 Now that you complete the register, I have posted the link in the chat, if you can't find the link, let me know. Then I can repost the link again. Welcome again to your exam preparation. The purpose of today's session, like I said, is just to go through the question that's part of revision and also preparing you to go write the exam. So it means I'm not doing any explanation of the concept or content. We just going to go through the question and answer the question and look out for tips and tricks in terms of how to answer question and what to look out for. The main purpose is for you, for the preparation, actually not the main purpose for the preparation for you to know that you are ready to go write the exam. You need to find as many as possible different types of questions, especially from the past exam papers. Or you need to go through a trial paper that your lecturer would have given you, especially on the platform or on my UNISA or my module, use that to practice because that will look almost similar to how your exam would look like. Today we're just going to go through some of the questions based on the past question papers. So then I will also tell you what you need to look out for. The other thing I need to mention is that for STA 1610, the structure of your questions for your exam will follow the structure of your study unit. So the first few questions will come from study unit one, followed by study unit two, followed by study unit three, until you get to study unit 11. You have 11 questions, you are writing out of 25 questions. Therefore it means every study unit will have at least two questions. With the exception of very few study units where you might get three questions, study unit one, two and three are almost close and similar to one another, they are linked. So you might find that questions from those that units appears multiple times from question one up until question six. So you just need to know how to identify the questions and be able to pick key elements within the question. Like we've been doing and practicing during the academic session way, you need to make sure that you understand the question, identify the facts given in the question. Then also identify the things that will help you answer the question like the formulas, the tables and so on. And then do the calculation if the question requires you to do the calculation and then look for the answer. Because you're writing a multiple choice question, the other thing that you need to always be aware of. You need to either evaluate every options given of all the statements that you are given to answer the question or they might ask you a question and you need to calculate and find the only one answer from the option. If you need to evaluate the question in the exam, not the same as when you are doing the practices in the exam. So the first question that you evaluate and it answers the question, it is that answer you move on. Unless if you are not sure about your answer, you are not 100% sure that you can evaluate all five statements. But in the exam, you need to be aware of the time because you're writing out of two hours. So be mindful of the time as you evaluate questions. Sooner you get to the answer, move on to the next question and always be prepared and be mindful of that. Okay, with that being said, let's do STA 610. Unless if there are any questions relating to the process of today. So this trial paper that I'm sharing with you, there is also a detailed solution at the end, but I'm not interested in the detailed solution. I'm interested in making sure that you understand the questions and understand how to answer those questions. So I expect you also to talk to me. I don't want the session to be about me. I have been talking since from March, like I said from March or April up until September. So my voice is very tired of me saying things again and again and again. So I expect you now to reflect on things so that I can help you and guide you in terms of answering the questions. So let's look at question number one, which one of the following or which one of this variable is not a quantitative variable. So you need to know what a quantitative variable is. Number one, the number of rings on a person's finger, the height of a person, weight of the person, a person's opinion and about legislation of marijuana, age of the person. So which one of this is not a quantitative. So the key things that you need to look out for is all that. Which one of those one, two, three, four or five. Four. Number four, not a qualitative variable. Question two, the following is a stem and leaf display representing the amount of gasoline purchased in gallons for a sample of 17 cars. Do you still remember a stem and leaf plot is the plot that is ordered in array and it's got the stem and the leaf. And when you read the values, this is not one, five, eight, it is 41, 45, 48. Right. Based on this 17 car sample size on displayed on a stem and leaf plot, which one of the following statement is incorrect. We need to find the statement that is not correct. The range, you need to be calculating what the range is, what the mode is, what is the fifth largest observation and what is the median. And number five, I've already gave you the answer. So do you know what the range is? The range is your highest value minus your lowest value. The mode is the most frequent number. The largest, so you will need to do the count up until you get to the fifth observation and that will be your, sorry, from the bottom. The fifth largest observation is the fifth number from the bottom. So you just need to be able to know which one is that. And then you need to first find the position by using the formula position by using the formula n plus one divided by two where n is the number of observations and divide by two. If your median is on a whole number position. So for example, if it's one, then your median is on position one and you go and read the value that you see. If your median position ends with point five, then you need to take the average where the value of the median position falls between those two values and take the average, meaning add the two values and divide by two. Okay, so because this is a question where we need to validate each statement, so let's go on. What is the range? What is your highest value? 73. 73 and your lowest value? 41. 41, so do the calculation. 73 minus 41. 32. 32, so therefore it means this is correct. What is the mode? Which number appears more than the other numbers? That's the incorrect one because 52 was an appearance. Karim? Karim? Mode in this case will be 52, 65 and 66. The mode, they are 52, 65 and 66 because 52 appears twice, 65 appears twice and 66 appears twice. So this is the incorrect statement, but because we are doing the practice because you will never know in the exam you might need to go and find all the other values. So let's continue. The fifth largest observation, so you need to count, you can count the leaves so that you get to the fifth number and then read out the stem and the leaf of that. So what is the fifth largest observation? Say one, two, three, four, five, which is 56. So that means this is correct. What is the median? So we need to first calculate the position so we know that they are 17 because I don't have to go and count how many they are. They told us in the beginning. So you say 17 plus one divided by two, 18 divided by two is nine. Therefore it means it's on position nine. So you go to your leaf and you count until you get to nine. One, two, three, four, five, six, seven, eight, nine. So it is the median will be 61. And I already told you that the stem and leaf plot is an ordered array or it's data that it's ordered from lowest to highest, which is also known as the ordered array. And that's how you will evaluate every statement. Easy. And so three, a graphical technique applicable to qualitative data is, remember, qualitative data is data that you can put into categories. Which one? One, is it a histogram? Two, is it a stem and leaf plot or display? Three, is it an ogive? Four, is it a data plot? Five, is it a pie chart? Which graphic or which graphical technique or which graph can you use to display data that can be placed into categories, which we call categorical data or qualitative data? One, two, three, four or five. Anyone? Pie chart, it will be a pie chart because a histogram is for numerical values and there are no gaps in between a stem and leaf plot. We just went through a stem and leaf plot. The stem and leaf plot is for numerical values. And ogive, you can develop an ogive from a histogram by just taking the median of the values and the cumulative frequencies because an ogive is your cumulative frequency polygon. And a scatter plot uses two numerical values, your x and y variable. And we use the scatter plot, remember, in study unit 11. So you need to know these things. Okay, question four. The following data represents the number of children in a sample of 11 families. Number of children is numerical value in a sample of 11 families. So it means they're giving you 11 data points. And the size of 11 data points. One, two, three, four, five, six, seven, eight, nine, 10, 11. 11 families. Which one of the following statement is correct? So we're looking for the correct statement. Therefore, it means we need to evaluate all the statements. Number one, the distribution is not symmetric. Therefore, what they are saying is the mean is not equal to the median. That's what you need to evaluate. If the mean and the median are equal, therefore it means the data is symmetric. Number two is the median. We find the position first by using n plus one divided by two and then we'll find the median value. It means before you do that, you need to sort your data from lowest to highest before you can find the median. Your data needs to be sorted. There is only, only the mode is equal to the median. You need to go and check the mode, the most frequent number, the median, you would have found it in number two. Are they equal? So you need to check that only the mean and the median are equal. So when you answer number one, you would be able to answer number four as well, because you need to calculate the mean and you need to find the median. Remember the mean to calculate the mean is the sum of all your values divided by how many we are. The median and the second quartile are not equal. Do you know what the second quartile is in terms of the quarter and do you know what the median is? So if you're not sure about it, the second quarter, you find it by first finding the position of n plus one divided by two to find the mean, the quartile value. And once you get the position, then you go and find the value. The median, you go find the position and once you have the position, then you're going to find the value. So there is no need for you to do any calculation because I've just given you the answer on number five. Okay, so with that, first I will suggest that you sort the data from smallest to highest. Zero is also a number. So I'll give you a few minutes and then you will tell me the numbers. I will write them on here. Are you done writing out the number in order? I don't know. Are you guys saying you can't find the link? Let me post the register again. I've just posted the register again on the chat for those of you who can't find it. Okay, are you done? Give me the numbers in ascending order. Anyone? No one. Okay, zero, zero, zero, one, one, one, two, two, four, four and five. That's how you will write them in ascending order. Now let's answer the question. The mean, can we calculate the mean? If you add all of them, is the sum of all of them? Twenty. Twenty divided by eleven. What is twenty divided by eleven? Lizzie. It's Adele. Hi Adele. May I please ask that students who cannot access the register put their student numbers in the chat, but we need to make sure because of the personal information protection act when students put their student number in the chat that there are a way that other students can also see their student numbers. Yes, I did mention that to them. Okay, thank you. Thank you Lizzie. Thank you Adele. Twenty divided by eleven. Let's keep it to two decimal. One point. One point eight two. One point eight two, okay. Now let's also go and find the median. I'm not going to answer that question. I'm just going to do the calculation and then we'll come back and answer all the questions. The median, we need to first find the position. So let's find the position. So that will be eleven plus one divided by two. Six. Which will be equals to six position. Therefore the median is one. It's one. What is the mode? Let's go find the mode. Zero and one. Zero and one. Okay now let's answer the question. The distribution is not symmetrical. So it means the mean is not equals to the median. Is that true? Our median is one. Our mean is one point eight two. So option one is the correct answer. But we can also go and look at the rest of the questions. So we know that the median is one. So it means that question is incorrect. Only the mode is equals to the median. We know that we've got two modes. So it's also not correct. Only the mean and the median are equal. We just proved that with question number one. That they are not equal. The median and the second quartile are not equal. They are the same. It's one and the same thing, right? So also that's not correct because they are equal because it's the same thing. So you need to be able to know how to answer every question based on the calculations that you have done. In the exam, the minute you get to question number one and you answer number one, you don't have to even worry about the rest of the other question, right? Using the value in number one, calculate the standard deviation. You can either use your calculator by putting your calculator to state mode and capture the information. Or you can use a formula. So your formula will be s is equals to the square root of the sum of your x observation minus the mean squared divided by n minus one, which should be very interesting if you need to answer it in this way. Those who know how to use their calculator to put your calculator to state mode, you can do that. So our first value, we know what the mean is, is 1.8. So zero minus 1.8 to squared s, zero minus 1.8 to squared plus one minus 1.8 to squared plus one minus 1.8 to squared plus until you get to the end, not going to go to the end, five minus 1.8 to squared. And then you divide everything. So don't forget to do two to all the ones and the twos and the fours as well. They in between and we say 11 minus one. I'm going to use a calculator but because I didn't open my calculators online, I'm going to use my actual calculator to put my calculator on state mode. So if you don't know how to use your calculator on this, there are recordings that we shared in the previous sessions that we had when we use our calculator. If you are using Acacia, I can tell you the steps, but you will have to follow what I do. You first need to put your calculator to state mode by pressing the mode button and then pressing the one minus bar. So you will press state and you will do state SD which is button zero and then you will press one for one minus bar and you will have a table with X and to capture the data, you just press zero equal, zero equal, zero equal, one equal, one equal, one equal, two equal, two equal, four equal, four equal and five equal. I hope I have all of them. One, two, three, four, five, six, seven, eight, nine, ten, eleven. Then you will have up until eleven and once you are done, your data is stored on your calculator and then you press AC button which now you are ready to calculate and to calculate, we can even start with the mean. To calculate the mean, you press the shift button and you press button number one and it will give you a menu and you will go to the menu where it has var and I think it's number 31, number four, I don't know, but you will think of on your calculator, I'm not using a cashier calculator, you will press on the var and when you press the mean, it will give you 1.81. The mean will be the X bar. If you want to calculate the standard deviation, which is what we are looking for, you just press the AC button and press shift, start again and then press the var again, I think it's button number four and you will press the S X. So on your calculator, it will be S X and when you press the S X and press equal and your answer will be 1.7786 which is the same as in four decimal, it will be 1.7787, which is option two. So that will be, I calculate the whole thing, it will be the square root of 3.16 which is the variance and when I take the square root of the variance, I get 1.778. Long way or short way? On your calculator, using the shortcut using state mode, it's quick and easy to do, to do the calculation. But anyway, you need to practice. Okay, moving on to question six. Question six, suppose that now we are in the basic probabilities, maybe I should be asked to write in this. Why am I writing POTMAS? Basic probability. Suppose A and B are mutually exclusive. So they are telling you, so by that time, you need to already in your mind know that when two events are mutually exclusive, then it means the probability of those joint events is equals to, you know, zero. It's equals to zero. So those are the things that you need to think. Identify effects given in your question. Right? So we do that. Such as the probability of A is equals to set 0.3, which is 30%, and the probability of B is equal to 0.2 or 20%. Which one of the following statement is incorrect? So it means we need to be able to evaluate all the questions. So we can start with question number one. Question number one is a compliment. So they are asking, for you to calculate what is the probability of a compliment of A, which means you need to say 1 minus the probability of A, which is 1 minus 0.30. So this statement is correct, right? Because 1 minus 0.30 is equals to 0.70. Oh, why am I giving you all answers? Oh, me and Bert. Okay. That was the last time that I drew all the answers. I'll just give you the hint. You will need to do the calculations. Number two, the probability of A and B is equals to 0. Is that correct or incorrect? That is correct. That is correct because we already identified it when we were reading the statement. So those two are correct. Number three, the probability of A given B is equals to the probability of B given A. How do I know? I don't know, but we can definitely evaluate that. So the probability of A given B is given by the probability of A and B divided by the probability of B and the probability of B given A is given by the probability of A and B divided by the probability of A. These are straightforward equations of conditional probabilities that you should know. So let's find out if they are both equals to one another. Substitute the values. What is the probability of A and B? It's 0 because we said it right there. Even A and B are mutually exclusive. Divide by what is the probability of B? I'll give you the answer. 0.20 0.20 0.20 And therefore it means it is equals to 0, right? What's about the probability of B given A? It will also be equals to 0 because the probability of A and B is 0. Divide by the probability of A of 0.30 0 divided by another number is the same as 0. So it means they are the same. They are both equals to 0. So they are both equals to one another. So that is correct. We're still looking for the incorrect answer. Probability of A or B. Now it means we need to go and evaluate what is the probability of A or B? It's given by 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. But now since we know that A and B are mutually exclusive. So therefore this will be 0. It means it doesn't exist on the formula. When event A and B are mutually exclusive, the probability of A or B or the probability of A or B will be given by the probability of A plus the probability of B and that is what the addition root states. So is that equals to 0.5? Let's check. Yes. Probability of A is 0.3 plus 0.2. Therefore it's 0.5. That is correct. And if that is correct, then it means we are left with number 5. So number 5 says A and B are independent. How do we know that events are independent? We need to evaluate them. The probability of A given B will be equals to the probability of A. If two events are independent, it means the other event, the given event has no bearing on the probability of an event that you want to calculate. So then it means conditional probabilities, the probability of A given B will be the same as the probability of A. Or you can use the probability of B given A will be the same as the probability of B. So let's check because we need to check either one of them. What is the probability of A given B? The probability of A given B is 0. We did calculate it, we found that it is equals to 0. What is the probability of A? The probability of A is 0. Therefore they are dependent on one another because they are not equal. This is if they are independent. So they are not independent. They are dependent events. And you can also check with the second one if you want. You can use either one. So the incorrect statement is E. Moving on, basic probabilities. I need to go back one step a little bit because if you look at this, question 5, 4, 3, 2, 1. They relate to study unit 1, 2, and 3. You can see that. Study unit 1 doing the introduction of statistics. Study unit 3 plus study unit 2 when we talk about visualization which they mainly if it's part of the visualization. But study unit 3 also touches on the measures, calculating the measures, numerical measures. And study unit 2 and study unit 3. So you can see that they are almost interlinked to one another. So basic probabilities. There are two questions. So question 1, we have dealt with something. I don't know what I put it wrong with. There we go. Question 7, it's also part of the basic probabilities. So you are given the contingency table. What is missing with this contingency table? So the table shows the distribution of 25 students according to gender and preschool experience. And they gave you the event. What is missing here that we will need? Totals. Totals, very important, especially when they gave you events. You need to make sure that you quickly calculate the totals because you will need the totals. Okay, so let's calculate the total. How many are male? 8 plus 6. It's 49 plus 1. 11. 14 plus 11 should be 25. It should be equal to the sample space. 8 plus 9. 17. 6 plus 2. And 8. And 8 plus 17 should give us 25. Okay, so now we've got all what we need. We can be prepared or we are prepared to answer the questions. So it means we need to evaluate each and every statement. So let's look for our incorrect statement. Remember the following. The probability of an event is given by observation satisfying the joint event divided by N. The probability of A and B is given by the joint event. Observation satisfying the joint event divided by the sample space. And our sample space is N. She's always the grand total, right? And we need to always remember the probability of A or B. It's given by the probability of A plus the probability of B minus the probability of A and B. And the probability of A given B is given by the probability of A. And let me not use the end because it looks like I'm writing a sentence and use the sign. A and B divided by the probability of C. So those are the main formulas that you always need to remember. Remember the data that they gave you inside is your joint event. So it's those with an end, the probability of A and B. The outside, like 17, it is your symbol or marginal events, right? And what do we mean by marginal events? Marginal events are addition of two joint events. For example, 17 is made up of joint event, male and preschool and female and preschool. So those are the things that you always need to remember when you answer basic probability questions. So now let's answer this question. What are the events satisfying preschool? What is the probability of preschool? So preschool, it got less of whether they are male or female because it's a simple event. You're going to use the event satisfying the simple event preschool divided by the grand total. I'm doing one and you need to do two. So that will be 17 divided by 25. Does it give us 0.68 during the calculation? Yes, it does. So that is correct. Probability of female, what is the probability of female? 0.44. Are the events satisfying? 11 divided by 25. 11 divided by 25. Does it give us 0.44? Yes. That is correct. So you just said 11 divided by 25 should give us 0.44. 20 event preschool and male. Are the events satisfying male preschool or preschool and male? Events satisfying divided by the sample space. So there are eight divided by 25. Is it equal to 0.32? Yes, ma'am. So that is correct. Number four, we are looking for no preschool of female. How do we define that? 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. So therefore we need to do the probability of no preschool plus the probability of female minus the probability of no preschool and female. So what is the probability of no preschool? 8. 8 divided by 25. Probability of female? 11 divided by 25. And divide by 25 minus the joint probability of no preschool and female? 11. No preschool and female? That's two. Divide by 25. So because they all have the same common denominator, you can say the common denominator is 25 and you say 8 plus 11 is 90 minus 2 is 17. 17 divided by 25? 0.68. 0.68, which is not possible. And the probability of preschool given female, if we know the formula says the probability of A given B is given by the probability of A and B divided by B. Therefore we can say the probability of preschool and female divided by the probability of female. And that gives us what is the probability of preschool and female? It's 9 over 25 divided by the probability. Can I ask that we ever play music, please do not switch off the music because that part of the song, if it's copyright song, you won't be able to publish the recording. So it means you would have made everyone who hasn't attended the session miss out on those who want to listen to the recording later on. You won't have, you can't publish with copyright music in. Okay, so the probability of female, so make sure that your music is off. Probability of female, it's 11 over 25. And if we apply math, this is the same as 9 over 25 multiplied by 25 over 11, which is the same as 9 over 11 because 25 and 25 will cancel out. In math, we apply, keep the fraction, the first fraction, change the division to a multiplication and flip your second fraction. What was at the top comes to the bottom, what was at the bottom goes to the top. So your numerator and the denominator, they swap places. And that's what I did. I kept 9 over 25, change the sign to a multiplication and flip my 11 over 25 to 25 over 11 and simplify 25 and 25 cancels out. And we are left with because it would be equals to one and equals to one, nine times one is nine and one times 11 is 11. So what is 9 divided by 11 should be 0,8218. Is that correct? Yes. Yes, ma'am. Hi. And that's how you will answer the probability question, but in the exam when you get to four, you stop right there and move on to the next question. So question eight, you can see that only two questions dealt with basic probabilities. Now we are in discrete probabilities. How do I know that? They gave you the table with x and the probability corresponding to that x observations. A BSE student in statistics applies for five different jobs. Let x represent the number of offers given below are the probabilities of the number of job offers. If they receive zero offer, one offer, two offers, three offers, four offers, five offers, so that BSE student is very clever to receive five offers. Which statement is incorrect with regards to this? And when you get to the discrete probabilities, you always also remember the sign, right? What does it mean? The at most, at least, the less than or equal, the between, inclusive and exclusive and so on. So those are the things that we need to always remember. So let's evaluate the statement. What does at least mean? In terms of a mathematical sign, is it a less than? Is it a less than or equal? Is it a greater than? Is it a greater than or equal or is it an equal? Those are very important to remember. What is at least? It's greater than or equal. It is greater than or equal. So therefore it means it must include two or any of the bigger number. So if we talk about that, it means we need to add all these values. So the probability, the probability of X greater than or equals to two will be the same as the probability of X is equals to two, plus the probability of X is equals to three, plus the probability of X is equals to four, plus the probability of X is equals to five. Or, alternatively, you can say the probability of x is greater than or equals to 2 is the same as 1 minus the probability of x, and I'm going to put it into bracket, x is equals to 0 plus the probability that x is equals to 1, because you just need to do the complement of those four. Instead of adding four numbers in the exam, you can just add the two and subtract them from one. They will give you one and the same thing. So, what is the un-term? 0.786. 0.786, which is not correct, right? Then number two, it says, what is the probability of x less than or equals to 5. So, it means it's all of them. What do we know? If it's the probability of all of them, it's the sum of all probabilities, right? And in terms of basic probabilities, the sum of all probabilities, they are always equals to 1. So, if I have to add all of them from 5 to 0, so it means I expect the probabilities to be equals to 1, because you can also double check it. It's 0.025 plus 0.138 plus 0.299 plus 0.324 plus 0.175 plus 0.039. I have to add all of them. It is not equals to, and so that will be incorrect. Number three, just going to reduce the font a little bit so we are able to see all the questions. Number three, it says the probability that x lies between two and four with two exclusive and four inclusive. So, what does that mean? That means two exclusive, because it does not have an equal sign. It means we're not going to include two, so it's any number above two, but it also says it should be any number less than or equals to four. So, therefore, it means from four, any number up to there. So, therefore, our answer there should be the probability of x is equals to four, plus the probability that x is equals to three. That's the only probabilities you need. What is that? Remember, we're looking for the correct answer. That is correct. And that is the answer we are looking for. I'm not going to do the expected value and the variance. I'm just going to give you the hints in terms of how you answer those questions, but you will need to go and practice on how to calculate them. So, question four, it says you need the expected value. So, the expected value of a discrete probability is your x observation times its cross-bonding, there should be a sum, is the sum of your x observation times its cross-bonding probability. What does that mean? It means you need to say zero times 0.39, which is x times px, which will give you zero. One times zero comma 175 will be zero comma 175. Two times 0.624 will be 0.648, and like that, until you get to the end and then you do the total. This will be the answer you get when you do the total, ending all of them up. It should give you your expected value as the expected value. How do we then find the next question, which is the standard deviation? The standard deviation, which is sigma, oh, actually it's not the standard deviation, this is the variance. Finding the variance, the variance is the sum of your x observation minus your expected value squared times the cross-bonding probability. What does that mean? It means you're going to say zero minus the answer you get from the expected value, square the answer multiplied by the cross-bonding probability, plus one minus the answer of the expected square, multiply that with the cross-bonding probability, plus two minus the expected value squared, multiply that with until you exhaust all of them and add all of them up, because this summation means aiding up. That will give you your next question. I'm not going to do that. You can practice. Moving on to question. I have a question. So say in the event they don't give us, if you can just move a little bit further yes. So say in the event they don't give us these probability values at the bottom, they gave us actual values, the whole numbers 25, 19 and so on and so on. Would we then have to work out what the probability of each one of those points are before we could go answer those questions or would we get a question like that? Yes, you will have to add all of those frequencies because they would have given you the frequencies and then once you have the frequencies, these probabilities remember is just a proportion. The frequency divided by the total will give you the probability and then you will create your own probabilities in your assignment. In your assignment, you had a question where they tell you they don't give you the probabilities, but they tell you that if they all have the same probability distribution. So it means if there are five, they will have all similar probability distribution, which means one, two, three, four, five, six. So here you have six. So you will just say one divided by six and that will give you your probability. They will all have the same probability of one over six and then you can use that to answer the same question. Okay. Thank you. On to question line. I'm looking at the time and I'm thinking we still have to get to question 25. We are way behind that, but you get the grasp of what is happening. So we are on discrete probabilities. So one question. So discrete probability is one of those study units that you might get three questions because in under discrete probabilities, you have discrete probabilities, the properties of the discrete probabilities. Then you have binomial probabilities and you have Poisson. So you should expect at least three questions from there. So we've dealt with discrete probabilities. Now we're going into Poisson, sorry, binomial. So this is by normal and they will follow the same pattern. So they will not put binomial first before they put discrete probabilities. They will always be like this. So it will be discrete binomial Poisson. So which one of the following statement is incorrect? So it means on this one, you need to know and understand your properties of a binomial distribution. So for the binomial distribution, there are n independent trials. Is that correct? Or are we looking for the incorrect statement? So you must always remember that the outcomes are always independent. So you cannot fail and pass at the same time, right? Because it's failure or success in binomial. There are two outcomes. So it means the trials needs to be independent. So that is correct in terms of the properties of a binomial. Number two, the probability of success, which is the probability of a binomial trial will always be represented by, oh, of success is pi and remains constant. So it cannot be constant. I don't know why they say it's constant. What does that mean constant? What does that mean? But the probability of success is pi. That is correct. The expected value of a binomial is your sample size multiplied by the probability of success, which is correct. The distribution is continuous. That is not correct because it's discrete probabilities. The distribution is discrete. It comes from a discrete process. Remember with numerical values or quantitative data, you have discrete and continuous. Continuous, it's dealt with in normal distribution and sampling distribution. You must always remember that, right? So the incorrect statement here will be number four, because the distribution should be a discrete distribution. There are two possible outcomes. I already touched on this. It's a success or a failure. So that is question right. I'm sorry I answered that question on your behalf, but I was looking at the time before I, yeah, we spend more time trying to figure out what's what. Then we come to another question. It's also binomial. Binomial probabilities as well. Home security has a reliable rate of 45%. In a sample of nine houses, which is n equipped, so this is your pi, equipped with this system experience and attempted backlaries. The probabilities are independent from one home security to the next. Let X represent attempted backlaries. Which one of the following statement is incorrect? So now you need to also use the table. So for those of you who haven't attended any of the sessions that I offer, under the acolyte sessions that we do for statistical literacies, there are nodes. Under those nodes, there are tables. It's written STA 1610 and 1510 statistical tables. Those are the table that you can use and you can also use the same table when you go write the exam. You are allowed to have the statistical table. So the table that I'm talking about looks like this. So you can use that. So you need to go to the binomial distribution table and it is written binomial probabilities if you need to know how to read this table. Very complex, but easy to understand if you practice. So the table has probabilities at the top from 0.01 to 0.5. And at the bottom there are also probabilities at the bottom, but they are on the second page. They also relate to this table. So 0.99. So this one is 0.99 and this is 0.50 at the top and the bottom as well. And the next one is 0.55. So there are probabilities at the bottom and at the top. So how you read the table, the probabilities at the top, you read it with the n values and x values on the left, top with the left bottom. You will read it with the right n and x values. And you can see that your x values at the bottom, they go from below to the top, right? They are in reverse. Whereas on the left, they start from the top, they go to the bottom. So you need to know how to read the table. So going back to our question, our probability of success is 45%, which is 0.45. So on the table, we are going to use 0.45, which is the last column. So since it's at the top, we're going to use n on the side, on the left. They are both on the side. It says a sample of nine homes. So on the table, we're going to look for n of nine in order for us to help to answer this question. So n of nine and at the top. So here at the top, there are probabilities, even though they are missing, you must also remember that there are those ones at the top. So we know that this is 0.50 and 0.45. So the second one should be 0.45. So let's see, second one was 0.45. So then it means we're going to use the last second column. So on nine and we're going to use the last column and our x values are from 0 to 9. We need to validate each statement and find the one that is incorrect. The probability of x is equals to 2. It means exactly 2. 9 x of 2. And you just go to the x. 0.0. Let's make it a little bit bigger. And the answer is 0.1110. 0.1110. So that is correct. The probability that x is equals to 1 equal to 1. It is the second row, which is 0.0339. 0.0339. The probability that x is equals to 0, you go to the first row, 0.0446. 0.0, going back. 0046, 04060. That is the incorrect one. The probability that x is less than or equals to 3, you will just need to go to 3. And let's remove this. Go to 3. And that is 3. And you're going to add all of them. You need to add all of them. 0.046 plus, 0.0046 plus, 0.033. 9 plus, 0.1110 plus, 0.2119. It's 0.3614, which is correct. The standard deviation, you will need to use the formula. It's the square root of your pi. Actually, we must start with n. The sample size times the probability of success times 1 minus the probability of success, which is equals to the square root of 9 times 0.45 times 1 minus 0.45. Square root of 9 times 0.45 times 1 minus 0.45, which is equals to 1.492, which is the same. Good. Are we happy? Are we good? Yes. I just wanted to quickly get the name of the table again. Sorry. This table, it's the binomial table. So in your past exam papers, if you went somewhere online and you downloaded the past exam paper, they are always at the back of the exam paper. You will find these tables in there. Thank you. This is Poisson. Poisson statements will be stated that you are now looking at Poisson. The binomial, they don't always say that it follows a Poisson distribution or something like that. You need to make it up. So this is Poisson, a law enforcement, and you can see that here we have four questions on discrete probabilities. But I guess because this is a trial paper, and probably because it's 2012, it was set by somebody else. But bear in mind that at least some of the questions you might have two or three questions from that study unit. You always get two questions from that unit. So a law enforcement agency claims that the number of times that patrol car passes through a particular neighborhood follows a Poisson process with the mean of three times the night shift. Three times, therefore it means our lambda is equals to three. Let X denote the number of times that the patrol car passes through the neighborhood during a night shift. Which one of the following is incorrect also for binomial? Please use the table. Oh, sorry, for binomial, use the table. What I forgot to mention is that you can also use the formula which is your NCR, which is your combination, times your probability of X times one minus the probability of success and minus X. You can use this formula to calculate the probability of equal. The minute you get to the probability of less than or equal, then it means you will have to use this formula three times. Because if you're doing the probability of X less than three, you will have two. How many did we say? One, two, three, four. You will have to calculate that formula four times to get the answers, all of these answers using the formula. Similar with Poisson, calculate the probability. You can use the formula if you have time. Otherwise, you need to use the table. Because if you calculate the probability of less than two, then it means it's zero, one, and two, three times. You will have to calculate this three times to get the probability of zero, the probability of one, the probability of two, and add them together. Otherwise, you use the table. Easy. Already they have calculated this formula for you. What you need to also remember is your lambda is the same as your mean is the same as your variance, but it's not the same as your standard deviation. The standard deviation is the square root of your average. So if we need to answer the question, which one of the following statement is incorrect? Number one, is it correct or incorrect? Think about time. Is it correct or incorrect? The statement is correct because I just gave you the answer there at the top. Right? I gave you all the answer. This is the variance. What is the probability that x is equals to three? So we need to go to the tables. Go look for Poisson table. And your Poisson is divided by the lambda. So every lambda table has its own set of x value. So you will see that with this table that starts from 0.1 up until one, the x value starts from 0 to 7. You need to pay attention to that because it's very important, especially when you calculate the probabilities of greater than or equal or at least or less or greater than. You need to take that into consideration so that you don't miss out. So we're looking for lambda 3. So it's table 3. And you can see that the x values on table 3 starts from 0 up until 12. It's very important because for 3 you can see that 12 has 0.0001. So you need to pay attention to that. So the probability that x is equals to 3, you just go to the table and the 3x is 3. You just read the value. That is your answer 0.3240. Is it right? Otherwise you can calculate. Yeah, by substituting the value of lambda is 3. So the probability that x is equals to 3, you will say e to the power of minus 3 times 3 to the power of 3, because our average is 3 anyway, divide by 3 factorial. And we will calculate that. But we know that this is correct. The probability that x is equals to 5, you go to 5. It is 0.1008. 0.1008. Let's go double check because the values can be very confusing. You can see that it is incorrect. So that is the incorrect statement and you stop right there and you move on. Okay. And I can see, I think this is the one way they have a typing error. Right? Because if you look at number 5, it says, oh no, there's no typing error because number 5 says the variant. What is the variant? Number 5? The variant is 3. So that is correct. The probability that x is less than or equals to 2, you go to the table, you go to 2. Let's remove all this. We have to go to x is equals to less than 2. So it means you add all of them. So we will say it is 0.04498 plus 0.1494 plus 0.2240 which is equals to 0.4232 which is correct. So option 3 is the incorrect one. Question 12. Now we are in normal distribution. 50% of the college students attend school within 50 miles of their homes. In a sample of 500 college students, the probability that the sample proportion will be between 0.45 and 0.55. Yes. So this is not normal distribution. This is sampling distribution. So on this trial paper, they didn't follow the normal distribution. So this is sampling distribution. So it's study unit 7. So probably because study unit 7 and study unit 6 are almost linked to one another probably maybe hence they are also interchangeable like this. So we're dealing with sampling distribution of the proportion. So they told you that 50% of their students, so it means our probability of success or whatever we call it, the population probability. In a sample of 500 which is our n and our p small p is given by that. So we need to find the probability that small p is between 0.45 and 0.55. So it means we're going to have to standardize this by using p minus the proportion divided by the standard error which is the population proportion 1 minus the population proportion divided by n. And you do the same p minus the population proportion divided by the square root of your population proportion 1 minus the population proportion divided by n. So we need to find the probability of 0.45 minus our population proportion is 50% which is 0.5 divided by the square root. Hey, why is my thing doing zigzag? Like now I've got arthritis. Hey, okay, maybe it's the pen, it's not me. 0.5 times 1 minus 0.5 divided by 500. 0.55 minus 0.5 divided by the square root of 0.5 times 1 minus 0.5 divided by 0.5. p will be equals to, we just need to calculate the values. Have you calculated the values? Let's see 0.45 minus 0.5 divided by the square root. I'm using a casual calculator, so let me not say the words the way I'm seeing it because I'm just doing my fraction calculator calculations. So if you are not using a casual, you will have to do things step by step. So on the 0.45 is minus, I need to keep only 2 decimal, 2.24 because the answer is minus 2.3606, but I need 2 decimal and you will see why we need 2 decimal. And on the other, I shouldn't have removed everything, 0.55 minus 0.5 divided by the square root of the fraction, just 0.5 and the answer this side is positive 2.23, 2.4, 2 decimals, 2.4. Okay, so now what you also need to remember when we deal with probabilities are the following. When we go to the table, the probability that z lies less than a value, the value we find on the table will be that probability that we are looking for. If it's the probability of z greater than a value, when we go to the table we need to say 1 minus the value we see on the table. If it is between, if z lies between two values a and b, we're going to say the table value for b minus the table value for a. So we're going to apply the same rule, right? The same rule that I'm applying here on this question because it is between. So this is our a and this is our b. So we need to go to the positive side on the table. So we go to the cumulative standardized normal distribution table. She's the first table and we're going to go to the positive side. It has the cumulative standardized normal distribution has two sides, the negative side and the positive side. So we're going to go first to the positive side and we'll find 2.24, 2.2 and then 4 at the top. So 2.24, we find 2.2 on the left and the last digit, we always find it at the top. So where they both meet, that is the probability we're looking for, which is 0.9875. So that will be 0.9875 minus and we go to the negative side and minus 2.24. On the negative side, we do the same. We will find negative 2.2 and 4 at the top where they both meet. That will be the probability we're looking for, which is 0.0125. Which is 0 comma 0125 and we just subtract 1 from the other. 0.9875 minus 0125, 0,0,0,1,2,5. And the answer is 0 comma 9750, which is option 4. That's how you will find the probabilities. Now with the next question, where is question dating, where they give you statements and the values, therefore it means you need to evaluate each one of them. So based on this rule that I just told you right now, the basic skill of how you retable values, especially for the normal distribution, you're going to apply the same. So number one, you're going to find the value on the table under minus 1.44 and subtract that from 1. So let's go and do that. Minus 1.44, so we'll find minus 1.4 on the side and 4 at the top and we're going to take this value and subtract from 1. 1 minus 0.0749 and that should give us the answer that we're looking for, which is correct. And you can stop right there and move on to the next. You will need to go and validate the rest of the values. So z of less than 2, I'm not going to do the rest of them. Z of less than 2, you just go to the table because the side says less than, that will be the table value, less than the table value, greater than 1 minus the table value. So you'll go and find 1.59 and the value you see on that table, you subtract it from 1. This one, you will use the table value to validate. Okay, moving on to question 14. I'm aware that we left it 12 minutes and I think if we can get to question 15, 16 because the other question is not long ago that we dealt with those chapters, like study unit 10, 11 and 12, you have dealt, study unit 11 and study unit 10. You did that recently, so it's not going to be a problem. Okay, so those were, I'm going to say this is also normal distribution, but it can also be sampling distributions. So at least there are two questions from there. Okay, it is important for an airline to know the appropriate total weight of the baggage carried on each plane. An airline researcher believes that the mean baggage weight for each adult is 60 kg. To test his belief, now we're testing beliefs, so we are in hypothesis testing. This trial paper does not follow the structure of your study unit, so I've already made that connection because now after doing sampling distribution and doing the sampling distribution and doing the normal distribution, you should be moving into confidence interval, but let me not state that this is a hypothesis because I need to read the entire question. It might not be hypothesis testing, it might be still normal distribution. To test his belief, he draws a random sample of 50 adult passengers and weigh their baggage. He finds the sample mean to be 57.1 kg. If he knows that the population standard deviation is 10 kg, the probability that the average baggage is more than 55, so it means we need to calculate the probability that the sample mean is greater than 57.1. So based on that, because we have the sample size n, so we're still in the sampling distribution. We're still in the sampling distribution, so therefore it means we need to calculate or standardize the greater than sample mean minus population mean divided by the standard error, which is the population standard deviation divided by the square root of n. And since they gave us the population standardization, so it makes everything easy. So calculate the probability of z greater than our sample mean is 57.1. Our population mean is 60 because that's what they gave you there. This is your x bar. Divide that by your population standard deviation of 10, which is sigma. Divide by the square root of 50. And the probability that the z is greater than and we just do the calculation. 57.1 minus 60. The answer is minus 2.05 minus 2.05. So because it is the probability of a greater that, so to find the probability of the mean greater than 51.7 or 57.1 is the same as 1 minus the probability of z less than minus 2.05. So it means I must go to the table to find minus 2.05. So minus 2.05 be 0.0202. 0.0202, which then is equal to let me double check the values at the top should be 5, right? And minus 2.0. So I did 1 minus 0.0202 is equal to 0.9798, which is option 1. Option 15. We are in hypothesis testing because I can see that we're doing hypothesis testing. So in testing the hypothesis, so you need to know the six steps of hypothesis testing. So but anyway, the null hypothesis states that the proportion is or the population proportion is 0.40 and the alternative is greater than 0.40. So I'm doing a one-tail test. This is a one-tail upper, upper test. And at alpha or level of significance of 5%, which is the alpha value of 0.05, if the sample proportion is 0.5 and the sample size is 49 and the standard deviation for the proportion is standard deviation for the proportion, it means you need to calculate the standard error. Standard deviation for the proportion is the population proportion divided by the square root of n. So our, oh, I'm talking about proportions, right? No sample mean. Let's go back. My bad. Sample proportion and here they gave you only the the sample proportion. So we're going to use the square root of your population or your sample proportion one minus sample proportion divided by n. So the square root of 0.45 times one minus 0.45 divided by our sample size of 49 and that will be, that is equals to 0.0711. Probably because they would have calculated this regularly as well. I'm going to assume that question 15 they would have said is number two. Let's see. Let's double check the access question. Sorry, my bad. I see where I went wrong. They did give us the population proportion. I used the sample proportion instead of the population proportion. So this will be your population proportion comes to your popular because they gave us the hypothesis testing where the population proportion is given. I'm thinking of your assignment where the population proportion was not given. So you needed to use the sample proportion. So that we delete the five zeros everywhere and seven zero. It would be 0.06999985 which is option two. Okay. And using the same statement, consider the following information on question 15 what will be the critical value. So remember now the critical value and pay attention to detail. We're doing a one tail, upper tail, one tail test. So therefore our critical value is z alpha. If we were doing a two tail, we will divide by two. So we're doing a one tail. So it will be z of 0.05. So it means we need to go to the z table inside the z table. Yeah, on the right place. Look for inside the table. Going to look for 0.05. 0.04905 will be one of those two values. And therefore it is an exceptional one, which the critical value will be one minus one point or because it's in the upper tail, it will be positive. 1.645. It's 1.645. And because it is in the upper tail, because it's greater than that, so it will be in the positive. 1.645. Your mean is at that point. Option two will be the correct answer. You need to pay attention to the information given. So if the sign here was a two tail, then you will have two regions of rejection. Your critical values, there will be two of them. But if you would have divided your alpha by two, if it was in the less than or the lower side, you would put it on the less than side and it would be negative. Pay that close attention to that. So our type actually is up. I will post this where the notes sections are, but at least this gives you some ideas in terms of how the questions are. They look almost exactly the same as what you did in your assignment. So if you practice, you won't go any wrong with anything. Okay, so before you leave, I just want to also share in their chat where you will find multiple desktop open. I need to navigate to my desktop so I can get the things I want to give to you. This is going to be one. Just want to share in the chat. For those of you who don't know where to find, actually I don't want to share the, because I want to share the recording. The recording. Okay, in the chat, I'm sharing the link to the way you will find the recording. But on that note as well, I want to go there so I can share my screen. I'm sharing it up there so that we can talk about that as well. So on the link that I shared with you, if it's for the first time seeing what I'm showing you, probably you should get excited because this is where you will find the recordings for all the sessions that the Western Cape Paro offices offers. So we've got tutorial classes. If you do any of the modules that are here, you can go and watch the recordings. If you do in the writing center, that's where the other colleagues in terms of academic literacies in the writing center for the English and other facilitation that they do, you can find also the recordings. So if you do English, you will find tutorial classes as well as the Akalit classes. So there will be two, so you can watch both of the recordings because they do different things. Tutorials are different from Akalit. And I'm in the numeracy center. So in our site, you will go to a basic statistical literacies, which is the session that we are on. If you click on that link and you will see that since from April, we had several sessions. All the recordings are uploaded. It takes time to reflect on the other recordings. So you will see today's recording as well. It will appear here next week or even during this week. So you should be able to see all the recordings for the previous sessions. I don't know why this recording is not working. I will ask them to check that. But all the sessions, you can see that we do have all of them. So when I was explaining, if you don't know how to use your calculator, this session, we are using a casual calculator, financial math calculator, and the sharp scientific calculator. You can come to this on the 29th of April to see how we answer those sessions. But those are the links I wanted to show you if you go up again. So on the same link, when we click on open class notes, you will find all the notes for all the sessions that we had. And I'm going to post inside here as well. So the ease, the tables, we are the tables. I know that I did post them. So here are the table. So you will find the table. You will also find a document on functions to use to calculate the mean, the standard deviation and all that. You can follow the document as well. It explains all different calculators, the casual calculator and the sharp calculator. You can find the information there. We do have also some templates that we used for study unit 10 and study unit 11. You can also use them if you watch the recordings on those sessions. You can also follow that by downloading this do not open this template here. You need to download them and use them on your machine so that you don't save and change and make changes to them. And I'm going to post here the file paper that I just used to upload it. So you will find this. So you will find it here. There is no need for me to send it to anybody. You can come here and download it. You saw that there are questions and then there are solutions at the end. What I would suggest you do is work through the trial paper. And once you are done, then go and look at the responses to see if you are doing right. Because that's how you will practice and know where you are going. And it is a detailed trial paper. It's got all the solutions and step by step on how you answer every option of the solutions. Of the questions as well. So those are the things I can share with you right now. I want to wish you all the best of luck with your exams. Remember if you need to ask any questions relating to STA 1610 content, you can send an email. I forgot. I need to also tell you, by email, actually, it is on the document. So I don't even have to tell you how to get there. So there is my email on every documentation that is uploaded, almost all the documentation. My email address is there. So you can send an email, but you need to copy CT and type it, unisa.ac.za, when you send that email. Otherwise, those who are on the WhatsApp group, we can always continue to chat, even though the WhatsApp group is muted, nobody talks, but you do have that opportunity as well. And on that note, it completes today's session and it completes 2022 sessions. I want to wish you all the best with your exams. Those who already have started writing exams, good luck with your results and those who are still going to write all the best. And goodbye, unless if there are any questions or comments. None at the moment. But thank you so much. If there are absence of questions and comments, goodbye. I will see you at the corridors of South Africa. And all the best. Bye. Thanks, Lizzie. Thank you.