the square root of n our our mean is 2.2 substitute 2.2 plus or minus our critical value we did find it is 1.96 1.96 times the standard error which is the standard deviation of 0.35 divided by our square root of n which is square root of 11 we can do we can solve the other side when I use my calculator from here not the expensive one I think I can still get away with this okay I think so so 0.35 please double check for me because I'm going to do you also get 3.31 multiply because it's a long number I don't want to write the long number multiply by 1.96 equals 0.26 so here I have please call out the number you can leave 4 digits so we'll have 2.2 plus or minus 0.20 0.20 0.68 so we can leave at 4 decimals so now since I have in this smaller form you can start splitting the equation by saying 2.2 minus 0.2068 not close bracket we're not closing the bracket as yet we're putting a semicolon and you go 2.2 plus 0.2068 close bracket so now go calculate what is 2.2 minus 0.2068 that's 1.9932 1.932 semicolon 2.2 plus 0.2068 2.4068 and that is your confidence interval we have found the confidence interval and we can make conclusions so I'm just going to come here so you can also write it in this manner or you can leave your answer in this manner so depending on how you find the options but usually the standard way of writing it is with the brackets and we know that this is the lower boundary and this is the upper boundary so if the question was asking you to only give the answer to the lower boundary so that you're going to use the minus if they asked you to find the answer in the upper boundary you're only going to use the plus sign so how do we then conclude we can conclude by saying we are 95% confident that the true mean resistance is between 1.9932 and 2.4068 OHMS and with that what we needed to learn today to remember that the key thing with everything is the population standard deviation is it given if it's given then we're going to find the critical value from the Z table the other key thing how we find the critical value if for example let's say they didn't say confidence interval but they said at 5% level of significance are you going to get confused what they mean if the question was determine a 5% level of significance of the population no because that's 1 minus a no level of significance what is level of significance alpha value it's your alpha value so if they give you 5% level of significance what is the next step what will be your next step find the Z alpha over 2 you will use the you start from yeah if they give you alpha you go just using the alpha value to go find the critical value if they give you 95% confidence then you start by saying 1 minus alpha so you just need to pay attention to that some questions sometimes they might not give you a 95% confidence interval but they will give you a 5% level of significance and you will notice this when we also do hypothesis testing they will use them interchangeably either give you the confidence interval or give you a level of significance but you just need to know where you are and what the question is or the statement is given or what are you given in your statement let's look at an example then we can go home or go to sleep Africa check is interested in the activities of fake news tweets from a sample of 50 tweets and 25 impressions on average assume fake news tweets activity is normally distributed and the population standard deviation is 25 impressions key words it's even highlighted in population standard deviation so we know that the population standard deviation is known and if it's known or we're going to use the formula plus or minus the critical value alpha divided by 2 times the standard error 90% confidence interval estimate for the population will be what is 90% confidence interval estimate for the population mean so they're giving us a 90% so what are we also given we can quickly do that we are given the sample size which is n of 50 you are also given on average you must also when you read the questions pay attention to the key things that they give you so yeah they didn't say mean but you need to know that the mean is the same as the average remember that that is what we did when we were doing study unit 3 the mean is the average so our mean from the sample with 100 impressions on average so it means our mean is 100 the standard deviation which is 25 remember we are given 0,90 1 minus alpha because that's what 95% confidence level is therefore our alpha is 0,10 remember that and we need to go find the critical value and our critical value z alpha divided by 2 which is z of 0,10 we know that that is z of 0,05 and if we go to the table remember on the table we set the critical value for z,05 it's 1,645 so we have the critical value we substitute the values so it's 100 plus or minus our critical value of 1,645 times our standard error which is 25 divided by the square root of 50 due to calculation 25 divided by the square root of 50 3,54 multiply that with 1,645 5,816 25 divided by 50 equals 3,5355 multiply by 1,645 equals do you get the same 5,8159 yes since it's 5,95 which will be 5, I don't want to round off quickly so I'm going to keep it up until 5 8159,5 I hope I won't remember that 5,8 9,51 I'm joking I don't know 5,81 5,81 5,95 5,95 yes, let's keep it up to there so now let's expand 100 minus 5,81595 semicolon 100 plus 5,81595 the other thing what ok so these are commas copy and paste they didn't fix the commas there so answer to whatever the number of decimals you have so on this one it has 4 decimals so it means our answer should be in 4 decimals as well so what is the first 100 minus 5,8159 9,4.184 0 9,4 comma 1,840 sorry 9,4 comma 8 no, 1,840 so option 1 let's take it slow 1,840 and on the other side 100 plus will give us 105 point 81 859 5,95 which is 8,960 if we round it off to 4 decimals because then we add we add one there ok so that will give us option number 1 that is the only activity I had but that is not the only activity we can do so let's see what time is it now because I just took it from here and that's where I set it's just a copy and paste I just copied from this document so we can go and find the follow up questions from there since we still have 30 minutes we can use this let's say such fine Africa was it not Africa with the Ka at the end this one found it so we just did that one so we can move to the next so the next one it says suppose there sample size increases to 100 so in our I removed it so we had our n is 50 now it's 100 our x is 100 our standard deviation is 50 let's just go back to the statement oh it's 25 it's 25 it's 25 and we did find our z alpha divided by 2 which for 90% I'm not going to repeat it because we did find it there it was 1,645 so calculate I'm just going to leave it to you now give it some time calculate and then we will come back and do it together do it on your own and then we'll do it together Miss Liz kindly reshare the screen apparently just joined from the other class I don't know the values Lizzie I think you know the chat features also turned off for this meeting there so if you're looking for answers on the chat we won't be able to post it I'm able to post on the chat it's fine I'll mind my own business here and watch you signed in as a guest the previous weeks were you able to post yeah I see I'm signing as a guest I don't know why it's done that to me it's unfair it's okay never mind it's okay I see my dilemma okay you must join using your my life email no no I thought I thought I've got a profile set up just for Eunice and I signed in with that account so I don't know why it launched me as a guest okay so let's do the answer then so our mean is 100 our standard our critical value it's 1.645 our standard deviation 25 square root of our N but is the answer 25 divided by the square root of 100 times 1.645 4.1125 4.1125 let's expand 100 minus 4.11 25 double colon 100 plus 4.11 25 the answer for the minus 95 8.875 8.875 for the plus 104.125 104.1 1.125 option number 4 is in it I think you might find it challenging when you need to find the critical value but always remember that when they give you a 90% confidence interval or a 95% confidence interval what are the steps that you need to follow to find that I don't think we will find another example because this is what we're going to do next week not next week on Saturday as you can see the first one was 90 this one is 95 so you just need to know how to use all those confidence so this one is 99 and you can see that even on this question you are asked some theoretical questions so for example what we just did now we increased the sample size so let's go back to our question that we did let's go back here so what they are asking in that question is if we had this one was 50 and 100 coming back to the question that they are asking now they say when only the sample size increases the confidence intervals narrow or widen when it decreases it becomes wider if it increases it becomes wider or narrower so you can see there they talk about the sample size they also yet talk about the confidence level but we didn't do any confidence level so let's look at the sample size what they are saying with the narrow it means let's say it was 10 sample size your n is 10 and let's say the confidence interval was 2 and 8 and when we increase our n and 200 then now this because when we increase the sample size what decreases is your standard error decreases so when your sample size is smaller the the standard error is is going to be bigger when the sample size is smaller your standard error will be bigger when the sample size is bigger your standard error will be smaller so this might be 1, 1,4 let's say for argument sake so what this question is saying is if your sample size increases then your confidence interval narrows so it becomes smaller if your sample size decreases it becomes wider only when that happens so we're looking for the incorrect one because yeah I'm doing my own assumption so let's look at the actual question that we answered because there we increased and decreased the sample size remember that so remember this was when the sample size was your n here was 50 and your n here is 100 so let's look at the difference between these two when the sample size is smaller the confidence level is bigger is wider you can see there it's 94 and 105 if I don't count the decimals 94 and 105 for a smaller sample size and here we have a bigger sample size and the confidence interval is 95 and 104 which is smaller which is narrower now so what it means is when the sample size so the two statements are correct when the sample size increases this becomes narrower it becomes smaller because the sample size is bigger when it's smaller it becomes wider because the sample size is smaller so now you need to also test if it's a confidence level problem so here it says if confidence level increases it becomes narrower if it decreases it becomes if it increases it becomes wider if it decreases it becomes narrower so you need to you can use one of the examples so actually we can use this example so to answer that question we can use the one that we did to answer the second part of the question we can use this because on question 3 and question 4 what they're asking you to do is change the confidence interval based on the answer that we will get here next on Saturday we will be able to answer those two questions so on Saturday we can answer these two questions because the examples that we're going to use on Saturday will be those two where we change the confidence intervals but this is just what I want to bring to your attention that you will need to know also how to interpret and do theory as well and then okay so this one as well we can do on Saturday or any other day as well because this one you just need to apply what you've learned with the confidence interval to say if it's 90 which one will be because this is less this is the mid one and this is the bigger one so which one will it be so if it's smaller confidence interval which one will it be which confidence interval will it take but we can look at that once we do the activity on Saturday I'm just looking for other options oh there we go so as you can see here the question is asking you just to calculate the upper limit remember on the equation when they ask you for the upper limit which one are you using the minus or the plus the plus use the plus side of things to answer this question but this will do on Wednesday next week because it deals with proportions okay and these are proportions okay so that's today's session then I will see you on Saturday when we do oh sorry I need to finish things properly I will see you on Saturday when we continue and do confidence interval or when we construct confidence interval for the mean when the population standard deviation is unknown with that what we have planned today is the basic concept of confidence interval we plan how to build up the confidence interval what is required in terms of the confidence level that we need to first find the confidence level by using 1 minus alpha and then use the alpha value which is also called the level of significance to go find the critical value on the table and we've learned that the critical value on the table we find it by using the probability inside the table and go outside and go find the z value so our critical value is the z value on the cumulative standardized normal distribution table we also learned how to construct a confidence interval by using the formula point estimate plus or minus the critical value times the standard error where our point estimate for today was the sample mean plus or minus the critical value z alpha divided by 2 times the population standard deviation divided by the square root of the sample size that any question, any comment Lizzie maybe just one, I don't know if you've shared it in any of your emails where can we get our hands on more question paper but I think the Unisa website only gives us two is there another place where we can download more of these question papers false? No, I don't know but you know in all the platforms Unisa platforms where it's got STA 1610 students you can just post there and say anyone with the past example posts then they can send it to you I will gladly also love to have some because I need to create more activities than examples I only have one that I downloaded from my Unisa and I'm using the old tutorial letter that I found yeah so if we can get more exam papers that will be great as well I know that some of some of the people might be selling them because they will sell you with the answers you don't have to buy the answers you can just ask for a question paper without the solutions and we can work out the solutions yourself it's easy you don't have to spend a lot of money on that and when we do revisions as well because if I have those question papers all of our revisions are based on the past exam papers so it will be very helpful that you can work through them okay if there are no questions you can stop the recording and with that