 Welcome to the session. So we're going to do a lot of activities. We have two hours. I've booked our session from six until eight, so that we can cover as much as possible. So how I want to pace the session. The first hour, let's concentrate on activities relating to study unit eight, and the second hour, let's concentrate on activities relating to study unit nine. So therefore it means if we get to seven o'clock and we still haven't finished study unit eight, do not worry, we can continue with it next week, or you can look at the questions and ask via the WhatsApp or email or on my Vanessa. So without wasting any further time, remember study unit eight is confidence intervals. I don't want to move. It's confidence interval and with confidence interval. Always remember that we find the lower limit by using the minus and the upper limit by using the plus. So if we need to find the confidence interval for the mean where the population standard deviation is known, we use X bar plus or minus the critical value, which will be alpha divided by two times the standard error, which will be sigma divided by the square root of N. And we know that this critical value times the standard error is what we call the margin of error. This is your margin of error. And, okay, so the whole thing. So this is the margin of error. And we know that only this sigma over the square root of N is what we call the standard error. So don't get confused by the two standard error. And a similar, so also the other thing that you need to be, yes. Can I? No, I can see that my slide still stuck on study unit online, seven, eight and nine. I don't know if it's as in me only. All right, at the moment, do you see? Yes, I see confidence interval activities, yes. Okay, so I was explaining the confidence interval, the plus or minus z alpha divided by two and standard deviation divided by the square root of N. And I said, this whole thing here is your margin of error. Right? But only these two, the standard deviation divided by the square root of N is called the standard error. So those are the things that you need to be aware of. The plus is your upper limit. If any question asks you to find the upper limit, you use the plus side. If they ask you to find the lower limit, you use the minus side. And this you can also do not only for, this is only for when the population standard deviation is known. So when the population standard deviation is unknown, we use the t table. So it will be plus or minus your point estimate, plus or minus t alpha divided by two and the degrees of freedom times the sample standard deviation over the square root of N. Remembering that also, t alpha divided by two and the degrees of freedom is your critical value times the standard error will give you the margin of error. And the plus upper limit, the minus lower limit. So this is for when the population standard deviation is unknown. When we do for the proportion of the standard deviation is unknown, we do for the proportion of the standard deviation. We do for the proportion. So for proportions, there are a couple of things that you also still want to remember. They will give you the sample proportion or if they don't give you the sample proportion, you have to calculate the sample proportion. And the confidence interval for the proportion, we use your P, which is your point estimate plus or minus your z alpha over two times your standard error will be the sample proportion times one minus the sample proportion divide by N. We use that sample statistics because we would only be given the sample statistics. And you also need to know how to find the critical values of the table. But there's in the nutshell, it's everything that you need to know about confidence interval except if they ask you questions or theoretical questions where you need to state some properties of confidence interval, like whether we use a parameter or we use a point, our point is to maybe use a sample statistic. Okay, so let's look at the questions. So that I can also relax my throat. I will also constantly rely on you guys to also do some activities and I will just write on the board. Otherwise by the end of the session, my voice will be gone again because my throat is still sore. Okay, so let's look at the first exercise. A random sample size of 100 results in the sample mean of 100. The population standard deviation is known to be 25. What is the value of the standard error? So the question is asking us to find the standard error. What you need to also do is go and identify what is given in the question so that you are able to calculate your standard error. So we are given the sample mean, which then it's the X bar. We are given the sample size, which is your N. You are told that the population standard deviation is known to be, so therefore it means they gave you a sigma. So in order for you to find the standard error, remember your standard error, it's given by your population standard deviation divided by the square root of N. So do the calculation and substitute into the question. Anyone? The answer is 2.5. Or let's do the substitution and then give the answer. Let's work it out. I think it's 25 divided by the square root of 100 and the answer is 2.5. And because this is two decimal, you can also just add zero. And the answer would be B. If you are lost while we doing the question in terms of how did we get 2.5 and you are not getting it, let us know so that you are not left behind, right? You want to make sure that everybody understand. So I'm just to demonstrate because then I will do this once. So because this is 25 divided by the square root of 100, I'm gonna do that manually. So I'm gonna say 25 divided by, and I'm just gonna use the square root of 100, which is equals two. And the answer would look like this. You just use the SD change button, but if you are using a sharp calculator, you will get automatically 2.5. And those with the casual calculator with a fraction, you just press the fraction and just substitute the values as you see them. The same way as you see them on the calculator. And sorry, if you make a mistake, there's a delete. You don't have to cancel. There's a delete. You use a delete on your computer. Okay, so now let's move to question number two. Exercise two. It says a random sample of 100 results in a sample mean of 100. The population standard deviation is no. To be 25, what is the margin of error for estimating a 90% of confidence interval? What is the question asking you to calculate? The margin of error, what have they given you? Your sample size N, they've given you the mean and this is the sample mean. So it's your X bar. They've given you the population standard deviation is no, therefore they have given us a sigma. And also they've told us that we need to estimate it for a 90% confidence interval. Remember that this is the same as 0.9 is equals to one minus alpha and we can find alpha by moving alpha to the other side will be equals to one minus 0.9. And alpha will be equals to 0.1. Now you can go to the table to go find your critical values or you can go and use the table that I shared with you during the session the other time which the table has already the critical values displayed. If you don't have, so we need to go find the critical value which we find it by using alpha divided by two and Z of 0.1 divided by two. It's the same as Z of 0.0, five. 0.05 and because it's 0.05 we need to go inside the table. So we need to go to, this is the old one. We need to go inside the table to look for 0.05. 0.020495 and 0.0495. Remember now because it is between those two the distance of five. So remember this is the one with the special case. The one with the special case starts with 1.6. So our critical value here will be 1.645. It's 45. 45, so the only one is the only special case we have. So since we have the critical value remember to find the margin of error we use our critical value which is alpha Z alpha divided by two times the standard error. And if you remember we did calculate the standard error before. So this is easy, we just calculate this by substituting 6.45 times 25 divided by the square root of 0. 100 or you could have searched instead of between 25 divided by 100 you calculated it the previous right we did find that it's 2.5 you can just substitute the A by 2.5 but calculate this and get the answer. What is the answer? 4.1125. 2.135. 1, 2, 5 which is three. Let's go to exercise three. In exercise three a random sample of 100 has the random sample size of 100 results in a sample mean of 100. The population standard deviation is 15 is known to be 15. What is the margin of error for estimating a 99 confidence interval? A 99 confidence interval it's alpha of 0.01. So what will be the Z of alpha over two Z of 0.01 divided by two that will be 0.005 right? Yes. It's double two zero. So when you get for those who have the case and you get the answer with the make to the power 10 to the power of a negative three do not panic. You just say plus one to add one and always remember that you have added one so that you can remove one when you read the answer. So you will get the answer that looks like 1.005. So you just need to remove this one and write it as 0.005. Now you can do the same or you can use the table that you have. We come here we look for 0.05. We're going to use the last one which is 2.58. I know this one. I know them because for you for those who don't know the answers you just need to go back and refer to the table just make even when you go write the exam make sure that you have that table somewhere written as your reference table for critical values. So the answer here will be 2.58 because that's the critical value for 99%. So calculate the standard error. Give me the numbers. 2.58. 2.8. Multiply the standard deviation is 25. I'm sorry 15. 15 divided by the square root of 100. And the answer? Answer is 3.87. 3.87. Looks like all my answers are B, B, B, B, B, B. There's also an E there. It's the same. Do you see that Lizzie? Yes, probably one of them should say 3.78. Okay, I'll fix this. I'll fix it after class. I'll update the notes as well. So it's either B, U, O, E. I'll change this one to be the same. Okay, so the following intervals are 90%, 95%, 99% confidence intervals. For the population mean in no particular order, these are the confidence intervals. The question match the confidence intervals with the appropriate confidence level. So now, because we have the confidence levels here, what we can do is we can use any of the previous questions that we know about confidence intervals. So let's see if we can use this one. So I'm going to use this question. The same question as we have here. It's at 99, and we know at 90, at 90 the critical value is 1,645, right? Yes. And we're going to use the same and calculate the critical value at 95%. At 95%, the critical value is 1,960. So now for me to demonstrate how you're going to answer that question that side, we need to calculate the confidence intervals for this question. So let's practice by finding the confidence interval for the 90%, for the 90%. Finding the confidence interval for 90%. So this 99 is 2.58. So we're going to use the same question. I'm just going to remove this. It's going to take time for us to get it there. So let's start with 90%. So the confidence interval will be sigma plus or minus the critical value of alpha divided by 2 times the standard. I'm not going to repeat the equation for all of them. So we're just going to do forward. So the mean of this is 100 plus or minus. The standard error is 1,645 times the standard deviation is 15 over the square root of 100. So now I want you to give me the confidence interval for that. So just continue and calculating that. And I'm going to do the same for 95. So you continue and answer the question so that we can give me those answers there. The mean plus or minus. Oh, I said I'm not going to repeat this. Let's just substitute the values because we use the same formula. So we can save space. It's just 100 plus or minus for 95. It's 1.96 times 15 over the square root of 100. And then you're just going to give me the upper limit and the lower limit for it. And then we go to the last one, which is 99. 99% 100 plus or minus 2.58 times. I can just use the 3.8. But let me just do this. 15 divided by that. And then you're going to give me those values. Okay. So now we're done with this section because I need to use the space I gave. Let me know when you have all the... I'm ready, Lizzie. Okay, give me. I have, Lizzie. Yes. So for 90%, the lower limit is 97,535. Okay. And the upper limit is 102,4675. Okay. 95 is going to be 97,06. And the upper limit is going to be 102,94. And 99, we are at 96,13. Yes. And the upper limit is 103,87. So now we've got all three of them. So let's plug them in order. So the biggest number is 97, right? The smallest number is 96. So we can start with 96,13. The other number is 97,06. And then we have 97. I'm just also going to keep only three decimal because I'm going to run out of space. Two decimals, 535. The other number, let's look. Which one is the smallest on this? It's one, four, six, or four, seven. Yeah. 102.47. And then 102.94. 94. And then the last one is 103. Yes. 0.87. 87. So 93, 96, sorry, 96.13 corresponds with 103. So this was 90, 99. 97 is 104.92. This is 95. Sorry. 95. And then we have those ones. Those ones which is 90 percent. Now let's analyze this. So it says, if I look at this, the smallest lower limit corresponds with the highest confidence interval. Can you see that? The largest lower limit, upper limit corresponds with the the highest upper limit. Can you see that? So therefore it means I can always look at the lowest and take it from there and say, let's use only the lower limit. The higher you go, the lower your confidence interval is, right? Because this is the smallest. This is the highest. Smallest. This is small. This is high, right? So the higher your, the smaller your lower limit, the higher your confidence. The higher your lower limit, the lower your confidence limit. So it works in opposite directions. So this gives me a guide to say, if I come here, if I already know those things, I know what my lower limits are. I can rearrange this in terms of the lowest, the mid one, and the highest. So if I look at this values, which one is the lowest? Is this one, right? Because 194 will be the first one that I will start with. 0.82. And then which one will come second? Is this one? Because I'm increasing as I go. 196.06. And this is 196.70. So by just looking at the lower limit, I can say this is 90. This is, sorry, this is 99. This is 95. And this is 90. Because that's what we had there. This is 99, 95, and 90. So it's the same thing. So then it means my A corresponds with 99%. I'm doing it all wrong. My A, which is 19.06 corresponds with 95%. My B, which is the smallest lower limit corresponds with 99%. And C corresponds with 90%. Then I can come here and choose the right one because it's ABC. So let's see. A is 95. So those two are correct. B is 99. That one says 90. So therefore I can also just stop right there. Because it says match the estimate with the corresponding confidence level. And that's how easy it is. So already from this, you should be able to do this on your own. You don't require any calculations because you use the previous knowledge you have. So this is the previous knowledge I've given you. Use that. And this is the previous knowledge because we calculated each one of them and we can safely allocate the confidence levels based on the lower limit. So now this is your question. The confidence interval estimates below for the population mean calculated at 90%, 95%, and 99%, not necessarily in order as they appear here. Now you need to determine which one of these three orders estimates a 95% confidence interval. So now your 95% confidence interval should have the highest or should have the mid one or should have the smallest. So based on this previous information, you know, can you identify that? B, B is the middle one, right? Because 95% is the one that is in between the two. So B will be the correct one. That's easy, right? If you didn't even have to do some calculations of some sort or scratch your edge because now we know. The confidence intervals, how they correspond with the estimates. The confidence levels, how they correspond with the confidence estimates. The Department of Higher Education alias finding from a survey of 150 students are just the average time of 140 minutes from home to FTE college at one of the FTE of interest. Assume the population standard deviation of students travel time is 72 minutes. Construct a 95% confidence estimate for the population mean. Now you need to pay attention to the questions so that you are able to identify what you are given so that you can use the right things. So let's look at what we're given. We are given N, which is our number of students, which are 150. What else are we given? The mean, which is the average, 114. The mean, which will be 114. What else are we given? And the standard deviation of 72. What standard deviation is given? Our population standard deviation. The population standard deviation, so that is your sigma. So that helps because if they didn't give you a population standard deviation, they would have given you a sample standard deviation. So that is very important to always remember that, which is 72 and we need to construct at 95% confidence interval. What is alpha over 2 at 95% confidence interval? You should know the critical value at 95% confidence interval. It's always 1,96. So now construct a confidence interval by substituting all the values. Okay. The lower limit. Yes. 102, 4776. Let's keep it to 2 decimal. It will be 48. And the upper limit will be 125, 52. Those who don't know how they got to the lower limit and the upper limit, the lower limit is the minus. So you say 114 minus 1,96 times into bracket 72 divided by the square root of 100. Let me just demonstrate that once so that we don't leave everyone and assume that everyone knows what we are talking about. So since I've got a case, I like to always take advantage of my case, your calculator. So 114 minus. So I need to always start with the minus side 1.96 open bracket fraction 72 divided by the square root of 110. And go with an arrow out and out again and close bracket and press the equal sign or you can press the answer and the answer is 102.48, which is what she got. Then I can use my arrow to go out, out, out, out, out, out, out. And delete and add the plus sign and say equal and the answer is 125.52. As you can see that will be option B. Are there any questions? I haven't even checked the chat because I'm not even paying attention to the chat. Let's see. Are there any questions? Let me not just continue and carrying on and ignoring everyone. There are no questions. My name is not a question, actually it's just a comment. And I write 76 and I got 126 then I realized that is 72. So just for everyone to be careful with the numbers when they input on the calculator. Yes, you are right. Sometimes it's also good to double wait, especially when you do your assignment and your exam. Just repeat the same things that you did. So once you got the answer, before you move into the next one, just recalculate it. Do that double check to make sure that you have all the information correctly written. And also just browse to go through the statements to make sure that the numbers that you are using are the correct numbers as well. That's a good comment to that one. Yes, I support that hundred percent. Okay, let's go to question two. Question seven. A labor survey based on the sample of 50 households in a urban town found the average salary of urban domestic workers is said to be 3,500 with a standard deviation of a thousand construct a 99% confidence interval. So what are we given? We are given the sample size. That money is easy to find. What else are we given? I just want to see if you are able to identify the things that you are given. There's no right or wrong answer. I will tell you if you're not. The average is the mean. The average is the mean. Yes. You are given S. How did you identify that you are given S? If it was, it might have said population standard deviation. Yes. But from this statement, how would if they didn't mention that how would you know that this is an S. And not a population standard deviation. Okay, how would you this is a straightforward statement that you can also make that assumptions that this is a standard deviation that comes from a sample because the whole sentence is one. Right. They say there is a sample of 50 and from that 50 they've got a mean of 350 and the standard deviation of 100. Even though they didn't tell you that this is the sample mean or the population, but because that the same statistics comes from the same sample. You can also make that decision to say this is a sample statistics. Sometimes you will notice when you read the question, you will see that they will say a population they selected this from the population with a standard deviation of this much. So they didn't tell you that this is the population, but something that they hinted in the statement will give you that sense that this is a population standard deviation. The same way as when we're reading this question, you should understand that the mean standard deviation that they gave you comes from the sample. So let's construct a confidence interval because we are given S therefore it means we're going to use T right. So we're going to find the critical value by using T alpha over two. And we're going to use the degrees of freedom so the confidence interval will be the point estimate plus or minus T of alpha divided by two and the degrees of freedom and you still remember what the degrees of freedom is. It's n minus one. Times the standard. So our T alpha over two will be the same as T zero comma zero one divided by two and our end they gave 50. So it will be 50 minus one 50 minus one so T of zero comma zero zero five and 49 so you need to go to the T table. So we have to go to remember also when you come to the T table the top patch. It's not we use the values next to the table where it says apartheid area and we're going to use the last value and we go to 48 49 and we so it's two comma 68 zero zero that is two comma six eight zero zero then just substitute our mean is three thousand five hundred plus or minus our critical value of two comma six eight zero zero times the standard deviation of a thousand divided by the square root of fifty thirty thousand five hundred. I'm just passing time while you're doing your calculations. Okay the lower limits. Give me a second. We cannot always assume that everyone knows what we're doing. So the lower limit is three one two zero comma nine nine and the upper limit three eight seven nine comma zero one. So the answer is option. Are we good? Yes we're good. See that a T distribution with a twenty seven degrees of freedom which one of the following statement is correct. So since we're talking about a T distribution remember we're still talking about confidence intervals right. A T distribution it means we need to find T alpha over two and the degrees of freedom. And they told us that the degrees of freedom is twenty seven so it makes our lives easier. But that doesn't end the right day. So if our degrees of freedom is twenty seven therefore it means our N is equals to twenty eight because the degrees of freedom is N minus one. So we've got a little bit of other information. Number eight we're looking for the incorrect statement. That's the other thing that the statement is asking you. So it means we need to evaluate each and every statement to find the incorrect one. So number one says a T value associated with that. Can I ask some questions? Why are you saying N is twenty eight? Because they gave you the degrees of freedom of twenty seven and the degrees of freedom is N minus one. Oh then you're going to say twenty seven plus one. So degrees of freedom is twenty seven N is minus one and we just take minus one to the other side. N is equals to twenty seven plus one which is equals to N is equals to twenty eight. Because the reason why I'm mentioning that it's because somewhere sometime they will be asking you about the things that they didn't even give you much information. Okay so we will have to get to that but I'm just mentioning that N is that in case one time they ask you. Okay so we need to find the T value associated. I'm also like in between in terms of this because they talk about lower tail area and all that but anyway it's fine. The T value associated with the lower tail probability which then confuses the heck out of me as well because we're talking about critical values here not T value. But since we are in the confidence interval section we can assume we just going to make our assumptions here. We're going to make our assumptions here that they're referring to critical value. So let's see the T value associated with the lower tail probability of zero comma one is minus two four seven three. So let's see if that is the truth. The lower side remember the lower side is the one with the minus and the upper side. So this is the lower and this is the upper so it means our critical value will be the same as minus the value we find on the table so let's go and see. So here we have T of zero comma zero one divided by two and the degrees of freedom of twenty seven. So we need to go find T of alpha divided by two is the same as T of zero comma zero zero five and twenty seven. It's the last column and twenty seven. Am I selecting the right value? Yes and twenty seven is two comma seven seven comma seven seven and there is a seven zero seven which is they are three decimal so it will be two comma seven seven one. Because it's in the lower side we will have a minus as you can see there is two comma four seven three so that is the incorrect one. This one says at ninety five percent the area will fall between those two. So let's see at ninety five percent ninety five percent will be ninety five percent will be T of alpha of zero comma zero five divided by two which is T of zero comma two five zero two five and the degrees of freedom of twenty seven. Sorry Izzy. No I didn't get the first one. Why didn't you get the first one? So why are you saying two comma seven seven one? Because there the probability is zero comma zero one or are you saying because they say it's the lower tail probability then we only go we don't use divide by two. Since they say it's the lower tail because if we don't divide by two then we go to zero comma zero one. Yes and it was that two comma four seven three. Then it becomes two comma three seven then it becomes the correct one right. Yes. So let's let's use that as the information given because they say the lower tail but also if this is based on confidence intervals as well. We will have to divide that by two so maybe probably this is not confidence interval is just wanting you to find the critical value based on the lower side and the lower side it's only looking at one side not both of them. So when it looks for both sides then we need to find divide by two. So let's let's test all of them. So if we only looking at that so therefore the answer here will be this will be correct. So for two distribution that falls between the two values. Then we divide by two we get zero point zero two five and decrease of freedom zero comma zero two five. And twenty seven and that will be this value here which is two comma zero five two. But now if you look here it falls between two comma seven. Let's just double check the other one T value associated with the lower limit probability tail of zero comma zero two five. So if we just go to T of zero comma zero two five zero comma zero two five. It is two point zero five one. Or five two. That will be correct. If we just use that. If we go to a ninety percent and it says it's between a ninety percent will be zero point one zero divided by two. I just want to understand how they do this the between and that. So let's see this will be zero comma zero five. So if I go to zero comma zero five zero comma zero five. It's one comma seven zero three. Which is correct. Yes. And if I go to eighty eighty will be zero comma two zero divided by two which is zero comma one zero. And twenty zero comma one zero. And twenty seven which is one comma three one three seven. Which is correct. So the only one that is incorrect is this one because zero comma zero two five. What did we get? It was zero comma zero two five. It was two comma zero five one. Okay. So when they say the lower tail they don't want you to divide by two. Okay. Now I see. So they apply what we do in hypothesis testing. Hmm. So probably this should have been part of the hypothesis testing questions as well. So for the lower tail area you just use T alpha and the degrees of freedom. And you will find that the correct answer is that so for the between you will have to divide by two. So that you can you can go and find the critical values. Okay. I see that it's nineteen oh three. We should be with the hypothesis testing. But let's look at the two last questions. We just going to look at the two last questions because I need to do the proportion. The proportion question and then the last one you can do on your own. The last two. A simple random sample size of forty six from a normal population. Results in a sample mean of three hundred and the standard deviation of fifty. Construct a ninety percent confidence interval estimate of the population mean and choose the correct answer from the list of options. This is still the confidence intervals for the sample mean. What are we given? N. N. What else? X bar. The sample mean. Standard deviation. The sample standard deviation which is your S. And you can see that on this one they specifically told you that sample means sample standard deviation so that it's explicitly stated on the other one they didn't state that right. And they say you must construct this at. Ninety percent so it means we're still going to use T. Alpha divided by two and the degrees of freedom. Our T at ninety percent is zero comma one zero divided by two. And our degrees of freedom is N minus one which is forty six minus one. So our T. Of zero comma zero. Five. And the degrees of freedom of forty five. Forty five. We go to the table. We need to go to forty five. It's on here. But we need to go find zero comma zero five which is the third column one two three. Forty five. It's one comma six seven four nine. One comma six seven. Four nine. Am I writing it correctly? Six seven nine four. Four nine. So then go and calculate your X bar plus or minus. Your T. Alpha divided by two eighty degrees of freedom. S divided by the square root of N. Our X bar is three hundred plus or minus. Critical value. One comma six seven nine four. Standard deviation of fifty divided by the square root of forty six. You have the lower limit. Yes. I have the upper limit. You have the upper limit then you hold someone has the lower limit. So what is the lower limit? Two eight seven. Coma six. How many decimals? Two. Six two. So we only have one. The upper limit. Three twelve. Three one two. One three eight one. Okay. So the answer is three. Let's look at the last one and then we move to hypothesis. In the sample of hundred and fifty schools from twenty municipality. Hundred and fourteen reported the decline in the number of learners absent. Construct a ninety percent confidence interval. She made for the population proportion of schools that reported the decline in the number of learners absent. So what are we given? We given N. What else are we given? The number of schools which are our X. So it means we are able to calculate P. Because P which is our sample proportion is X over N. It's one one four divide by hundred and fifty. And what is our N? Our P. Zero comma seven six. Zero comma seven six. So then calculate P plus or minus. Z of alpha divide by two times P times one minus P divide. We working with Z. Remember ninety percent is one of those special. Special cases is one comma the critical value is one comma six four five. Right. Because it's a ninety percent. And ninety percent which is zero comma four five zero comma zero five. So this will be zero comma seven six zero comma seven six plus or minus one comma six four five times. The square root of zero comma seven six times one minus zero comma. Seven six divide by N. I'm going to do this with you. On my calculator. Zero point. Seven six minus. One point six four five. Again bracket. Square root. Production. One seven six times one. Nine times point seven six. Close brackets. Down. And right. A little. Again. A little. This is sorry isn't N one fifty. Oh yes N is one fifty. You are right. You see pay attention to. Because now I'm losing a hundred. To that. Did it. Fifty. And zero. That's right. And equal. Zero comma seven zero three. Which will be that the first one. And if I change the. Minus two plus. Zero comma eight one seven. Zero comma eight one seven. Which. Number eight. Number eight is correct. So that concludes. Our confidence intervals you can do the last. That are there on your own. So this is asking you to just do the apartment. You know where to do how to do the apartment is. The plus side. So they have given you the proportion. So there is your P. And then you can go and calculate this. There is your N. And this is at ninety five. Z of alpha divided by two. It's one comma nine six. You just. To be plus or minus. Z of alpha divided by two. Times the square root of P. Times one minus P. Divide by N. I cannot give you all the answers of the activities. Then I'm not teaching you anything. So you can go and do that on your own. So in the next last. Fifty minutes. Or even less. In the sample of the six. The sample mean. Is eighty four. It's also known that the population standard deviation. Is sixteen now. With hypothesis testing. You always need to remember that. You need to remember. To know all the six steps of hypothesis testing. Stating the null hypothesis. State what you are given. In terms of alpha. And N. And then. And then. Other information like for example whether you given the population standard deviation or not. And then state whether are you doing a T test or a Z test. And then. Find the critical value. Based on your. Identified hypothesis testing you need to be able to determine. Whether you're doing a one tail test or a two tail test. But. And that will determine how your critical views. You will find them and and find the region of rejection whether it will be one sided. Or two sites for region of rejection and also it will help you also when you go find the PV. You also need to be able to calculate whatever test statistics you are calculating using the. Sampling distribution formula for Z or T. T value. And then you need to be able to make a decision and remember. Your decision you can make use use of your critical value and your test statistic. And make your decision where whether it falls within the region of rejection. Or you can make a decision by using the P value. Which means you need to use your test statistic especially for Z only P value for Z. Where you take your Z value and go to the table to look for the probability on the table. If your probability or your P value is less than your alpha. You reject the null hypothesis and those are the decisions you can make. And hypothesis testing so now here they give you the statement. And they give you the sample size of N of 36. Your sample mean of 84. Your population standard deviation. This is very important to identify these things as early as possible. And they say you are required to use the information to test the hypothesis. So they have given you your first statement. Your null hypothesis and your alternative hypothesis. You need to make sure that you understand what kind of information you are given here. You can ignore reading that because it doesn't give you much remember with your null hypothesis. This value here they can always use an equal sign. They don't have to put the greater than or equal. It can just be equal for the rest of the hypothesis. So the one that is very important is this one in the alternative. The side placed on your alternative tells you whether are you doing a one. Side it. Side the test and this will guide you whether you're going to waste your original rejection. So it means. When you go into the decision you're going to create a region of rejection in the lower boundaries. So that is where it will be. It says it's less that so it will be in the lower boundaries. Anything that falls in the red shaded area. We're going to reject. The null hypothesis and anything that falls in the white area. We do not reject the null hypothesis. So which one of the following statement. About the test is incorrect. So it means you need to know all those six steps of hypothesis. Let's start with the first one. I've already gave you some of the answers. Number one, is it correct? Oh, incorrect. Based on the information given, we are given the population standard deviation. Remember when the population standard deviation is known. We use Z when it is unknown. We use T for proportions. We always use Z. So based on the other information that I have given you right now. Number one, is it incorrect or correct? Correct. It is correct because it's a lower tail area. Z test because the population standard deviation is known. It's a Z test. So this is correct. Number two, the test statistics is 1.5 zero. So here they're asking you to calculate. So go and calculate. Your sample mean minus. What I forget to put there, it's Z state. The sample mean minus the population mean divided by your standard deviation over the square root of N, which is your standard error. So let's substitute to the values. Your sample mean is 84. 84. I'm sorry. Your sample mean is 84. Your population means it's 80. It's given in the statement because it says your population mean is less than 80 or greater than equals to 80. Divide the byte. You're sorry. Can I go drink some water? It can continue to answer this and then I'll be back with you. Divide by 34. And number two is correct. It's 1.5. Divide by the square root of. What is it? Yeah, it is correct. Yes. So our population standard deviation is 16. It's not writing 16. Divide by the square root of N. And it's 36. So what is the answer? 1.5 zero. 1.5 zero. So that is correct. And number three says at 10%. The level of significance. The rule is to reject the null hypothesis. If the P value is less than zero comma one. So I also picked at number four. It says the P value is the sum because they are not asking about the critical value. So we can go and find the P value. And we're going to find the P value by using the Z test. So I'm sorry. At least I'm dying while I'm doing what I like. I like doing. So you guys, if I collapse and you don't hear me, know that I enjoyed being in the session with you guys. Because my throat is killing me right now. You are now freaking us. Come on. Yeah. So in my memory, you just come and you say, oh no. Yeah. She was doing what she likes. And then we'll get this thing from chapter one and so. Yeah. Okay. So we need to go to the Z table. We need to. We need to. We only pass the study unit one to nine. Yeah. Okay. So we're going to look for the Z value one comma five on the left and zero, zero, zero at the top, right? You still remember that? Let's go to the Z table. Yes. So on the Z because the answer there is positive. So we go to the positive side. And we look for one comma five and zero, zero, zero at the top where they both meet. So this is the answer. Zero comma nine, three, three, two. So our P value is zero comma nine, three, three, two. So we can answer the question. So now already I can see that the P value is incorrect, right? Because we did go and find the P value. It's that. So this would have been the incorrect one, but I'm just going to continue and do number three. Number three says the level of significance. It's 10%. So it means our alpha is zero comma one, zero. And because we're doing a one-tail test or a one-sided test. So we don't have to divide alpha by two when or we don't have to the P value. We don't have to add or subtract or edit by two or something like that or multiply by two. Because it's a one-sided test. The decision for the rule says if the P value is less than your alpha, you reject the null hypothesis. So let's see. Our P value is zero comma nine, three. Our alpha is zero comma three. Our alpha is zero comma one. I don't know where I'm getting zero comma one there. So our P value is greater than our alpha hence therefore we do not reject the null hypothesis. So let's read number three. At 10% level of significance, the rule is to reject the null hypothesis if the P value is less than or equals to zero comma one. So that is true because that's what the rule says. If the P value is less than that, we're going to reject the null hypothesis. So that statement is correct. The last statement. At 10% level of significance, the null value is rejected. The null value is not rejected. Because we did go and find the P value is then there are two statements that are not rejected. The P value is then there are two statements that are incorrect with this statement. The statement has two incorrect statements because we do not reject the null hypothesis because at 10% level of significance, at 10% level of significance, the null hypothesis is rejected, but it's not rejected. So this is also not true. Unless the statement here was supposed to say it's not rejected. So I don't know which one. So those are the kind of things. If you find questions like this in the assignment or your example, please let us send an email to your lecturer as soon as you notice there's an error in the statement. And this comes from last year's assignment. I think last year's assignment. So there might be an error somewhere with this. Or they would have given them a free mark or an error for this question. Okay, so moving on. We don't waste more time. A laboratory tested the random sample of dating chicken eggs and found that the mean amount of cholesterol per egg is 285 milligram. And the standard deviation is 20 milligram. If the population or the null hypothesis states that the population mean is equals to 280 and it's tested against the null, the alternative hypothesis that states that the mean is not equals to 230. At 5% level of significance with the assumption that the cholesterol of the chicken egg is normally distributed and suppose that the test statistic is z is equals to 1,37. What is the p-value? So after they have given you all the statements, so many that you read all of this, the only two things that are important in this statement is your alternative hypothesis. And the test statistic. So they're telling you that your z state is equals to 1,37. They also told you that this is a two-tailed test, right? They're asking you to find the p-value. Since we're finding the p-value, remember those things that we said when we were doing the concept recap on Sunday. The previous we said if your test statistic is positive and you're doing a two-tailed test, you will need to subtract the value you find on the table from 1. And then you need to multiply. So in order for us to find the p-value, for a two-tailed test where it is positive, we're going to say 2 times 1 minus the table, the table value. Because of the value is, the z value is positive. Because that value is positive. So we need to go to the t, not the t, the z table. So we go to the table we're looking for. You still remember what we're looking for? 1,37. We're looking for comma 3. Let's remove the old ones. 1,37 where they both meet. So it is 0,9147. So we say 2 times 1 minus 0 comma, forgot. 1,7. 0,9. You also forgot. Let's go back. 0,9147. 9147. There we go. So that will be 2 times whatever the value. Just give me the answer. 0,17. Are you giving me 2 times the answer? You are already giving me 0 comma. 1,706. 1,706 which is option. If you go to the table, you would have found this and sometime you would have said that will be the correct answer or you would have said 1 minus that and it will give you one of these answers which won't be correct. So you just need to make sure that you pay attention to the information given. So moving on. How many tissues should the Kimberley-Clack Cooperation Package of Kleenex contain? Researchers determined that 60 tissues is the mean number of tissues used during the cold winter. Suppose a random sample of 100 Kleenex users yielded the following data on the number of tissues used. During the cold, the sample mean is 52. The standard deviation, the sample standard deviation S is 22. Suppose the alternative you wanted is to test that the mean is less than 60. So it's a one-tail test and it's on the lower side. State the correct rejection region for alpha of 0,5. So the most important thing is this. State the correct rejection region. Well, what they want is going to find the critical value. What test are we doing? We're doing a one-tail test. It's very important because then if it was a two-tail test, we would divide alpha by two because we're doing a one-take. It's just alpha and the degrees of freedom. So our alpha is T of... That will be T of... Our alpha, they told us is 0,05. And the degrees of freedom, we use N. N is 100. That's what they gave us. So it's 100 minus one. So T of 0,05 and 99. So we need to go to... We need to go to the table. T table. We're looking for 99 to be somewhere right at the end and 0,05. So if I forget 0,05, it's on 1,2,3 column number 3. 1,2,3. 1,66,04. You need to write it down. So if I forget, I'm going to blame it on my illness. If my memory forgets this number. 1,66,04, you have written it down. You will just let me know. 1,66,04. Pay attention to this. It will be on the negative side. So your original rejection will be somewhere in the negative side. So therefore it means this one have a minus. So the question is, state the correct region of rejection. So if your test statistics falls in. So if this is your T value or your critical value right there. If your test statistic falls below... If it falls below the region of rejection, then you're going to reject your null hypothesis. So we're going to say we reject the null hypothesis. If your test statistic falls below this. So if your test statistics falls this side. Remember, we reject the null hypothesis. So the only answer that is correct is option 2. If it was a two-take and the critical values were this, we would have selected either one of them. Or no, because this one said Z and that says T. We didn't have selected that. But this is if this, the one-take was greater than, we would have selected that one because it would have been on the right. And if it was for two-take, let's assume that this was T. You will have it, the answer will look like this. But the only correct answer is option 2. Moving on. Most schools reported a decline in the number of absences during the Department of Education, LENA Transport Program. In a sample of 200 schools, 76% reported a decline in the number of LENAS. The district manager is adamant that the true population proportion of schools that reported a decline is different from 78% as reported previously. To test this, formulate the hypothesis testing for the true population proportion at 5% level of significance. So now, because they said to formulate a hypothesis testing, the question here is which one is incorrect? Before I can even look at all of them, what is it that we are given? We are given N, which is the number of schools. We are told that in those, your sample proportion, which is P, is 76%, shows the decline. And we are also told that the population proportion is 78%, which is different from... reported a decline is different from that. So the district manager is adamant that the true population proportion of schools that reported a decline is different from 78%. So therefore it means we are doing A, is it the one day or two days? It's very important to know these things. Because they will determine how you go find the PV. So answer that question. And how you're going to make a decision. So because they talk about different, if they would have said it's less than or it's greater than, it would have been a one take. Because they talk about different, therefore it means it is not equal, not different, different, not different. So since our alternative will state that the population proportion, will state that the population proportion is not different to 78%, then we can assume that. Since I've looked at, I browse through the questions and I can see that we have the test statistics, the PV and some decisions. So I don't have to worry about the rest of the other things. So I need to go and calculate the Z-stat for the proportion, which is P minus population proportion divided by the square root of population proportion times one minus population proportion divided by N. Our P is 0.76 minus our population proportion 0.78 divided by the square root of population proportion 0.78 times one minus 0.78 divided by our N is 200. Calculate it. Negative 0.68. I don't have to calculate it, right? You did find it. Negative 0.68. So which means number A is correct. So number A is correct. And because we're doing a two-tailed test, we need to go and find our P-value. So our P-value will be the same as our table value plus our table value. Or we can say two times our table value. Remember that. It's a two-tailed test. So you can add the table value, the value you find on the table. Or you can say two times the value you find on the table. And you can only do that because the answer here is negative. So we go to the Z-table. We look for Z of 0, minus negative 0.6. Negative 0.68. So we look for negative 0.6 here. And then we'll look for 8 is the last second column. 0.8, which is the last second column. And that is the answer. But it's 0.2482. So it means you can say 0.2482. 0.2482. Thank you. Or you can say two times 0.2482. It should give you one and the same answer. We can clearly see that this is not correct. But we need to go to the next question. So the next question says the alternative hypothesis is that the true proportion is not equals to 0.08. That's what I just said there. And number D says the null hypothesis is not rejected. So let's test that scenario. So is the P value less than our alpha of 0, did you find the answer? What is the answer here? It would be 4. 0.4. 0.4964. So this one, we know that the rule says P value. If it's less than alpha, we reject the null hypothesis. So let's see 0.4964 less than alpha. Where am I putting less than? I still don't know. 0.05. So let's see. It is greater than, right? Because it's 0.4, which is 0.5, which is bigger than that. So we do not reject. We do not reject the null hypothesis. So this statement is correct. And we conclude that the true proportion of children that they decline in the number is not significantly different. From 0.78. So that also is correct. So the only answer that is incorrect is B. So let's see in the last 10 minutes or less. I don't know how many more questions we have, but they almost like repeating. So I don't have to do all of them. Suppose the calculated test statistic for a one-sided lower tail test is this. It's minus 2.74. Suppose further that the population standard deviation is known, which makes life easier. Therefore, it means we're using z steps because they told us that. And the z step they told us is minus 2.74. We are also told that we're doing a one-sided test. So if we need to go find the p-value, your p-value will just be the value you find on the table. And because it's in the negative, it's still fine because we're doing a lower tail test. So what is the p-value? We need to go to the table, to the z table. Look for minus 2.7 and 4 at the top. Minus 2.7 and 4 at the top. So we need to go to the negative side table. Come on. And look for minus 2.7, minus 2.7 and 4 at the top where they both meet. That is the answer you are looking for. 0,0031. That's how straightforward it is. If it was positive, it doesn't really matter. You would still go and find the p-value there because it says it is a lower tail. If it was upper tail, then you will have to subtract from 1. This one is a two-sided test and it's positive. We did a similar test. But let's see if you did listen to me. So we're fine because the population standard deviation is known. We're using a z-stage. So we are told that it is positive 282. Let's see. You still remember how to find a two-sided test for a positive value? The z-stage is positive. So you should know by now that the p-value is 2 times and because it's a two-tail test, right? If it was a one-tail test, we don't have to do 2 times. 2 times 1 minus the table value. So we need to go to 2.82. 2.82 on the positive side. 2.8012. That will be the answer. Let's double-check 2. That is 0.9976. 2 times 1 minus 0.9976. And that 2 times 1 minus 0.9976. That's one. The answer will be 0.0048, which is option 1. So this one you need to find the critical value. It says, consider two-sided hypothesis testing with 5% level of significance and the degrees of freedom of 25. If the population standard deviation is unknown, then it means they've given you s. Then it means we're going to use t, a t-test or t-critical value at some point. So because we're looking for the critical value, so we need to find alpha divided by 2 because also they told us that it is a two-sided test. So you need to be very careful when you read the statement. If it was a one-sided test, we wouldn't find that alpha divided by 2 will find alpha. Because it's a two-sided test, we do alpha divided by 2. And we do degrees of freedom, which is n minus 1. So t alpha, our alpha value is 0.05. We need to divide that by 2. And our n, they told us we have a degrees of freedom. So we don't even have to calculate n minus 1. Our degrees of freedom is 25. So this will be t of 0.025 and 25. So you go to the t table and look for 25 and 0.025. When they both meet, that is the answer. Happiness? Happiness. Yeah, but I didn't write the number. 2 comma 0.595. Did I write the number right? 2 comma 5. 2 comma 0.5. 2 comma 0.595, which is the same. If we round it up to 2 decimal, it will be 2 comma 0.6. So now it's a two-sided test. So we do plus or minus, which is option number 8. Left with four minutes. Let's see in the next four minutes if we can squeeze the other one. I'm not going to do this. You can do it on your own. Similar to what we just did, we need to find the critical value. So yeah, we find the critical value. The difference is this one was two-sided test. This one is one-sided test. That is your homework. I can't give you all the answers for all the activities, right? It means I'm not teaching you anything if I just do everything myself. So this you can take and do it as your homework. If you have challenges, you can let me know. And I think this is the second last, but you can also do 21 on your own time. When you get to this, the information is uploaded on radio on Teams and also on my UNISA as well. You can find the notes. It's activities for study unit 8 and 9. You can also do this. This is almost repetition of everything that we just did. Pay attention that you're doing a p-value. And this is sample proportion. So you must just remember that to calculate the z test and then use the z test to go find the p-value. Because it's less than the value you find on the table will be the value you are looking for. It should be easy. So you just use the z set of p-population proportion divided by the square root of population proportion times one minus population proportion divided by n. And use the z-value that you find here to go find the p-value. The value you find on the table because it's less than is the p-value that you are looking for. This is the hypothesis testing. You just do the same. It almost looks like the same question that we did. But this one is using, we use 76. This one uses 80. You should be able to answer the same question whether you give it 76 or 80 or any of those. The last one, you can also calculate this. They just want you to calculate the p-value. It is a greater than. So therefore it means the p-value you're going to find, you're going to subtract it. So you're going to calculate. So they have given you a z-value. And your z-value is positive, right? You're just going to take one minus the table value. And that will be your p-value. Regardless of whether this value is positive or negative, it doesn't really matter. So the value you're going to find on the table, you subtract it from one, because of the size, it says greater than. Remember the table, the z-table contains the values of less than values. In that case, that's it for today. It's eight o'clock. The only other thing that I didn't touch was this one, right? Because they've got so much context to it, like num letters. So I'm lazy to read the whole thing. But all what they want from here is, are you able to use this information given? Pay attention. The standard deviation there comes from the sample. So you are given a sample standard deviation. Therefore, it means you will use t. So yeah, you will use the t-value, t-statistic, and t-alpha, depending on the question. So yeah, it says less than. So it means you're just going to find the critical value by using t-alpha and a degrees of freedom. Now, yeah, because it says less than, therefore it means it will be on one side. And your rejection area will be there. This will be your critical value, t-alpha and degrees of freedom. If your test statistics falls here, you reject the null hypothesis. So you make decisions based on that. So that will give you information. You are given enough information to make that. So number three, you cannot choose that, but you're looking for the correct statement. So that is the only one that I didn't look at in the paper. That is one of those. You can do this on your own and also if you want, we can have that discussion on WhatsApp or you can post how you answered it on WhatsApp and we can help you in answering the question. And that concludes today's session. I will see you on Sunday. There is nothing for me to recap because we were just doing activities. One thing that I just want to bring to attention when you do this, you need to make sure that you pay attention to the information given, especially when you're doing the hypothesis testing. Read the question. Check whether are you doing a one-tail or a two-tail. That makes a huge difference. Read the question carefully so that you are able to identify the sign that is required to be placed on your alternative hypothesis so that then you are able to guide where your original rejections will be. The other important thing is when you find the p-value, especially when you're using the search statistics, whether you're doing the search statistics when the population standard deviation is known or when the population standard deviation, sorry, for the proportion, you need to make sure that you understand how you read the table, especially for the p-value. If your Z-test statistic is positive and you are doing a two-tail test, you need to make sure that you subtract the value you find on the table, especially for only for the positive. You subtract the value you find on the table from one and then multiply that by two. If you're doing a one-tail test and your Z-test is positive, the value on the table will depend on whether are you doing a one-sided left-sided test, which is the less than sided test. If it's the less than, the value you find on the table will be the value of your p-value. But if it's greater than, the value you find on the table will be one minus the value you find on the table. If your Z-test statistic is negative and you're doing a p-value for a two-tail test, then you multiply the value you find on the table by two to get your p-value. If you're doing a less than, the value you find on the table is the p-value. If you're doing a greater than, always remember from the basic probability we said, not basic probability from the probability. We said, if from the Z-scope, if your Z-value, you want to find the probability of Z greater than, you always remember to subtract the value you find on the table from one. Because your Z-test statistics or your standardized normal distribution Z-table contains only the less than values. Those are the things that I can just give you some hints on. Other than that, practice makes perfect. Find more questions to do and practice with those. Feel comfortable with them and nothing can go wrong. By the end of this session today, you should be able to submit your assignment four with ease. Remember the top four assignments where you scored highest mark will be used to calculate your year mark. If you didn't submit at least one of the assignment right now, you need to make sure that from now on you submit all your assignments in order for you to set yourself up for a good year mark. That's it from me. For today, I will see you on Sunday unless if there is any question or comment, I will stop the recording right now. Thank you very much, Lizzie, and Gateway. We hope to see you. I hope by Sunday I will be able to talk much better.