 I was waiting for the thing to start and welcome to your final exam preparation session. And today will be the last, last, last, last one because after today, there won't be any other exam preparation sessions that I will be hosting. But hopefully if you participate and take part, we might decide that we will have another one during the week before you go out the exam. But let's see how today progresses and then we will start. I will discuss at the end what's next. So we're going to look at the mock exam, which is one of the past exam paper that I created just to give the students that are in my E-Twitter group some exposure or practice into a timed environment when they do their exam so that you get a feel of getting a totally new question that you've never been exposed to, but that is related to your module and how are you going to be dealing with that? So we're going to look at that just now. So I created this mock, which is one of the past exam paper. So let's see, we're going to do it together. Some of the questions and see how we answer them. So I expect you to participate. Let's see. The first question, the following information is collected from an application form or a call to a certain bank, which variables below are quantitative? So A is total of monthly expenditure in rent. E, street address of an applicant. C, gender, D, mental status. E, types of job. Quantitative variables are those variables that you either can count or measure. Which one? A. Do we all agree? Let's see if it's A. And A is correct. Sorry to be a pain. I think someone's not muted again. Can you just make sure that you're muted, please? It maples with the sound. OK, I see there is a hint and I don't see anyone who's unmuted. Koji, I don't know if I'm pronouncing your name correctly. I think you are the person, your mic is unmuted and there is always sound coming up in your background. Can you make sure that you are muted? It says muted, but. Hi, ma'am. It's Matthew here. I couldn't be able to to see the questions that we have shared. Are you not able to see the question? But I'm sharing my entire screen. And now. Lockout and log in back. Is there anyone who also is experiencing the same challenge? Nobody? No, we can see your screen. You can see the screen. OK, so please log out and log back in. Pickle. OK, so let's move on to the next question. Which of the following statement is not green about the mean? A, it utilized all the values in its calculation. B, it is not affected by extreme outliers. C, the value of the mean times the number of observation equals the sum of all observations. G, in a symmetric distribution, the mean, the median, and the mode are all equal. E, it is the best measure of central tendency when the data is not viewed. A, B, C, D, or E. B. It will be B. It is not affected by extreme outliers, are you sure? Which one is not? So yes, definitely. Because we know that the mean, it is affected by extreme outliers. If your data is viewed, your mean will pull towards the bigger data. If it has very big numbers, huge numbers in terms of big numbers, it will be pulled towards that. So B is those who are in my E-Twitter group. You can take, after we've done with this, you can also go back and try this MOOC exam paper as well. Because every time you try, you will get a new question. It's not only these questions that we are looking at. So you can try as many times as possible to get a variation of questions. And when I talk about variation of questions, it's not the same question like in terms of this. It's totally new question that you will get. So please try to use this to practice as many times as you can. Hi, Lizzie. I have a question on that one. So I think it's possibly my understanding of the word observation, but could C be right too? No. Remember C, because I cannot go back now. What C says is I cannot write there. What C says, it says, if you take the sum of all the observation, I can't even now remember, but let's go back to the mean. So this will be your mean and your observation. It says, if you did it say divide or multiply, I think it says, if you multiply the mean with the sum of the observations, you will get the number, the observation. What it says is if this is the sum of all observation divided by the x bar will give you the mean. That's what C was saying to you. It says the number of observation will be given by taking the observation, dividing them by the mean. Sorry, because how I read it is that it was saying that if you take the mean itself, then you multiply it by the number of the observations, then you will get the sum of all observations. Yes, that's the same thing. If you take the mean, so if I take the mean and multiply, cross multiplication, mean multiply by the number of observations, I will get the sum of all observations. That's what it says. Yes, we get the means by adding everything together and dividing by how many people. So it's basically a reverse. It's the reverse because if I need to make the sum of observation, the subject of the formula, I'll multiply with the answer of the mean. The mean times n will give me the sum of all observations. The sum of all observations divided by n will give me the mean. Okay. Sorry, the question was not correct, right? It was to say the choose one that's not correct. Not the correct answer. We already moved on that one, so I can't go back because I made it linear so that you have the same feeling I can't go back. There is no go back. It's only next page. Okay, so let's look at this one. So if you have a question before I press the next button, please ask, don't wait until I press already on the next button because then we lose that question. A graphical technique applicable to qualitative data is A, is it a scatter plot? B, is it an ogive? So C, is it a histogram? D, is it a stem and leaf display? E, is it the pie chart? Remember, this is categorical data, which is qualitative data. It's to be. It will be E because A scatter plot is when you have two numerical values and ogive is when you have a cumulative frequencies and you put them on a graph. And histogram, it is a pie chart for numerical data. It's a stem and leaf plot is when you have numerical data and it can be a 10th stem and leaf or hundreds stem and leaf or a decimals stem and leaf plot. And E is a pie chart where you break your categorical data into different categories. And the answer here is E, check. Are there any questions before I move? If not, then we move. The following data represent the number of children in the sample of 11 families from a certain community. And there they give us the data. Which one of the following statement is incorrect? A, it says, the median and the second quartile are not equal. B, only the median and the mean are equal. C, only the mode is equal to the median. D, the median is five. E, this distribution is symmetrical. Oh, it's not symmetrical. When it's not symmetrical, it means the mean and the median are not equal. So what do you need to do on this question to answer this? Let's come back to our, let's wanna make it smaller. We go to our data. You need to sort your data. One, sort your data in ascending order. Number two, calculate the median by using the median position first. N plus one divided by two. And find the answer to your median. Three, calculate the mean, which is the sum of all observation divided by N. So let's do that together. Let's sort the data. You have three zeros, one, two, three. Did we have the mode somewhere? Let's see. Oh, yes, we did have the mode as well. So we'll need to also, number four, we'll need to identify which value is the range. So how many zeros we have? Make this bigger so that everybody can see. Three zeros and how many? Three one, one, two, three. And two, two, and two, fours, and five. Okay. So they should be 11, one, two, three, four, five, six, seven, eight, nine, 10, 11. So let's find. Let's find the median position. So they are 11 plus one divided by two, which means it's 12 divided by two. And it's on sixth position. So if it's on sixth position, we count one, two, three, four, five, six. That is your mean. Calculate, sorry, that is the median. Calculate the mean and all of them divide by how many they are. Okay, the mean is 1.8, 1.82. One. 0.8182. Okay, we can just keep it to two decimal. And the mode, I don't even have to worry about that. The mode. It's by model. It is a by model one, but the mode is zero and? One. And one. So now let's answer the question to choose which one is incorrect. Let's do my answer because when I click some way it, okay. So we have all the information we require. So let's answer the mean, the median and the second quartile are not equal. So what is the second quartile? So let's go to number five. We can say quartile two, we find the position by using n plus one divided by two, right? So the median and the second quartile are not equal. So are they? And also remember we're looking for the correct answer, right? So number one is, is it correct or incorrect? In. It will be incorrect. Only the mean and the median are equal. So median and the mean one and one point. So it's also incorrect. Only the mode is equal to the median. That's, do they, are they incorrect? It's incorrect. The median is five. Is it correct? It's incorrect. It's incorrect because the median is five, is one, not five. And the distribution is symmetrical. So we can check. It's not symmetrical. So is the mean equal to the median? The median, no, no. If they are, it will be the median. If they are equal, right? So are they equal? No, they are not. So our correct answer on this one is, E. It's E. Now we need to calculate the standard deviation. So to calculate the standard deviation, the formula to calculate standard deviation, I can put it right here, because this is the sample. So the standard deviation is the square root of the sum of your X observation minus the mean, squared, divide by N minus one. So we already calculated what the mean is and we found that the mean is 1.81. So you can use that information to calculate here. So you can say the square root of two minus, because it's the same data set, two minus 1.81 squared, plus zero minus 1.81 squared, plus until you get to the end, I'm gonna get there, two minus 1.81 squared and divide that by 11 minus one. So you can do that. Or alternatively, you can put your calculator to state mode. So let's do that with a case. You are not muted and it creates an egg. So let's put our calculator to state mode. I hope you're all using the case your calculator. If not, if you are using a sharp calculator, I will also give you the steps to the sharp calculator just now. But let's start with the case show. So we first put our calculator to mode state two and one, because we're only working with one variable, one minus one, it's for one variable. And we have the table. We capture the information by putting the data and pressing the equal sign. Zero, zero is also a data point, right? Don't forget about that one. Zero equal, five equal, one equal, one equal, four equal, zero equal and two equal. They should be 11 dataset captured. Once you have captured all your information, you can use your arrow to check if your data is captured correctly. And once you confirm, you just press the AC button. And you press the shift and you press one first step because we're reaching for that orange button. And you get this because we're interested in the standard deviation. If you're interested in the summation, I just want to demonstrate to those who didn't know the summation are all those sum X. If one time they ask you to calculate the summations, there is the sum X squared and the summation of X. So shift always, step and we interested in four. So you press four and we will understand the deviation. Otherwise we can check the mean, which is number two to see if we did calculate it correctly. So the mean equal 1.8, 1.81, we did calculate it correctly. So let's go to the one that we want. So it's the bar, which is four. And we're looking for the standard deviation. This is the population standard deviation. This is the sample size. So we're interested in the sample standard deviation. You need to be very careful. Sample standard deviation, population standard deviation can also look almost similar. It's sigma equals to the square root of the variant, which is your sum populations, your X minus your population mean divided by capital letter N, which is the population. You can see that the only difference is the minus one. So you need to pay close attention. Don't choose the incorrect one. So let's choose our standard deviation, which is four and equal. And the answer is option number C. 1.7787. Okay, I've done that for you. Those who are using Casio or Financial Calculator, just give me a second. That's the problem. I needed to open all the calculators that I have. Like I said, I cannot share my entire screen as I mitigate. Just give me a second. You can use the Shop Calculator. So with Shop, you also do the same whether it's Financial Calculator. So those who are using Financial Calculator, you will use the Enter button. And those who are using the Shop Scientific Calculator, you will use the M plus. So we first put our calculator to stick one mode and we press zero for SD. And it will look like this. Your calculator will be on state zero. And to capture the data, you're going to use the value and Enter if you are using a Financial Calculator. If you are using a Scientific Calculator, you're going to use M plus. Scientific, you will use M plus button, which is on the same position as those who are using Shop Financial Calculator. Calculator, you will use the Enter button. They are on the same location, right? They are above the close bracket. So to capture your data, you say to minimize you first, because I can't see my data, you will press two and you will press M plus. And you continue zero M plus. So you will say to enter, zero and four M plus. One M plus, zero M plus, five M plus, one M plus, one M plus, four M plus, zero M plus and two M plus. And it will say data set 11. If you make a mistake, not a problem, you just press second function mode, which will clear your data and you can start capturing again. On your case, if you make a mistake, you just go mode, shift, and you follow the same steps and capture the data again. Okay, so now you are ready. So you press the on and off, your data is stored in your Calculator. To calculate the mean, you will press button number four to calculate the sample. Standard deviation, you will press button number five to calculate the population. Standard deviation, you will press number six. If for some reason they ask you to calculate the variance, right, which is something that we didn't touch with the previous one. So the standard deviation, or your S on your calculator, you will use sigma X or SX. If you want to calculate the variance, this is the standard deviation. To calculate the variance of the population or the sample, you will just press the X squared button. That will give you the variance of either one of them, right? So it's easy. You will press this X squared button. On the case, you will press the X squared button. It will give you, for example, if I press the X squared, it will give me the value underneath the square root, which is the variance. Okay, so now let's calculate the variance, which is five. So because it's written in green, you will press alpha and button number five and press equal and that is one comma 7787. If I want the variance, I just press the X squared equal and it will give me the variance. If I want to go back to the standard deviation, you just take the square root of the answer and that will give you back. So it's just moving around and knowing which one is which. Okay, so that is correct. Are there any questions, any comments, any query, anything you wanna ask before I move, before I press the next button. Are we good? Are we happy? Let's see, are there any? No, nothing on the chat. Okay, so let's move to the next question. So the next one, I'm gonna give it to you to answer without me giving you the answers. The following data represent the salaries of a sample of dating employees in a firm and they are in decimals. As you can see, calculate the coefficient of variation. So the coefficient of variation, I can just give you the formula, coefficient of variation. Let's put it here. CV is your standard deviation divided by the mean multiply by 100. So all you can do is put your calculator to state mode and then calculate your standard deviation and the mean. You can do that together and put the data bigger so that everybody can see them clearly. Shift and mode, I need to go back. Mode, stat, it's two and one. And I can keep capturing the data, 26.5, 23.5, 29.7, 24.8, 21.1, 24.3, 20.4, 22.7, 27.2, 23.7, 24.1, 24.8, and 28.2, there should be 30 of them. And I can just double check. That I've captured all the values correctly. 2.7, 20.4, 24.3, 21.1, 24.8, 29.7, and 23.5, and 26.1. Okay, I'm gonna give you a chance while I also capture using the other calculator. Second function C8 should clear out all the numbers, 26.5, and plus 23.5, and plus 29.7, 24.8, and plus 21.1, 21, 21, and plus 24.3, and plus 12345620, 24, and plus. The challenge with the sharp calculator is you don't know where you are. If you make a mistake, you start from scratch. 22.7, and plus 27.2, and plus, so you will need to be very careful with how you capture your data with the sharp calculator. I hope I did capture them correctly. 24.1, and plus 24.1, and plus 28.2, and plus, because I cannot see the data or go back to the data. Whereas with the Casio, if I'm looking at the Casio, if I double check my data and I see that number one, I captured it incorrectly, it's still fine, I can replace the value by just pressing what it should be, and press equal, it will replace the same value there, and so on. With the sharp calculator, you cannot. So I hope you all have your information in, so let's calculate the coefficient of variation. I will start with the Casio calculator, so I'm gonna clear. On the Casio, we will shift, start, and remember, let's go back to the formula. Remember, our formula is standard deviation divided by the mean multiplied by 100. So we'll do the same. So four, we're going to get the standard deviation, which is button number four, divide by shift, that, four, and button number two, and I can say equal, multiply the answer by 100%, and that gives us 10.8, 10.7891, so which, nine, two, if we leave it to four decimal, into three decimal, it will be 10.789, 10. So the only one is D. So let's look at the Casio, should give us the same answer. So with the Casio, we also go on and off, and we go alpha, five, divide by alpha, four, equals, and multiplied by 100, equals, and that gives us 10.78919, which is nine, two, which is D. You need to practice, in order for you to know how to use your calculator properly, you will need to practice. If you do have a template, Excel template that summarizes or calculates this, you can use that, but you cannot wake up on Wednesday and hope that on Thursday, when practice on Wednesday and hope that on Thursday, when you write your exam, you will know what is happening. You want, it needs you to start practicing now, find more questions, especially in study, you need three to calculate using your calculator, especially for the mean and the standard deviation. You will have to do that. We go to next. Are there any questions? Hi, Lili, just a quick question. Regarding what your statement was earlier regarding Excel spreadsheets and calculating the formulas, is there a lot, I'm just assuming, because I've done a lot of Excel formulas for everything from the critical values to Poseidon and everything else in Excel. So if I can use that, that'll help me a lot in trying to calculate those formulas. Yeah, if you do have a template that has all that calculating everything else, you can use that. Fantastic, thank you very much. All right. And if you would like to also share that with the rest of the group, it will also help because what we're trying to do is to share as much information with everyone as possible so that everyone is equipped because we don't want to leave anyone behind. All right? As much as I support everyone, I also want everyone to support everyone, to support each other and not rely only on me. Okay, so. So it was a one more question. Yes. In the exams, I haven't tried this on the mock exam, but I have obviously multiple screens where I do my work. You know, one screen is for answering questions on multiple choice, and otherwise we have my Excel speech open. I've used IRIS before and IRIS obviously requests only one screen share, but the mock exams I see, it doesn't do anything with screen recordings, only your camera that it does everything with. All right, so what I will suggest you do is check with the mock exam that the lecturer has given you. Yeah. Fly with that and see if there are some discrepancies or anything that you pick up when you are doing, when you move across the screen as well, right? If not, then why not? So you can always even minimize and create a split screen like this, if I can put it this way. You can do it that way, where you have, the challenge is only this piece of other thing that is popping up, but you can do it this way because your camera is only to always stay on the side, right? So your camera will be somewhere on the navigation side or at the bottom, if you minimize your screen like this, it will show right here at the bottom, which as long as your face is not blocked, I don't think it should be any problem. But yeah. But you must try with the mock exam and see, because I think the more you try and see different ways that you can use your platform, the better to get prepared for your exam so that you don't get surprises during the exam. If you get a notification to say your screen needs to be maximized all the time, then you just make sure that you don't repeat that. Your screen needs to be maximized. Yeah. So let's move on to the next question, which is probabilities. Also remember that your questions in the exam will follow the structure of your study unit. So the first few questions will be from study unit one, two, three, and then you move to study unit four, study unit five, study unit six. There won't be mix and match where you've answered first the study unit 10 and then study unit, it will follow the structure. So if you know your structure of your module, you will know when you add different questions. So let's look at the basic probabilities. Suppose that A and B are mutually exclusive, such that the probability of A is 0.3 and the probability of B is 0.2. Which one of the following statement is incorrect? So you need to validate each statement. You cannot get away from that. You will have to test and validate each and every statement to see if it's correct. So let's do that. We'll start with A and move to other statements. Let's start with A. A says A and B are independent. So the first thing that we need to also take into consideration is what is given in the statement. So I'm gonna write there what is given in the statement. So the first step, we are told that the probability, oh, sorry, even A and B are mutually exclusive. So in your mind, somewhere you should know that if it's, A and B are mutually exclusive, therefore the probability of both of them being together, the joint probability of A and B will be 0. What else have they given you? They told you that the probability of A is 0.3. They also gave you that the probability of B is 0.2. So those are the information provided to you. So now let's use that information to select which one of this is incorrect in the exam. As soon as you get your answer, you move on. So we're looking for the incorrect so that I always remember what we're looking for. So A, how do we test that A and B are independent? So we need to check that. Let's use another. So we need to check that the probability of A given B is the same as the probability of A. If we can find that they are the same, then A and B are independent. So let's check that. Let's start with the probability of A given B, which is the probability of A and B divided by the probability of B. So in order for us to test that, we know that the formula, the probability of A given B will be given by the probability of A and B divided by the probability of B. So what is the probability of A and B is 0 divided by the probability of B, which is 0.2, which is equals to zero because any value divided, zero divided by any value will just be zero. So they are not independent. So number one is incorrect. Probability of A given B is not the same as the probability of A because the probability of A is 0.3 and the probability of A given B is 0. So event A and B are dependent. So we're looking for the incorrect answer. So therefore number A is the incorrect answer in terms of this. Number B, it says you need to find the probability of A given B is the same as the probability of B given A. We know that the probability of A given B will be equals to zero. Similar, the probability of B given A will be the joint probability divided by B. And we know that the joint probability is zero. So this one will be correct, right? Because the probability of B given A is given by the probability of a joint probability of A and B divided by the probability of A. The probability of A given A and B is zero divided by the probability of A of 0.3. Therefore, it will be equals to zero. So both of them will be equal. The probability of A given B, it's the same as the probability of B given A based on the information given. So that is correct. The next one, four. It's correct also. Yes, because the probability of A or B is given by the probability of A plus the probability of B minus the probability of A and B. And we know that this is zero and this will be zero comma three plus zero comma two, which is equals to zero comma five based on the information given, right? So number C is correct. Number G, we answered it because based on the information given, right? We did do that. The probability of A and B is zero. Okay, number E, what is the complement? It's correct also. One minus. The complement will be one minus the probability of A, which is one minus zero point three, which is equals to zero point seven. So the only answer that is incorrect is A. In the exam, if you find that number A is your answer that you are looking for, you just choose it and move on to the next question, right? Are there any questions? Before I move to next, you see no hands, no question in the chat so we can move on. Don't be shy to ask. Then we move on to the next question. Question eight, it's also probabilities. Also remember, you have 11 study units so you're writing out of 25 question, I think. If there are 25 questions, therefore it means every study unit might have at least two questions per study unit with the exception of few study units that might have three questions. So expect at least a minimum of two question pairs, study unit or every study unit. So there should be study units that you know better than the others as well. Okay, so the following table describes the breakdown of roller skaters or roller skate sale by age and sex. So age at the top and sex on gender. So they have given us. So what do you need to take into consideration? What are these venues that they have given you? Probabilities. There are probabilities. When you are working with probabilities and they give you decimals, you must know that those are probabilities. If there were whole numbers, you will know that those are your events. Yes, your events. Now on this table, what is missing? Is your totals, right? Yes, is your totals because on your totals, that's the way you calculate the probability of a simple event. So if our question there, let's go there. Just wanna move this to the side so I can have space. So if our question is like this and they tell you that you need to find the probability of F or O, you just need to rewrite this in the formula. The probability of A or B is equals to the probability of A plus the probability of B minus the probability of A, not O, N, the probability of a joint. NB, so we can rewrite this in this format. So let's rewrite it in this format. So the probability of F or B, F or O. So it means we can say if we take ASF, so it will be the probability of F plus the probability and we replace B with O minus the joint probability of F and O, see how easy it is. So because we don't have the total here, you will have to add the totals. So what is the probability of F? So you go to not F complement but F, female. You will take 0.109 and 0.112. What is that? Do the calculation, the total? 0.221. 0.221 and the probability of O. So you go to O, not the O complement, the O. It's 0.4130. Plus 0.413, okay? Minus the probability of a joint of O and F, which is 0.109. 109, so calculate that and see which one is correct. E. And the answer would be 0.525, which is E. That's how you will find the answers. Is there one more? We have a question. I've already pressed next, yes? No, it's fine, you don't have to go back. I just wanted to find out the probability of O. Why did you choose the one on the first? Because on the table they gave it to you like that, right? They told you O and O complement and they gave you F complement and F. So you just look at the letters that they've given you, the way included in there. All right, thank you, I include the O complement. Yeah, now we are in discrete probabilities. Which one of the following is not an assumption of a binomial? So you need to know also theory on some of this. So binomial, we're looking for the one that is not. So A, the number of successes in the trial are counted. You must remember that binomial is one of the discrete processes. So it comes from a discrete process. So you need to know what a discrete process looks like or a discrete variable looks like. All trials are independent. Think about a coin. If you toss a coin, when a coin falls on a head, does it have an influence on when it falls on a tail? Think about it in that way. Each trial must be classified as a success or failure. I don't even have to explain that. The name says it all. It's a buy, means two outcomes. All trials must be identical. And E, the probability of a success, is equals to 0.5 in all trials. Think about a dice. Think about if I have a survey that looks at how many number of people own a car in a household. Or how many cars every household owns. Will the probability of success in all the different scenarios I gave you be 0.5? Think about it. OK, so which one? Oh, the following are not properties of a binomial distribution. It's E. E equals E, because A, the trials comes from a discrete probability of process, which means discrete means counted processor. The trials are independent, because one outcome does not affect what the other, or does not have any influence on the other. And there are always two outcomes, a success and a failure. All trials are identically able. And E is the only option that does not fit the binomial distribution property. The number of cars owned by each of the families in the city is shown in the following table. So we are given a table with the number, which will be our cars, or will be our x value. The number of families, our frequencies, what they didn't give you is the probabilities. You will need to calculate the probability of each and every one of them. So we'll suggest what you do is your x will be given by 0, 1, 2, 3, 4. It will be our table, and you will just need to calculate the probabilities. In the exam, you don't have the time to calculate the probabilities. Go and look at what the question is asking, and only calculate those events that they are asking you to calculate. Calculate the probability that any family selected at random owns a minimum of one car. What is minimum in a mathematical symbol? It's less than. It's greater than or equal to. Less than or equal to. Minimum is less than or equal to. So if they say a family owns a minimum of one car, therefore it means we are only interested in those ones in this category. So we need to add all of them. So what you will do is to find the probability of x less than or equals to 1, you will just add, but you will need to add all of this value so that you can calculate your total. You need to have the total. Sorry, that is the other thing that I missed. So you will need this value here, which is the total, and you will need those two values and add them together. Then you will need to calculate the probability of x is equals to 0, plus the probability that x is equals to 1. So go ahead and add all of the values and tell me how many they are. 487. Wait, wait, wait. 6487. That is the total. So you can say 27. Divide by 6487, plus 1422. Divide by 6487. Or you could just say 27, plus 1422. Divide by 6487. And I hope everyone has calculated and you found that that is the total. The answer is D. Are we happy with B? Moving on to the next question. My system is very slow. The law enforcement agency claims that the number of times that a patrol car passes through a particular neighborhood follows a poison process with the mean of three times. Nightly shift, let x denote the number of times that a patrol car passes through the neighborhood during night shift. Which one of the following statement is incorrect? So it means we need to validate or evaluate every statement. So we need to do that. There are a couple of things that we will need to have handy. Your piece of paper, your tables. We need to go to Poison table. You will need to go and use your Poison. And those are the things that you will require to answer the question. So let's start. The first two questions that we need to evaluate, we're looking for the incorrect answer. The first two, we need the Poison table and the last one. So let's use the Poison table to answer one A, B, and E. So our lambda from the data, it says it's three times. It's given to you the mean, the lambda history, which is the same as your variance and so on. I'm going to put it that way. So let's go find the probability that x is less than 2. So that is the probability of x less than 2 will be the probability that x is equals to 0 plus the probability that x is equals to 1 plus the probability that x is equals to 2. So we'll have to add three things. So let's go to the table. We're going to look for lambda equals to 3. And there is our lambda equals to 3. We're looking for 0, 1, and 2. So we're looking for the first three. Add them together. Question 0.4232. Is it 4, 3, 2, or 4, 1, 3, 2? It's 4, 2, 3, 2. 4, 2, 3, 2. 0.4232. So let's see if that is our answer, which is correct. Now let's go to x is equals to 5. x is equals to 5. It's 0 comma 1, 0, 0, 8, which is correct. Let's go to e. x is equals to 3. x is equals to 3. It's 0 comma 2240. 2240. So a, b, c, a, b, and e are correct. Now let's see. I'm going to pull up my, the variance is equals to 3. Yes. And the average is equals to 3. Then there is no incorrect answer here. No, there is. Sorry, number 2 is incorrect. Sorry, I read it wrong. Number 2 is it's 0 comma 1, 0, 0, 8. I already draw. Sorry, my bad. This is the incorrect, b is incorrect. b is the incorrect value. And that will happen in the exam. So you need to pay attention to all the numbers, because they might be a little bit tricky. That's why when you write your exam, your stats exam, especially because it's in the morning, you need to have a well-groomed interest so that you don't make mistakes, silly mistakes. It's a game of concentration. Okay, we only have one hour. We are left with one, two, three, four, five, six, seven, eight, nine, 10, 14 questions that we still need to go through. Let's see how well we can do. I should have stopped the session at 4.30 so that it doesn't become long, but it's fine. I will split it before I upload it. Question 12. We are now in normal probabilities. Now, when you are in normal probability and when you move into sampling distribution probabilities, you will need to understand the key weights and key things that are given in the question as well, because the two look almost exactly the same. So a max on the chemistry test follows a normal distribution with the mean of 65 and the standard deviation of nine. Approximately what percentage of students have a score below 50? What is below 50? Less or equal to 50? Below? Less than 50. It's less than. So they're asking you to find the probability that X is less than 50. They have given you the standard deviation. They have given you the mean. The mean is 65. The standard deviation is 12. And you just need to find the probability that Z, we standardize that. Z is less than X minus the mean divided by the standard deviation. That's what you need to do. That is less than X is 50 minus 65 divided by the standard deviation of 12. Do the calculation. It's negative 1.25. Now, the other thing you need to remember always, like I always say, we will go to the table, right? The cumulative standardized normal distribution table. The probability of Z less than a value, which is a value in this case minus 1.25 will be the value we find on the table. The probability that Z is greater than a value, we will say 1 minus the value we find on the table, right? Will be 1 minus the table value. If it's between, if Z lies between two values, A and B, we say the table value for B minus the table value for A. Those are the things that you always need to remember, right? When we work with probabilities. So let's go to the table, cumulative standardized normal distribution table. The one with negative and positive values. So we're looking for minus 1.25. So we go there. We look for minus 1.2 and the five at the top, where they both meet, that's where we are. The answer is 0.1056, let's see. The answer is 0.1056, but it says what percentage? So you will need to multiply that by a hundred. So when you multiply that by a hundred, you get 10.56, right? The answer will be 10.56%. And if we round it up to one decimal, it will be 11%. 11.56. So we're looking for minus 1.25. It will be 11%. 11.56. 11% is E. We can forever to respond. Let's go to 12. It's still connected. Let me swap my connection, right? Which one of the following statement is incorrect? Is the session still recording? Yes. Which one of the following statement is incorrect with regards to standard normal distribution? A, the standard normal distribution has the mean of zero and the standard deviation of one. A normal distributed random variable X can be standardized according to the Z score. C, the standard normal distribution is symmetrical or is symmetric around the mean. D, the normal distribution is a discrete. E, the area underneath the curve is always equals to zero. One, which one is incorrect? Think about normal distribution where we talk about, you can see that we are adding numbers together, right? So from normal distribution distributed with the mean of zero and the standard deviation of one. That is correct, correct? And this is symmetrical. That should be correct. And the area underneath the curve, we always say this side is 0.5 and this side is 0.5 to the left and the area to the right is 0.05. Therefore, it means the sum of all probabilities will be equals to one. So A, B, C and E are correct. So therefore it means normal distribution, it is a continuous distribution. Agree? So if that is the case, then the only incorrect statement here would be option number would be D. So you always need to remember that normal distribution is A, continuous variable or a continuous distribution. Come, come, come, come. I don't know why it's so slow. You might be experiencing load shedding maybe around your area maybe or next to you maybe. I was load shedding at that point. Now load, it came back, it just came back not so long ago. I shouldn't be load shedding. I'm back from load shedding. And I don't know why, maybe because I'm connected via my phone and it's holding with only two bus and my Wi-Fi doesn't wanna connect. It's its own type. 50% of all college students attend within 50 mile of their home. In a sample of 500 college students, the probability that the sample proportion will be between 0.5 and 0.55 E. So we need to find the probability of the sample proportion being between 0.5 and that. So our population, based on this information, what is our population proportion? It's 0.5. It's 0.5 because it's given in the statement right there. So we need to find the probability that the sample proportion lies between two values, 0.45 and 0.55. So you can calculate them separately. So we can start with 0. We can start with the second part, which is z of, let's calculate it together anyway. So then we do all of them together. I'm just gonna make this bigger because I need to use the probability of z lies between your sample proportion minus your population proportion divided by their standard error, which is the population proportion one minus the population proportion divided by N. And we do the same on the other side. Let's substitute the values. Sample, 0.45 minus our population proportion, 0.5 divided by the square root of 0.5 times one minus 0.5 divided by, and our sample size was 500 divided by 500. So you do the calculations. 0.55 minus 0.5 divided by the square root of 0.5 times one minus 0.5. 0.45. Sorry, I'm not gonna stop at 0.55. What is 0.55? 0.55. 0.55 minus 0.5, then the second one is gonna be 0.45. Remember, I'm just converting the standardized. So I'm still on the standardized. I'm standardizing this side, which is this side. Standardize that side. I haven't gotten to the minus the other. So once you give me your values, so I'm still having them as in this way. I'm still having them in the format, the probability that Z lies between A and B. I'm still having them in that format. But this is 0.45, right? And this is 0.55, right? I've substituted them correctly. 0.5 is my population proportion. 0.5 is your population proportion. So give me the answer to the 0.45. It's negative 2.236. Only two decimals. Oh, negative 2.24. And on the 55. 2.24. It's a positive 2.24. Now I have A and B. I can do the probability of Z less than 2.24 minus the probability of Z less than minus 2.24. So let's go to the table and find on the positive side, 2.24. The positive side, we're looking for 2.24, not 2.24, 2.24. And sure this, so let's see, I think it's supposed to be 2.23 because the next number is two, not bigger than five. Yeah, let's go there. Let me double check, since I get two different answers. So let's use me as the break, the person who breaks the tie, which is 0.5 times one minus 0.5 divided by 500 equals, it's 2.4, the answer. Can you see? Negative 2.24. Yes, yes, the first one. Yes, and then let's do the second one. The positive. Let's do this, second one is minus 4.24. Must check how you calculated the second one. Oh, okay. Okay, so let's go to the table, 2.24 is 0.9875. I keep on forgetting, I'm using the white box, 0.9875, I hope, 9875. And then we go to the negative side, negative 2.24, negative 2.022, negative 2.2 and four at the top where they meet, which is 0.0125 minus 0.001.25. 0.0125, okay? And that is, do the calculations. 0.975. 0.297, that is the answer, so let's see, which one? 0.975, which is option C, happiness. Are we good? Especially, are we good? Especially with my method that I'm using, you could have, immediately at this point, split it into two from there, and did the probability of 0.55 minus 0.5 divided by the standard error, minus the probability of, minus the probability of 0.45 minus 0.55, it will still work it out the same way, because the answer will be like this, because this will fit into that. So it's the same process that I followed. They didn't break any rules. Okay, so let's move on to the next question. Today we didn't even take a break. Oh, gosh. Now it's even worse. Taking forever. And I just double check something quickly. It's gonna disconnect. You see, are you back? Am I back? Are you able to see my screen? Yes. Yes. I have two network connectivity, and it still refuses to connect with my Wi-Fi. I don't know why. Lizzie? Yes. Would it be possible, the file that you shared earlier, would it be possible to also add it to the eTutor site? Yes, I will add it to the eTutor site once the session is done. Did I create a folder called exam? I think so. Yes. I'll edit another exam preparation folder. Yes. I will add that, and I will also add the other document that another lady shared on the WhatsApp group as well. Okay, perfect. So moving on to the next one, after some couple of minutes of waiting, the proportion of eligible voters in the next election who will vote for the ANC is assumed to be 0.55. That is our population proportion. What is the probability that a random sample of 500 voters less than 0.49 will say they will vote for the ANC? So what they are asking you is to find out of the way and out what they are asking us. So we know what the population proportion is, is 0.55, and our sample size is 500 voters, and our P, they want us to find the probability that a random sample, not a sample, not X, but the proportion, which is P, is less than, less than is just a less than 0.49. We need to calculate that. So we normalize the probability of Z, less than P minus the proportion divided by the standard error, which is the population proportion, one minus population proportion. And this question is unfair because then I've got two population proportion questions. But when you do the trial on the actual paper, you might get a different question as well altogether. So in sampling distribution, we also have the standard deviations, right? And the means is not only the proportions. So it's just that on this, on our options that we got now, it only gives us the proportion P of Z, less than our P is always in the question is 0.49 minus our population proportion of 0.55 divided by the square root of 0.55 times 1 minus 0.55 divided everything by 500. Let's calculate and do the calculations. What is the answer? Minus 2.697. Yeah, round it off to 2 decimals. Minus 2.710. We don't have negative 2.70. So we need to go to the negative side of the table to go find our answer. So let's do that. So the negative side, we're looking for 2.70 and the answer is 0.035, right? 0.035. Is that what we will get? Because it's less than. Always remember that, right? For the less than, the value you find on the table will be that probability that you are looking for. I'm going to ask that we stop now and take some few minutes so that then I can sort out my network and also stop the recording and then we come back and we do the rest of the questions. As long as how long it will take us, are you guys good with that proposal? We take only a five-minute break. So let's do that. Before I move to the next question, I'm just going to leave it at this point. Let's stop the recording. Because I might switch off as well. I'm just going to stop this.