 Then we can continue from where we left off. The recordings are uploaded on YouTube on year. So I just want to show you those who don't know a way to find them. Check it. Let's do it this way. Can I share my entire screen now? So most of you joined the team from your lecturer site. So if you are in this, when you go to recordings, all the record when you click on the recordings there, you should be able to see all the recordings for all my sessions. There are 34 recordings or 35. But the year is your exam prep part one that we did earlier today. I will also upload the other one. You will be able to see it from there. Those in my group, you also have the same access because those who are linked to my group, you also have the same recordings because the link is the same. Otherwise, those who did not join teams and they are not in my group, the recordings are on YouTube. I can go to YouTube and show you where to find it. Just Google my name, Elizabeth Lizzie Boye. You will find my channel. Look for the channel called STA 21. Why look for my channel and look for a playlist for this group. The same group as my tutorial group. That's where you will find all the recordings, which are the same as these ones that are uploaded here. They are the same as the ones that are uploaded everywhere else. Okay, so going on and since I don't want to share my entire screen, I will go back to sharing only what we require or what we will need. So we ended up on this question. Let's see, did we do the, yeah, we did part 11. Now we are on part 12. So part 12, we're done with discrete probabilities. Now, the other thing I want to also mention before we move, because on this mock exam paper, they only included binomial and Poisson. Sometimes they will include discrete probabilities. Remember a discrete probability. How will you recognize that? You will see that by them giving you a table with the X observation and the probabilities. That you need to remember that that is also discrete probabilities and you can also apply the same information as that, the greater than or less than and so forth. The other thing you need to also remember is how to calculate the expected mean for the discrete probabilities, which is your, the sum of your X times your corresponding probabilities and how to calculate the variant or the standard deviation. I'm not going to do the variance because the variance is the square root. The variance is the square of your standard deviation. So yeah, you will have your sum of your X observation minus the expected mean because you would have calculated the expected mean squared divide, no, I'm lying. Sorry, no dividing, multiply by the, multiply by the corresponding probabilities. But you will remember that if you are lost and you don't know which formula to use, remember this document that we're using for the tables, it has formulas. So you can go any wrong, any way wrong. So those formulas are also included in some of this and you will also make use of the notes. Remember the summary knows that you need to prepare between now and before you go write the exam, which you don't have enough time. Make sure that you collect all that information, including those formulas. Okay, so moving on. Let's look at part 12, which is normal distribution. Why do I see a normal distribution here? It's because I can already recognize I've moved from calculating probability of poison or bilomial and yet they are mentioning things like the mean and the standard deviation. So part seven, they are asking what is the probability that a random, that randomly chosen recipient will receive a payout of at least, at least 1400. Now with normal distribution, also remember the following. The sign matters. If the sign is greater than or less than the probability you find on the table, it will be that probability that you are looking for. If the sign is greater than or equal or greater than, going to find the probability by using z minus the value you see on the table. If it's between, maybe I'm also doing it right for you. So let's do it the correct way, the way I'm used to doing it. If you find the probability of z less than a value, then the value you find on the table will be that probability. If you are given the probability of z greater than a value of z, usually I see on some of the questions that have been posting on WhatsApp, they use a small z, a small z. So I'm going to use a small z so that you can understand that. So the probability that z score is less than or equals to the z value, you find it on the table. The probability that your z is greater than a z value, you will find it by using one minus the probability you find on the table. And if you find that it lies between, your z lies between two values, small z one and small z two, two values, then we're going to find the probability of z less than small z two minus the probability of z less than small z one. I hope you are able to see that. So it means we're going to find the probability of the second z value minus the probability of the first z value. It's because this one will have the bigger probability value and this always has a smaller probability value. Okay, so that is how you're going to answer all the questions relating to normal distribution from here on going forward until we do hypothesis testing that you need to always remember. Okay, so now we need to find the probability that x, which is the payout x, it's greater than or equals to 2,460, that's what they want you to calculate. So you can either do it inside or you can just go and find the probability, the probability that z is greater than or equals to x minus the mean divided by the standard deviation. Information is given to you there, so you need to go and calculate the probability that z is greater than or equals to our x value is always the one that is given on the question. So it will be 2,460 minus the mean is 3,000. Am I recording? I forgot. Yes, you are. Yes, you are. Yes, you are. Thank you. Yes, it's recording. 37330 divided by the standard deviation of a thousand. So now go ahead and calculate since you guys have calculators and give me the values. Negative one comma two seven. Negative one comma two seven. Okay, so now we need to go to the table, but we cannot go to the table with a greater than. We need to go to one minus the value we're going to find on the table because the table contains the less than values. I'm just going to use less than. The table contains the less than values, so we are going to use the z value that we have. Yeah, and go to the table. So going to the table, we go to the z table on the negative side of the table. We z negative. We're looking for one comma two seven. One comma. And then we need to go to the top and let's do this. Go to the top and look for seven at the end. And that is the value we're going to use, which is zero comma one two. Zero one zero comma zero zero comma one zero two. So you're going to say one minus zero comma one zero two zero. And you can just let me know what is the answer. Zero comma eight nine eight. Which is which is eight. And that's how you answer the normal distribution probabilities. Sorry, can I just also. The sign changed from at least here. Sorry. Yes, we can hear you. Yeah, I'm saying that the sign changed from at least to at most. Okay, not necessarily, but yes, I've changed the sign because on the table, you always going to find those probabilities, the probability of a less than. You're not going to find the probability of a greater than. That is why we do a compliment of this very. Which then a compliment will be a less than. And the table, the Z value table, this table only contains the probability of a less than. Contains all that even when you go to the positive side, it's all the probability of a less than. And you can also see from here, when I was doing this one as well, you can see that I'm going to the probability of a less than probability of a less than as well. Whereas on the side, it was Z 1 greater than Z. At the side change now. Okay, now this one. So on this one, now let's go back to this one. Can you please mute yourself because I muted you and then you unmuted. Let's go back to this one. Now remember what we did here. We went to. The table to go find. This probability. This probability of 0, 8, 9, but we actually went to on the table to find that probability. So we went to say the probability of Z less than minus 1, minus 1, 2, 7 is equals to 0, 1, 0, 2, 0. Now, if I remove this value and I put Z here or small Z, let's make use of the same, if I put a small Z there, it tells me that the value of Z that I find here helped me to find this probability. So the same way I can take this probability because we went and found it by using the Z value. We can use this probability to go outside to go find the Z values. And that is what is required of you to answer this next question. Suppose that a Z value is less than and since because it's less than, remember any probability of a less than the value you see on the table is that value. If it was greater than we would have done 1 minus the value we see as an answer there. So this says go to the table, find this probability on the table. Then once you have that probability on the table, go out and go find the corresponding Z values. And that is what we are looking for. So it means, yes, it means we need to come on here and look for any value that is close by. So if you look at these two values, which is very difficult because they're both dating differences away or say almost similar. So if you look at your answers here as well, they have them as three digit numbers. But we know that our Z value is only three decimal numbers. But based on those two values that are close, you can choose one or the other. Just look at the examples. If they gave you a Z value and they only gave you two digits or two decimals, then I would say you choose the one that is close by, whether it's 6 or it's 5. So if I look at this, let's say I'm taking any one of those. So they correspond to negative 0.2. And if I go to the top, it corresponds with 0.5. So it's negative. Can I just come in there? What I did was I took both of them and added them so that I can get the 2.55. Because if you add, let me go to my table, if you add, so you added both of them and divided by 2, which is the same as just getting 0.255 because it's halfway between the two. It's like one of those, you remember when you do the critical value there is one way, it is 1,45. It's 1,645. It's one of those sections that you can also do on this one as well. So the answer will be 0, it's on the negative side of the table. So you must also remember to keep the negative value as well. So the answer is E. There was nothing necessary for you to calculate anything. The only thing you needed to do is to recognize that this is the probability, come to the table, look for that probability inside the table and go out to look for the probability that you are looking for. That's all what you need to do. Okay, moving on to number 14. Consider once again the UIF, coronavirus, blah, blah, blah, with the mean of 3,730 and the standard deviation of 1,000. Suppose furthermore a random sample of 10,000 recipient were selected. Calculate the standard error. What is the standard error? Is your standard deviation divided by the square of n? Sorry, Lizzie. I'm so sorry. I just wanted to do a follow-up question, especially on the last order from the previous. Yes, especially, I came across with this question similar to this one, but the statement was reading as it at most 0.6. It's the same as this. Remember at most is less than or equal. When we deal with probabilities less than or equal doesn't really matter. What's important is it's less than. You can either use the less than or equal or less than. The same way as yeah, at least it really doesn't matter. For when we calculate the probabilities, we don't even need to even worry about the equality side. Even when it says at least. You can always ignore that, but this is only for normal distribution probabilities, whether you're doing sampling distribution or when you get to the hypothesis testing or when you're doing this. Normal distribution, chapter 6 or study unit 6, only then. When you do. Discrete probabilities, the sign is very important. You must also consider that. OK, so if it says at most, yeah, if it says at most, it's less than or equal. So it's the same as less than you treated the same way as this. If they would have said at least. Yes, OK, then yeah, you you would have taken this number, the value 1 minus 0,4 and you will get 0,6 and then you use this to go to the table to go find it. Because if the site they the wedding was saying at least it means the answer that they would have gotten yet they did 1 minus the value you found on the table and it gave you that answer. It would be the same as what we did here. So in this instance with the probability of Z. It's greater than or equals to minus 1,27 that probability 0,8980. But in order for us to find this probability, we actually had to go and find the probability of less than that. So you just need to make sure that you know how to apply all these rules when you're working with normal distribution probabilities. How to move back and forth, how to move from from this. From getting an answer of a probability by using the sign to using the table and so forth. OK, now coming back to our number 14. Sorry. Elizabeth. Yes. I'm sorry to disturb. Part 30 is only shown on my screen now. So I can't really recall how we solved that one because I've seen it before and I remember that I had trouble solving this very same thing. So can you quickly just recap? Yeah, I haven't been seeing this page of the first time I'm seeing it but I've been hearing you speaking about it. So if you could just quickly recap on how we could solve this, I'm sorry to take you back. Solving this we go. Yeah, we find the values on the table where it's closer to 0,4. You go find the probability Z value. So it's 0,2. But because those two values we cannot decide, it will be between the two of them. So it's halfway. So 5 plus 5 divided by 2. It's 5.5. It's 55. So we just use the same 0.255. And that's how we got the answer of negative 0.25. Coming back to 14. It seems as if like 14 doesn't want to come. OK, how do we? Oh, not how do we? Because I already showed you the formula. So substitute your standard deviation of 1,000 divided by the square root of 10,000. And what do we get? 10. And the answer is 10, which is C. 15. With confidence intervals, there are three things that can happen. Confidence interval for the mean, when the population standard deviation is known, or when the population standard deviation is unknown, or for the proportion. You need to be able to identify those things when you read the question. Are you giving sigma or population standard deviation? Are you giving S, which is the sample standard deviation? Or are you giving the proportion? Are you giving the proportion? So you need to be able to identify those things. Because to find the critical value, if the population standard deviation is known, we're going to use the Z critical value. If the sample standard deviation is given, therefore the population standard deviation is unknown, we're going to use T. For the proportion, we always use the Z value. You always need to remember those things. And when you go and find the critical value, always divide your critical value by 2. For confidence interval, whether they ask you to find the upper limit or the lower limit, always divide by 2 your critical value. So in terms of this question, you need to go find the critical value at 95% confidence interval. Because they've given you also, you need to take into consideration the statements that they are giving you as well. You are given the sample standard deviation. So it means the critical value will find it on the T table. So you will use T alpha divided by 2 and the degrees of freedom, which is n minus 1. Always remember that. Now let's go find the critical value. Alpha, you remember how to find your alpha value from a confidence level. So if a confidence level is 1 minus alpha, which is 0,95. To find the value of alpha, you can just say alpha is equals to minus 1 plus 0,95. I'm applying meds and I'm going to assume that you all know meds. And we can multiply it throughout by negative number. Therefore, this will be alpha and negative will be positive and positive will become negative 0,95. And this is equals to 0,05. I didn't have to do all these steps anyway. I didn't have to. Okay, so our alpha value is 0,05. So our T of 0,05 divided by 2 and the degrees of freedom of 70 minus 1. Just give me a sec. Just to erase all these things that I wrote. Okay, so therefore our critical value here will be T, 0,05 divided by 2 is 0,025. I'm just going to put the 0 there and this will be 69. So we need to go to the T table and go look for 0,025 and 69 degrees of freedom. Looking for the T table. We will look for T distribution tables. You can find it anyway on your table. Okay, so the other thing with the T table as well, depending on where you're finding your T table, it's very important that you standardize your tables as well. And I saw that your lecture site, you can use your study guide or something like that. What did they say? You must use them. So in the study guide also it's a 3 decimal. Study guide, right? So it's 3 decimals. So we need to use the one that has 3 decimals. Okay, so we go and find our alpha divided by 2, which was 0,025 and we need to look for 69. So I just want to remember that column. Okay, so we are on the next page. We're looking for 69 and the 69 and where they both meet. 1,995. That is our critical value. 1,995. Okay, so that is a critical value. That's not the answer that we're looking for. We need to calculate the confidence. So calculating the confidence interval, we use point estimate plus or minus the critical value times the standard error, which will be the sample standard deviation divided by the square root of n, which is times the standard error. So we just need to substitute the values we are given. Our sample mean is 3,730 plus or minus. Our critical value we did go find it. It was 1,995. Your standard deviation is 1,000 divided by the square root of your sample size, which is 70. Now we can go ahead and calculate. Calculate the site first. When I mean this site, I mean only the part. Just give me the value of that. You get 238,4481. 2? 238,4481. 238,4481. Am I, did I hear you right? Yes, correct. Cool. Now we need to create our upper limit and our lower limits. So you can take 3,730. You need to be very careful with this. You need to have your lower limit this way. The negative sign is 8. So 3,730 minus, let's make this bigger. 3,730 minus 238,4481 and 3,730 plus 238,4481. So for my handwriting now, which one? Option 5, E. So this was 3,492,3968. Is that what you are telling me? So that is option E. Happiness? Peace. Peace. Next one, how would you know whether are you doing the population, the confidence interval for the mean or for the proportion? For the mean, you will be given things like the mean and the standard deviation. If we look at this question, are you given the mean and the standard deviation? It also says construct the confidence interval. So when you read the question, you should be able to identify what are you given. So this is for the proportion. What are you given with the proportion? You are given N and you are given the observation that satisfies some sample proportion somewhere. So with the population proportion as well, sometimes you will not be given the population proportion that you will be given or you need to use the point estimate. So the sample proportion, but you might not be given the sample proportion. You will be given observation that satisfy that sample divide by and your sample size. Then you can calculate your sample proportion because you will need the sample proportion to construct your confidence interval, which is alpha divided by two times the standard error. You need to use that formula. So what is your sample proportion? 0.4, which is 80 divided by 200, which is equal to 0.4. So we're not done. We also need to find the critical value. Yeah, we need to go to the z table. So z of alpha divided by two. Now what is our alpha value of 99% confidence level? Alpha is 0.004 divided by two. We need to divide that by two. 0.004. That will be 0.0. How many zeros? Two zeros? Two of the decimal one. So we need to go to the table. Now we need to go to the z table. So this is the probability. This value you see here is your probability. Oh, if you go to, I'm not even sure now, I think when we were doing the confidence interval, we did have a table that has the confidence levels for your z. Yes, you gave us a table. Tomorrow that table should become very handy for you to use because then you need to take this value, go to the table, to the z table. It's going to be time consuming, whereas you have the information in front of you, you can use that. So you need to come inside this table, look for a value called 0.005. So 0.005 should be somewhere here. I can use this one, the one where it says 49. So I'm going to take negative 2.58. But I also know that that is the z score. So that will be negative. That is 2.58. I don't even have to worry about the negative value that is there. So I can just take that as my z value. So you just come here and take the closest value. Even this one is close, but I know that the critical value is 2.58. So that is why I'm choosing the 2.58. So now I have my critical value, I have my proportion, my point estimate. So I can just substitute 0.4 plus or minus 2.58 square root of 0.4 times 1 minus 0.4 divided by our n is 200. Also similar, you just go and calculate the values. You can do step by step, if you don't know how to use your cashier calculator, the fraction one, you can do this step by step. But if you know how to use your cashier calculator, you just use the same. So when you get to this one, you can put it into the brackets. So you just say 0.4 and then you start with the minus first, minus 2.58, open bracket, square root, or you can say multiply by, or you can open bracket, square root, put the square root sign and then put the fraction button and then do 0.4, open bracket, 1 minus 0.4, close bracket and then go to the bottom and put 200, then go to the outside and put the bracket and then enter the negative side of things and then enter the positive side of things. What do you get? Which option? Option A. Option A. Option A. Those who don't know, speak now or forever, you won't understand. Okay, okay, Elizabeth. Yes. So can I use level of confidence on page, on study guide on page 158? So if I don't, if I don't have to go long process. Yes, let's see where page 158. 158. Yes. Hmm. Where am I now? No, I'm still not. Way, way, way, way. Way is 150. Oh, there we go. 158. There we go. Yes. This is the same table that we shared in the, yes, you can use this because there it is with alpha for a 99% confidence, confidence level. Okay. You can just come here and use this. So when we get to, when we get to the hypothesis testing, then this, we can also use this. Actually, they should have just did it like this without that. It's an error because it's a one thing. One thing we don't divide by two. Yes, you can use this definitely. So make note. Thank you. Make note. Make note. Or you can even write it some way on your summary notes. Okay. We are in somewhere on the borderline. I think now we are in hypothesis testing. Why do I know that we are in a hypothesis testing? It's because they mentioned things like one-sided lower tail. In competence interval, we don't talk about the one-sided. We talk about upper limit and lower limit. So the minute they mentioned things like this, you must know that now it's hypothesis testing. With hypothesis testing, we do two tail tests or one tail test. Okay. So suppose that the calculated test statistic is one-sided tail test in the lower side, which is in the negative side. It is minus 2,82. Suppose further that the population standard deviation is known. So if the population standard deviation is known, the test statistic that they would have calculated would have been there. Is it T or Z? Z. It would have been a Z. So therefore it means this value corresponds to negative 2.82. Now they say in the lower tail area, since they say in the lower tail area, we can also assume that it is less than. So which makes things easier for us. Because the question they're asking us is what is the p-value? Now finding the p-value also is determined by the side. You use the greater than. It means we will say one minus the value you find on the table and so forth. And because this side says it's a lower tail area, which is the less than tail area, which is in the negative. So we can go to the negative side of the table and look for negative 2,82. Because we're looking for the p-value. And the p-value, we must always remember that this is a probability. p-value is a probability. So use the Z-value of 2,82. Negative 2,82, negative 2,82, which is that. Yes. Which is option number C. Is it? Okay. So if it was said it's two-sided, so how do you solve that question if it is two-sided? If it's two-sided, we multiply or you can add 0,0 or 0,0024 plus 0,0024. If it's only for two-sided, for two-sided test. Two-sided test, we multiply the p-value, the value we find on the table by two, because there are two sides. Are we taking into consideration that there is that side and this side? We take both of them. For a one-sided test, we only look at one value. Okay. Number 18. Also remember, when we talk about hypothesis testing, you need to know all six steps of hypothesis testing. Remember them. Stating the null hypothesis and the alternative hypothesis. You need to be able to state them correctly. Remember that the null hypothesis, we do not use the sample statistic, we use the population parameters. Also, the null hypothesis always has an equal sign. So it can either be less than or equal, greater than or equal or equal. Anyway, it doesn't really matter. You can also just leave it as an equal sign. What matters the most is the sign that you allocate to your alternative hypothesis. You need to remember that the sign you place on your alternative hypothesis will tell you whether you're doing a one-tail test and whether is it an upper tail area or a lower tail area or whether you're doing a two-tail test. That is very important because if it's a two-tail test, then it's not equal. If it's a lower side area, it's a less than. If it's an upper side area, it is a greater than. And all those things, they affect how you find your critical value, how you find your p-value, how you make a decision. So it's very important to know how to state the null hypothesis. So it means you also need to know how to find the critical value and how to calculate the test statistic and how to make a decision. So those are the hypothesis testing things that you need to know. So for example, like this question, where they ask all six steps of hypothesis as options. So we need to find the incorrect statement. So we can go through each statement one by one. So number A, you will tell me whether is that statement is correct or incorrect, but we will do all of them. Number A, there is a 5% level of significance. Look at that. Correct. That is correct. This test is a one-sided test. Look at the full statement. Correct. Because it's more than 1,000. So that is correct. The critical value is one for five. Now ask yourself, what am I given here? We're given the population standardization, so therefore it means we're going to use z. And are you going to divide z by alpha or is it just a z scale? Just z. It's just a z alpha. So it will just be z alpha and we can go to our study, our guide. So let's go back. We are given our alpha of 0,05. So therefore it means z of 0,05. And a z of 0,05 is the same as a 95% confidence level. So come in here. You also need to remember to put your alpha value, because they didn't give you your alpha value. So this is that way, 0,05. So that you extend this table a little bit just to give you assistance. So we're looking for 0,05 or 95%. So let's go there. One-sided test or one-tail test. It's 1,645. So that is also correct. Right. Number G. The alternative hypothesis is that the average cost of a low-end smartphone is more than 1,000. Incorrect. Incorrect. Correct. Because the null hypothesis states that. So the null hypothesis can be the same as the alternative. No. But remember, in the null hypothesis, you can never have a null hypothesis. You always have an equal sign. And if the statement from the researcher says more than, and a more than is a greater than. So that cannot be in a null hypothesis. So your null hypothesis will state that the moon is greater than or equals to 1,000. And the alternative will state what the researcher wants to prove, because that's what can only go into the alternative. Because in the alternative, the statement should not have an equal sign to it. You can state it in that way, or you can even just ignore that and just put an equal sign there. It will still be fine. But you must always remember that your null hypothesis, in your null hypothesis, there is always an equal sign. In an alternative hypothesis, there is no equal sign. So the signs that you can put on, the signs that you can put on your null hypothesis are equal, less than or equal, greater than or equal. The signs that you can put on the alternative is not equal. It's less than or it's more than or greater than. So those are the only signs you can put. So this is also correct. Now, the last one says you need to find the test statistics. So yeah, it means we need to do some calculations. So you need to calculate your Z value, your Z score. Z is equals to the sample mean and we clean it nicely because all these inks are still there. So we have Z is equals to our X minus the mean divided by the standard deviation divided by the square root of n. Our X is that state it's always, oh sorry, actually our X is our X bar because it's the sample mean. This is the population mean because it's the one that is stated in the hypothesis testing. The sample mean is 1200 minus 1000 divided by standard deviation of 800 divided by the square root of n is 100. In the exam, I'm also going to show you should cut as well. In the exam, you don't have to go and calculate this because I can see already there it says it's negative. The answer you get at the top determines the sign as well. So if it was 1000 minus 200, then I will say continue working out. But because the answer you get at the top will be 1200 minus 1000, you get a positive answer there at the top. You can just then determine that this is the incorrect value. But just go ahead and calculate the test statistic. That is for tomorrow when you write the exam. I get 2.5. 2.5. So this is the incorrect one because that is negative 2.5. Lizzy, it's California. Can I ask a question? Yes. Tell me if it was less than the answer will be correct if it was less than 1000. The alternative hypothesis is if it was less than 1000, that means the test statistics were supposed to be negative. Am I correct? No. It doesn't really matter. The sign when it comes to the test statistic, it doesn't matter. You only need to know about the sign for finding the critical value and making the decision and finding the P value and determining whether you're doing a one-sided test or a two-sided, a two-tailed test. The test statistic, whether the answer we get here is negative or positive, whether the sign was greater than or not equal or it really doesn't matter. It will not influence the answer you get. It influences how you use that answer. Okay. Okay, thanks. 19. A two-sided hypothesis testing for the mean is a known population standard deviation. Well, with a known population standard deviation yields a P value of 0,03. Things are highlighted in both just to give you an indication of what you need to look out for. Two-sided test, known population standard deviation, then it means this is the Z value. We can use the P value, yields a P value of 0,03. It's a two-sided test. So if it yields a P value of 0,03, therefore it means this P value is divided into those two parts. So it means in honor for them to find that P value, this sign was 0,015 and this side is 0,015 because both of them would be equal to 0,03. That's what is happening. Which one of the following statement is incorrect? Okay. Now, there are a couple of things that you need to also take into consideration as well. Number A, it says the test statistic is 1,85. So what is 1,85? We need to round it off to two decimals because our Z value needs only, we need only two, we need only two decimal places. So this is 1,89. So we need to take this 1,89. Let's go to the table. Now, another thing you need to take into consideration, when we look at the P value on the table, if the Z value is positive, then only when we do the P value, if the Z value is positive, then we're going to say the value we find on the table, we're going to minus that value from 1. So we need to go and find that probability and subtract it from 1. So let's go. We're looking for 1,89. On the positive side, we go look for 1,8 and 9 is the last column and that is the value we are looking for. So that is 0,9706. I think on this question actually as well. So, but anyway, let me not preempt the answer. Let's go 0, what did we get? 0,9706. And the answer you get is 0,0294. Sorry, Ms. Liz, why are we doing 1.89? Why are we doing 1.89? It's because we need to use, remember, we are giving what the P value is. We have to go to the Z value table. Yeah. We need to round it up. So because on the Z value table, we only have 2 decimals. So the value to the left or to the right, it's greater than or equals to 5, we add 1. So I added 1 to that. Thank you so much. If you look at this, you can see that it doesn't even correspond to any of the answers that we have. So number 8 is the incorrect one because it should be either 0,3 or actually should be 0,015 because then we need to multiply to get to that. Yes. Since this is a two-tailed test and then the Z-status positive, are you supposed to do 2 and then in brackets 1 minus the table value? Yes. You can do that because it's a two-tailed test. So this value that we have should be one of this value. And that's what I'm saying. So this, it does not correspond to this value. So we can also immediately from here, you can also eliminate it because you need to say this value because it should be those ones. You need to take this value and multiply it by 2 like we did there. Remember? You need to remove it. Yeah. Well, for two-tailed value, we take both of them. So by looking at this value, it should be 0,15. It's not 0,15. It is 0,024. So this means it will not deal that value. So that is our incorrect value. That should be correct because our population standard deviation is known. So that should be correct. We're using a Z-test. The null hypothesis is rejected at 5% level of significance. So it means we can say, remember the decision. So here, we don't even have to worry a lot because the decision says, let me call this decision, the decision rule states that if the p-value is less than alpha, we reject the null hypothesis. So our p-value in this instance is 0,03. I don't have to use the values that we are calculating now, but I just need to use the values we have there. And our alpha value is 0,05. So our p-value is less than. So we reject the null hypothesis, and that's what it says. Now, number D says the null hypothesis is not rejected at alpha of 1%. So alpha of 1% is that. So the p-value, we'll also need to change the sign. So the p-value here will be greater than. So we're not rejecting the null hypothesis. So yeah, we'll say we do not reject the null hypothesis, which is correct. So the only option that is not correct there is number one, because number one should have given us 0,05. Or when we multiply it by 2, it should give us 0,03. But it's giving us. So if we multiply this by 2, it gives us 0,048. Okay. In the exam, you don't have to go through all the statements and all that. Just make sure that you are able to identify the things that you need for that purpose and move on. Because time is of essence. That you must complete all the questions because you might be missing out when you get stuck on one question. So in the exam, don't do all options to try and verify all of them. If you find the answer, move on. Now we're on a contingency table. So yeah, we actually almost closer to the end of the exam when you see another contingency table popping up. So this is Chi's way. We're doing a test or a hypothesis for the relationship of two categorical variables. So also remember, in the exam, if they didn't give you the totals, on this one they gave you the totals. If they didn't give you the totals, you need to go and calculate the totals. Otherwise, you know, our famous Excel template, you can use the Excel template that we always use for doing the Chi square test. So you can also use that. Copy the values into that Excel. Look for the correct template. There are multiple templates. Look for the one that talks to the data that you are looking at. Okay. So this is a contingency table. What is the number of degrees of freedom? So yeah, the degrees of freedom for a contingency table is number of rows minus one times number of columns minus one. Now please be careful when you read the number of rows and when you read the number of columns. Do not read the total columns and rows. Do not read the header. Only read the input, this one and that one. So you only need to read those ones. Not the total, not the header, or the labels. So how many number of rows do we have? We have one, two. How many number of columns? There we go. Do not read this column and that column. How many number of columns do we have? We only have two. I just said it now. Don't read the labels and the headers. So one times one is equal to one. So the answer is B. What the degrees of freedom is? We did calculate it. We found that the degrees of freedom is one. We need to go find the critical value. So a square of alpha and the degrees of freedom, we find it by using our alpha value. They gave us the alpha value of one percent, which is zero comma zero one. Zero comma zero one. And the degrees of freedom is one. We need to go to the critical values of chi, table, and that is the critical values of chi. And we're looking for one and zero comma zero one. So you just use the alpha value and one and just come to the table and find it easy. Moving on. Okay. On this one, they also gave you the observed values, which are your FOs and your expected values. So you don't have to go and calculate the expected value on this one. But I don't know in the exam whether they will give you questions like this. So if the MOGA exam resembles what the exam paper looks like, then you don't have to worry too much because then they have calculated the expected values. So you just substitute the values into the formula and work it out. But I will suggest also because sometimes in time consuming as well, you can use the template. Rewrite the only the observed values onto the template and only use a two by two table because this is a two by two. So if in the exam they give you a three by two, use the three by two. If they give you a four by four, you will have to sort it out yourself because I don't have a four by four. But you just need to know and remember how to use the template. I'm not sure if everyone has the template, but let's say the entire screen again. So this is the template that I am referring to, which is one of the other cheat sheets that you can use. So we're doing a two by two. This is a two by two. And as you can see here on this template of ours, we've got pre-edited things. You don't have to use them. You must replace them with the correct information that you are using. Just want to open that and minimize it a little bit. Miss Lizzie, cheat sheet is such a horrible word. It's called a visual aid. Oh, then you can use the visual aid. Just want to make it also smaller. I can close this one because we're done using it. Okay. So there are so many things happening on my screen right now. Okay, I need to go to the two by two. So even in the exam, I'm not sure if you are able to minimize your exam. You can minimize it. Or if you have two laptops, you can open two laptops because your exam, you don't use invigilator. So you must take this opportunity because next year it might not be the same. Next year, if you plan to write next year, they might use the invigilator app, the iris, all those things. So take this opportunity where you can write your exam freely and use it. Okay. So we're using the two by two. I'm just going to replace the values I have. I'm not going to worry about the labels for now because that will not affect my answer. I just need, and if you notice, I'm only capturing only those values because at the bottom, the table weight says expected values. All those expected values get captured there. And the only thing I need is the chi-square test statistic. And that is our chi-square test statistic that I'm looking for. So I've got my answer, which is, it is in two decimals, so I can also just move it to two decimals. Is it two decimals or three decimals? It's in two decimals. So 11.228, which is that option. Hi, Elizabeth, how do we get this visual aid? Okay, let's take this one. Your fellow students must send you the visual aid. There are two visual aids that we use. I can take them all, guys, or send them to you. Sorry, can you repeat that, please? I'm saying your fellow colleagues can email or send you or what's up you the visual aid because you see now, oh, people are going, you see, that's why I don't like sharing my entire screen. Are you able to please put it in the chat for us? I will put it also on the chat. Some people can access the chat. Let me get out of here because I will do that for you. Let me put it on the chat. You can go on. Oh, you can do that. Thank you very much, Edian. You can't access the chat, though. Yeah. Okay. We've sent it on the WhatsApp group. You're on your groups. Here we've got all the groups, so they will send it to you. Don't worry. You will get it. I'll put it on the group end here. So those guys are getting on Lizzie's WhatsApp group and share it with the other guys that are not on that group. Yeah, so they will share amongst themselves. Don't worry about that. You will find a way. Okay, so what did I do now? I stopped sharing. Lizzie? Yes. Can I ask that the spreadsheet we were using in the first session as well be shared on the group? I closed that one. I didn't save it. I will create one, but I didn't save that one. I will see what else can we do. Okay, so it's getting late. Let's finish up. So we've got 22, and I think we left with one last question, if I'm not mistaken. Oh, two or three or more questions. How many questions are left? You're writing out of how many questions? Twenty-five. Okay, so we left with three questions. Okay, so also this one uses the template as well. So you can use the template, but some questions you don't even have to worry about templates. So let's look at the question that we have here. So based on this information that we have, they've got scatterplot, A, B, C, D, E, F, so many of them, and they're asking you to find, consider the sample data below and develop a scatterplot. Which one of the scatterplot A to F best described the data? Because there are so many of them, and it says scatterplot B, scatterplot A, F. Okay, F it's coming up twice. Okay, so D, E, F. Okay, and they are not in order. As you can see, A corresponds to scatterplot B, so you must also pay attention to those options. So let's look at these scatterplots. So we can take one point. Let's take the first one and see if we are able to make sense of it. Sorry, let's say I'm not sure if it's on my side. I can see your screen. Are you sure? We can see it. Yeah, we're good. It's appearing on my side. Yeah, we can see it. I can also see your screen. Yes, I can also see it. Okay, maybe it's the problem from my side. You can go out and come back in. Okay, so now we can take that first point and go look for it. Sorry, Lizzie. So it says yes. I'm saying for this one that we can use the same method that we use for when we were doing that for the assignment. I can't even remember. We take the minimum and the maximum values, then we were eliminating. Okay, so we can do that. So what is the minimum value of this one? It's 14. So that will still work the same way. So it's fine. So if I take X of 14, so it should be somewhere, yeah, and Y of 17, it should be somewhere there. That can be true. There should be a point somewhere, somewhere here for C. And for this one, X17, there is, okay, wait, actually, let's even look at all this. Wait, I don't know. I'm so confused right now. Wait, we can use that point. X of 14 and Y of 17. Yeah, this one does not even have 14 and 17. So we can just even ignore it because it starts at 20. And this one starts at 15. So it doesn't have 14. So you don't even have to worry about that one. This one actually starts at 20. For X, where is 14? There is no point that corresponds to 14 X. Yeah, so we can also eliminate that. This one is a potential because 14 can be somewhere here. And one of those values can be either one of them. We can put the question mark there for now. 14 and 17. And this one also, it can be a potential, but that is above. It's scatter plot E. Yeah, it's E, scatter plot E. So, Ms. Lizzie, what I did with this, it's actually quite, well, for me, quite quick. If you plot all those points in an Excel graph and you select them, and you just say insert scatter plot, it draws you exactly the dots that you're looking for. Very, very quick. Yes, that should be very, yes, you can do that if you know how to use your Excel. I'm looking at scatter plot E. That also can be correct. Do we have any value of X that is more than 40? Yes, we do have 41. Do we have any value of X that is or Y? 10. 41 and 10. That is correct. Okay. And on this other one, that won't be correct. I pasted that Excel output on the chat if you wanted to look. Yeah. So, do you also, it's not correct. So, E is the only one that I can also think of that can represent the data well, because this one doesn't have X of 40 should be some way there. And there are no values that for X that are more than 41. So, this one, these two values can be correct. This one can be correct. So, that can be correct. That can be, can also be correct. Oh, but B started quick. So, this one, I'm going to say E. So, E corresponds with G. Scatter plot E. G is the option. Okay. Don't go and choose E from the answers, because you need to look at the scatter plot title. Okay. Question 24. E tests all the summation values that they gave you, and they're asking you to find the slope. So, in this instance, the slope, there are formulas. So, the slope which is B1. So, you can use the template on this one. So, the slope is your sum of your X and Y minus the sum of X times the sum of Y divided by N divided by the sum of X squared minus the sum of X squared divided by N. Some books or templates, they have B1 is your SSXY. Did they give the SSXY? No, they didn't. So, you can use that, because then that is the same as this. So, the top part is sum of X and Y over the sum of X, which is that. So, you can use that. Some books in terms of N, they can just take this N and multiply it here. It still works the same way. They are represent one and the same thing. So, you will still get the same answer, whether you take N multiplied by that. So, let me look at your tutorial letter, the formula that you can use on here. On this one, it uses, oh, it uses this one way you can. Okay, we can also use that way. We just divide by N, not here. Do the same, divide by N there, divide by N. So, you just substitute the values that you have. So, we are given, let's make it bigger. We are given all the values. There is your sum of Y, 11502. Oh, come on. 11502 minus the sum of X465 times the sum of Y, 613, divide by N. Our N is the sample size, which is 30. That is our N. Divide everything by the sum of X, which is 9455 minus the sum of X must pay attention to this. This is not sum of X, which is not that. It is the sum of X, which is 465 squared, divide by 8. 0,59. 0,59. I say 0,89. 0,89. That is option 8. Hi Leslie, quick question. What happened to the 613? What happened to the 613? What 613? There next to 465. What happened to it? Yeah. Or where did you calculate that all? Oh, that is the sum of X, the sum of Y. Here is the sum of Y, the sum of X. Okay. You just substitute into the formula, the values you get. You are given. Okay. Okay, sorry about that. So you just need to substitute into that formula. So you need to be able to identify the correct formula. So as you can see, all these formulas, there is nothing written on them that says what is what. But you can use this to give you guidance because from that, it's your sum of your sum square measures of X and Y. And if you look at that sum square measure of X and Y, that is that formula. The sum square measure of X, which is this. And that is all what we use to substitute into B1. So when you calculate things like your R squared, also they will give you all these sum square measures. Your SSR is your B0 times the sum of Y plus B1 times the sum of XY minus the sum of Y squared N divided by SST, which is the sum of Y minus that. All these formulas, you can use them. So you just need to keep in mind. Yes. Yes. So it is that you will just train the session now. If I may ask, when are you going to upload the session on my UNISA? On my UNISA, I will do, okay, on my UNISA. On my UNISA, I will do session one. And then while this one is uploading on YouTube and all that, then I will, once it's done, I will upload it. So the sooner we finish with the session, the sooner I will upload the sessions for you guys. Okay, thank you. So we are actually almost done. Yes. Can you see? Yes. Are we going to get both sessions on YouTube or are we going to get one from my UNISA and we for you? No, both of them are one session one. The earlier one is already on YouTube. Upload it. Oh, okay, okay. Yeah, if you subscribed, you should have already received the notification. Okay. So even with this one, you will receive a sub-air notification once I upload it. Okay. All right, last, last question, last, last question. Then we can entertain any other questions outside of the session. Suppose the relationship for the store sales Y and the food traffic X is explained by this regression line of Y is equals to 6.64 plus 0.89 X. So what you need to also remember here is this is your Y hat is equals to your B0 plus B1 X. You always need to remember that with the coefficient of determination R squared of 0.6099. Which one of the following statement is incorrect? So with regression line, you need to be able to know how to estimate the value, how to interpret the slope, how to interpret the coefficient of correlation. And if you given the coefficient of correlation, how to find the coefficient of determination. Or if you given the coefficient of determination, like in this instance, how to find the coefficient of correlation, which is R. So now, when interpreting the coefficient of determinant, which determination, which is R squared, we always say whatever percentage of the total variation, because this is a total variation in Y is explained by the variation, the variation in X, sorry, in X. So in this instance, then it will be total variation in sales is explained by the variation in the food traffic, something like that. We can also take our R squared in order for us to find the value of R. We just take the square root of R squared. That will give us our coefficient of correlation. And coefficient of correlation, we interpret it in terms of strength and direction. Remember that? How we interpret the coefficient of correlation, we have a negative or a positive, a negative positive, a negative strong relationship. Strength is the negative direction. What am I talking about? Strength is either weak, moderate, or perfect. Direction is either negative or positive. That's how we interpret R. Interpreting the slope for one additional unit increase or decrease in the value of this will increase or decrease in the value of that. So a one unit increase in value of X will have an impact in terms of a decrease or increase in the value of your Y. And that's how you interpret your slope. All right. And your slope also can give you the direction of your correlation, whether if it's negative, remember the slope. Let's go to our scatter plot like this one. It's on the scatter plot. It tells me that this is a positive. This is a negative. It gives me my direction. So therefore, it means on this one, my B1 is positive. On this one, the value of B1 is negative. So that's how you can know what direction your slope tells you. Okay. So now let's look at the statement. We need to choose the one that is incorrect. We are given R-square of 0,6099. So how do we interpret it? 78% of the variation in sale is explained by the variation in the food traffic. Is that true? Incorrect. That is the incorrect one, because it should have said 0, or it should have said 60.99% of the variation. Yes. So that is the incorrect one. In the exam, you are done. You close your session, you say, you done, pens down when you get to the final question. But let's see if we can answer the rest of the question. So in terms of the slope, it's positive. So that is why that question says there is a positive linear relationship. When there is no traffic sale, oh, that's the other thing as well. So remember, if your x value is equals to 0, so it means if we substitute this value with 0. Therefore, our estimated y value in terms of sales, our estimated sale will be, or will increase by just, or will just be 6,64. It will just be that value, because this one exists if x is 0. So and that is what they are saying here. But remember also, this is the rent, it's in thousands of rents. So 6.64 times 1,000 will give you 6,640. So how we interpret that? When there is no food traffic in the store, that is x is equals to 0, then the store will only sell online and the estimated sale for that store will just be 6,640. For when there are no food traffic. And that's how you interpret. Your y intersect. D is the coefficient of correlation. All you just need to do is what I just did there. Take the square root of this value, square root of 0,6099 should give you 0,7810. And that you can interpret it by saying there is a strong positive correlation between the store sales and food traffic. And that concludes our business of the day. Thank you very much. And all the best for tomorrow. Let me stop sharing. And let's see if someone wants to ask a question. Miss Liz, could you please post this on YouTube because my laptop crashed about 20 minutes ago. So you missed quite a lot. Yeah, no, no. All of this will be on YouTube as well. All right, thank you so much. Okay, no problem. Any questions? Hello, Miss Elizabeth. Yes, I do have a question. Yes. The way you were doing like showing the other lady in the previous one how to use the Casio. I was following until on the X bar where you said we must press 2. That's when like below that I didn't know like how to go about finding the last number. So I was following the shift 1, 4, 4, division shift 1, 4, X bar 2. From there going forward, I don't know how to do the next step. Can I ask those who attend my sessions? Did I, in the notes, do I include the steps? I can't even remember these things. Yes, you did. There we go. Let me just check. Yeah. I just have to check which. No, I have only four. Oh, gosh, that is not for Casio. Those are for. No, I don't have the status issue. Okay, but at the end of this, the study guide, let us say information about two concretes, Casio and Shab. Okay, thank you. What is the previous video as well? Yes. No, you can also use this document. That explained everything from the beginning till the end. Yeah. So you can use this document. I'm glad that they actually started using it. So you will get the steps for the Casio in the second column and the steps for the Shab calculator. So you will see that it also not only does the stats, but it also shows you how to work out like your factorials, how to work out your univariate. You just need to go to the ones that are relevant to you. The univariate stats, which is this one. So you should look at page 216 of your study guide on the steps on how to calculate the mean and the standard deviation, which is this. Okay. Thank you. Thank you so much. Yeah. So there are the steps to show you how to calculate the standard deviations. All right. All right. All right. I'm going to stop sharing and I'm going to stop recording and