Poisson Distribution and Normal Distribution

Okay. So, uh please also remember to uh okay maybe I must start with good morning everyone. Please remember to um complete the register which will be in the chat. Let me quickly see if I posted it now on the chat. just copy the links again and repost them on the chat and yeah and then at the end of the session don't forget to complete the feedback form or the evaluation form. So welcome to this week's session. uh we will continue with discrete probabilities but only focusing on the poison and then we will move in into today's session which covers the normal probabilities. So poison like I said uh we will still need to be using the table formulas as well as you need the calculator to calculate the probabilities. So what is a poison probability? A poison pro uh distribution is when you are interested in the number of times an event occurs at a given opportunity which means you will be calculating what we call an average because it will be at like you can't be counting them individually like 1 2 3 4 5 as we know that discrete probabilities are follows a counting process. So with poison as well once you observe those uh moments or events happening you will need to come up with an average of those uh number of times that those events are happening. So that will also enable you to calculate what we call an area of an opportunity which that area of an opportunity is just the average of that time that you observed or it will be either a volume or it might be a unit. So let's say for example the number of scratches in a on a car paint like for example how if you are working in an insurance firm you want to know how many of the the cars that comes for repairs on average um they come with scratches. So you count how many of those and you can average it for that period for that time period that they have been like in an hourly or in a weekly or in a monthly or something like that. The number of mosquito bites on a person. So you can also count how many they are and average for each individual persons that you saw for that week for that month. You can count those average number of mosquito bites and so on and so forth. So poison follows that pattern as well of a discrete uh uh distribution as well. With poison also there are some characteristics we will be able to calculate the average which is the mean of those. So it means taking the mean or the average of those uh number of times things happening or we can calculate but the variance or your standard deviation. So with poison the average that you get will be your mean and your variance will be the average and your standard deviation is the square root of your mean or your average. So it's fairly easy. So the average you receive is your mean. The mean is also equivalent to your variance. Your standard deviation is the square root of your average. Easy. If for example, this is what we need to calculate. A local police station receive on average 3.5 emergency calls per hour. So they would have counted every hour. this had the number of emergency calls they receive and they average it and they find that every hour they receive around three and a half calls. So those calls they follow what we call a poison distribution. If the question asks what is the mean of the number of calls that we the the police station receives that is easy because on average we know that it receives 3.5 so the mean will be equals to 3.5 or my pen is not writing today Just give me a second. Sometimes it need me to to activate it. Sorry, I should have prepared this. Yeah. It just needs to be waken up a bit. So it means the average is 3.5. And if they ask you what is the variance or the distance far apart from uh the mean and that variance will be equals to the same as the average which will be 3.5. And if they ask you what is the standard deviation? So the standard deviation will be the square root of your 3.5 which then will be equals to the square root of 3.5 will be equals to 1.9 1.9 and that's how easy it is. Poison is straightforward, easy, simple to calculate and find some of the characteristics of the uh what do you call the the parameters of the measures of poison distribution. Now here is your exercise. [clears throat] Suppose that the number of daily fake news post is a poison distributed with the mean of 0.2 per day. What is the value of the mean and the standard deviation? So which one? Which option? Number one, number two or number three. Since we know that the mean is the same as your lambda or it's your mu. >> I think it's option number one then. >> So is that option number one? Because the mean which will be a poison the mean is your lambda and here they're asking you to calculate the mean and the standard deviation and we know that the mean is the same as your average and your standard deviation is the square root of your average and then will be 0.2 and this will be the square root of 0.2 which is equals to 0 4 44 and we round it off to one decimal and only number one. That's correct. Easy. So you won't forget to the properties of poison. If they ask you to find the mean or the standard deviation, it's easy, easy, easy. When it comes to the probability, it's a different story. For a probability then you need to calculate the probability of an event happening which follows that average or the poison distribution. And the formula is calculated with uh the exponent to the power of the negative average times the average to the power of x / the um x exclamation which is factorial x factorial. So like we did with binomial uh distribution, the same can happen with poison. You can use the formula to calculate the probability of a poison distribution or you can use the table. When you use um the formula, please do not type in the number on your calculator. There is an uh an ebase or a base natural lock function and it is on the on it's written in orange. If you're using a cashio calculator or if you're using a sharp or something it's also going to be written in orange it will be on top of the lin function. So in order for you to reach that function you will just press the shift button. The negative is not a minus. There is a negative that sits in the bracket. So you will have to use that negative sign. And like we did with the poison last week or sorry with the binomial last week the factorial there is an x factorial or an n factorial depending on which calculator you are using. You can just use those functions to calculate. Right? So how do we then calculate the probability of a poison? The other catch or the other thing that we also need to remember uh are those um um mathematical functions like less than or equal, greater than or equal, uh between and so on. You need to know and still remember to apply those logic when you are calculating. So what does that mean? It means if I have to calculate the probability if I need to calculate the probability that the mean average was greater than 1 where then it means I'm going to be calculating the probability that x = 1 plus the probability that x = 2 plus the probability that x = 3 up until I get to whatever the number that I need to get to. What does that mean? It means this formula you will have to apply it on each and every probability that you have. So you will repeat this formula for where it's one, repeat this formula where it's two, repeat this formula where it's three and it makes it easier with the table because on the table they give you the probability and that's it. So that is just what I wanted to explain in relation to this because I remember that last year I didn't explain how then the formula you will it becomes so complex when it's a long formula that you need to do for each and every probability of an equal. So using a table easy. So if I have to calculate uh the probability let's assume that I'm going to be calculating the probability where x = 2 where the lambda is equals to 0.5. Right? So on the table um we will get to the table just now but uh bear with me I'm going to show you the actual table but the table is broken down by lambda. So you will have multiple tables that are broken down by lambda from the value of x's that starts from 0 to whatever the number based on the lambda distribution. So the next table will start at 1.0 or something like that. um and then it continues and continues and the number of x will also maybe will increase to 8 or 9 or 10. You just need to look at the average or the lambda value and then apply the x values based on that lambda value. So let's find the probability that x= 2 given that our average is 0.5. So on the table it's easy. We go and find the lambda where it's 0a 5 and then we then go and look for where x = 2 where the two meet. This my sign is not on the right thing. The answer will be 0758. So easy to find the probability. if I need to use the calculator uh and use the formula. So therefore it means I'm going to substitute. So e is a natural log. So you don't do anything to that. Our lambda is 0 5. We substitute that. And uh x is 2. So we make it to the power of two. And this is two. So on your casual your calculator you can put if you have a casual calculator you use your fraction uh formula and you put everything in and uh and the answer you will get you will see it will give you the same as the um the table formula. So now let's do another activity. What if then instead of saying the probability of X is equals to 2, what if they say the probability of X is less than less than 2. Uh we still keep the average the lambda as 0 comma uh 0a 5. What that does what does that mean? Then it means I need to find the probability of X less than 2 will be equals to the probability of X = 0 plus the probability of X is equ= to 1 because it's less than so it does not include two so I need to only take those probabilities for X of 0 and one. So if I'm going to use the table, easy because I just go to the table and add those two values. So I'm just going to add 0a 6 0 65 plus 0a 3 0 33. And that will give me the probability 0 65 plus 3033 which is the probability will be equals to 0 comma 90 98. Easy. If I'm not using the table and I'm using the formula still I need to go and find the probability of X = 1 which is equals to E to the power of 5 * 0a 5 to the power of 1 / 1 1 factorial which then on the calculator let me See if I can get my cashio calculator online as well. Okay, I'm not going to find it here. I will find it on my desktop, I guess. Then I need to go to the desktop. Sorry, my bed. Let's open it up if I still have a license and then I can show you and demo. I think my my license expired. See South Africa. Where are you? Oh yeah, I need a new like a a new license activation field. You have already reached the allowable. I don't know. I can't open my my cash license now. Oh, I don't have that. Okay. So, I'll just have to use my phone or my calculator to do that. Um, so anyway, it didn't allow me to. So, let's do that. my fraction thing. shift e to the power of 0 comma 5 and my arrow to the side multiply by or I can even use the bracket the way it's written.5 to the power of 1 not to the power of two to the power of one um I arrow down and one shift uh factorial and the answer I get is 0 comma 30 2653 which I can then just round it off to three. As you can see that it gives me the same as what we have. And now if I want to do for zero I just change everywhere where it is is one I replace the one. So x = 0. Therefore it's e to the power of 0a 5 * 0a 5 to the power of 0 / 0 factorial. Uh we know that anything to the power of a of of 0 is one uh and 0 factorial is one. So we just need to calculate shift e to the power of 5 which is equals to 0 comma 0 comma 0 uh 6 0 6 5 36. So the number to the right of five is three. So therefore it means it's five. So it also gives me that. So I just add the same number and I will still get the same. So using the table can save you time but also it's not always the solution because sometimes in the exam you are expected to also know the formula so that you can answer any question. If they give you a formula that is incomplete or the formula that is completed and they ask you for this question, choose the right formula. You need to be able to know how to do that. And we're going to see it in the um in the uh activity. So I'm just going to skip all of those and then come to this so that I can demonstrate the importance of also knowing the two methods. It's not like I'm saying only learn how to use the formula or use the table. I'm saying learn both. The table saves you time in the exam where it's time constraint and knowing the formula helps you to answer questions that are relevant to the poison distribution. So let's look at this question. It says the average number of adults with ASD consulting with a neurossychologist per day is poison distributed with the average of 1.5. So it means we know that our lambda is 1.5. What is the probability that on a given day a neurossychologist will consult with only one? Therefore, it means it's equals to because it's not saying greater than, it's not saying less than or it's not saying more than or at least or any of those. It says will consult with only one. Therefore, it means it will be we need to find the probability that X is equals to 1. What is the answer? Is it option one, option two, option four, or option five? Oh, I forgot before you get to uh option one, option two or option three, let me also do this. So on the tables there is a table called poison distribution table is table E should be after the the binomial table E7 as you can see it's broken down by the lambdas so you can see that for a table that starts with lambda of 0.1 to one it's up to 7 but the table with lambda from 1.1 to 2 uh runs up until 9 and so on and so forth you can see there so it means if you have to calculate the probability of a greater than three and the uh the average is 0.6 six, therefore it means I must add up until 7. But if I'm on 1.5 and I have to do greater than three, therefore it means I must do up until 9 and so on and so forth. So using the table we can quickly find the answer we are looking for. So we looking at 0 make the table smaller again. So we know that we are looking at this table because our average is 1.5. Right? So the process of elimination becomes easier. So you first go to the probability or the sorry the average which is 1.5. We know that we are using this table. Now we need x = 1 and we go there where they meet the answer is 0a 33 47 and you look at the answers you got here there is it's none of those answers. So therefore it means they're asking you to check if the formula is correct. Right? So then we go and we complete the formula. So let's see if we'll get the same x = 1 will be given by e to the power of lambda lambda to the power of x / x factorial. So then let's substitute e to the power of lambda is 1.5 and your lambda oh sorry your lambda is 1.5 so in if I look at this formula they using a multiplication so we can do that 1.5 to the power of 1 uh we know anything to the power of one we don't have to necessarily put it up right put to the power of one so we can leave it and this will be one factorial or we can put it so let's look at the formula does it relate 1.5 to the power of 1 we've covered that time e to the power of a negative uh 1.5ide by 1 factorial. Therefore, that is the correct answer. As you can see, the table doesn't necessarily help you answer the question in quick, but it can help you save a lot of time because you could have used the formula calculated calculated and still not find the answer and then get there. Or you could have used the formula, substituted and find the answer immediately. You'll never know how they asked the question. Are we good or do you want to practice more on the poison? >> I'm good, man. >> Cool. Now, let's move to the next section. The continuous distribution. Uh I didn't even check if we have more people joining the session. I can't see when I'm presenting um and see when I'm presenting. So normal distribution. So by the end of the session we should learn how to uh understand the basic concepts of normal distribution. what makes a normal distribution. The reason why also I I know that this are the like literacy sessions uh we should be doing more of calculation giving you tips and tricks but also in the exam or in the assignment they still ask you questions related to the content like the basic concepts of normal distribution. They might ask you question about what makes a distribution a normal distribution. So you need to know those properties. So that is why I also do share them with you so that you can know what those are. You should also be able to learn and know how to calculate the probabilities of a normal distribution. Now with the normal distribution it will work slightly different to how we did it with the binomial and the poison and the uh discrete probabilities purely. Uh here because with normal distribution we are not working with a probability at the point we are working with continuous normal distribution probabilities. So therefore it means we're working with cumulative probabilities as opposed to working with a um a point probability like with a discrete probability where we can find the probability of equal to or less than or equals to. So with normal distribution there are no point probabilities are always cumulative probabilities. So we will deal with the probability slightly different. So you should be able to calculate the probability of a less than a value, a probability of a greater than a value and a probability that a value sits between two values and also use the formulas. So we will have to calculate this Z value because we're not calculating a probability of an X value but we need to standardize the X value so that we can use the standardized value to find the probability and I'm just going to explain that that's why I said it's very different. So a continuous random variable is a variable that can assume any value on a continum and it can assume an account an an an unaccountable number of values because we are looking at cumulative value. Like for example the thickness of a item uh because if you if you look at um some item that are shaped like they are thin thin thin thin and then they become bulky bulky bulky. So the thickness if you continue with looking at let's say a bottle um a a juice bottle you can clearly see that it doesn't have the same um thickness across from the bottle top to the bottom top anyway. So those are the things that we look at when we look at continuous. So the continuity of how things appear, the time required to complete the task. So it might slightly differ. It might take longer longer and then the more you know about how to do it, it becomes shorter and shorter and shorter. And that the temperature of a solution uh in the beginning it becomes hot and then as it stand it becomes colder and cold and colder. So it doesn't stay static uh as in like uh it will always be 100° throughout. All right. Uh the height in inches and so on and so forth. This can be potentially taken on any value depending on the ability or the precise or the accuracy of that measure that you are using. So it means uh when we look at normal distribution you can have um the the distribution follow a different format and those formats are based on two measure or parameters. We can call them parameters because at the beginning of the sessions like at the beginning of the uh units you learned about the difference between um a sample and a parameter. Oh sorry a a statistics and a parameter. And here we are based we are basing the normal distribution on a parameter because we're looking at the population of interest. So the shape of your normal distribution depends on two things on the mean and the standard deviation. So varying the the the different means will give you a different shape of your uh normal distribution. If you vary your standard deviation, it will also give you the different uh shapes. So for example, if I look at um the pink um the pink uh distribution and also the normal distribution is also called the belly shaped curve and we're going to talk about it just now. If you look at the pink distribution there, clearly you can see that it has um the standard deviations are wider because in the middle of every um uh standard deviation is your mean. Sorry, of your normal distribution is your mean. And the distance between the mean and the end of your standard deviation the the end of your uh distribution is what we call the standard deviation. So this is your standard deviation. So looking at the pink the pink uh distribution you can see that the distance between the mean and the edge of your distribution it's bigger. So it means the standard deviation there is big. So this one has a bigger standard deviation and it forms a flat belly. So the bigger the standard deviation the flatter the normal distribution will be. If I look at the yellow and the um and the uh turquoise, I don't know whether it's turquoise or what. Uh this color, the blue color, you can see that clearly they've got the same mean almost. They have the same mean. What is different? What is different between the two is your standard deviation. So the standard deviation for the yellow and the standard deviation for the turquoise are different. So what does that mean? It means the smaller your standard deviation the bigger your distribution will be. Then I um so if your standard dev standard deviation is smaller then your um your peak of your your graph will be too large. So it will be taller and if it's normal like 1.5 um uh standard deviation away from the mean uh and then it forms a normal uh distribution. So it the shape looks much better and it looks what we call normal. Not that the others are not normal. They are normal but they've got narrow and also flatter shape depending on your mean and your standard deviation. So if you move the value of your mean, it shifts your um maybe I'm also talking ahead of time. So the more you when you move the mean it moves your distribution from from left to right. So depending so if you move your mean if your mean now comes here so it means your graph will move to the left. If your mean moves here then it means your graph will move to the right. Your standard deviation increases the spread. So it tells you whether uh if your standard deviation is smaller it will be flatter if it's bigger sorry if it's bigger it becomes flatter and if it's smaller it becomes taller like we said it uh so it means the changing or the the increase and the decrease of your standard deviation will change the spread of your distribution. the change or the shift in your mean will move your distribution from left to right. So those are the basic things that you always need to remember how the mean and the standard deviation influences how your normal distribution shape will look like. So what then do we do? So if for example I've got observations like our X observation at any point that I am taking and I record and monitor those observation over a period I need to in order for me to find out whether they follow a normal distribution I need to be able to standardize those points. So that is why we talk about a standardized normal distribution. So any normal distribution with the mean and the standard deviation can be transformed into a standardized normal distribution where we use the value of a zed and we're going to talk about that zed and we call those zed values um we're going to call them zed scores or zed values. So a standardized normal distribution always has the mean of zero or so it means the mean will always be equals to zero and the standard deviation of one. So that is the property of a normal distribution. So if you have a num a a distribution that has the mean of zero and the standard deviation of one, you can safely say that is a standardized normal distribution. So, we're going to transform our X units, which are our X values, into our Z unit, and we're going to use those Z units to calculate the probability of those uh X units that we we want. So, a normal distribution, like I said, it's a belly shaped curve and it's symmetric. What does symmetric mean? We did this in study unit 3. We said uh if uh the mean, the median and the mode are equal then it means the distribution is symmetric. So it's when the mean and the median are equal. In order for us to standardize the units, we then use this formula which is the Z formula or the Z score formula which is your X unit minus the population mean divide by the standard standard deviation. And we always remember and know that for a standardized normal distribution the mean is equals to zero and the standard deviation is equals to one. That does not mean in this formula we're going to put there the mean zero and the standard deviation zero and standard deviation one is just means for this Z distribution we are making sure that the mean is equals to zero and the standard deviation is equals to one. Therefore, it means we will have to know the population mean that we want to standardize and the population standard deviation which will be the values that we substitute and we need the x value that we are given that we need to standardize. Okay. So let's look at this example. If X is normally distributed with the mean of 100 and the standard deviation of 50, find the Z value for where X is equals to 200. So they have given us the mean which is the mu and the standard deviation which is our sigma of 50. We just go to our Z formula and we substitute into that Z formula. So x is 200, mean is 100 divide by our standard deviation of 50 and then we get the z units of two or our zcore of two. Therefore it means for x of 200 it is two standard deviation away from the mean of 100. So there are two increments. So if I have to draw that I don't know how to draw properly. The other thing that you also need to notice when you draw your normal distribution the line at the bottom does not always have to touch the distribution. So it's always almost it doesn't touch they don't touch. So they always do that. So we say we had this we know that this is the mean. Uh so if this is one and this is two. So what does that mean? It means from here to here on this example that we are given h the distance between the mean and the edge is only just two standard deviation two increments of 50. So the if then it means this was 50 and this was 50 which makes it two standard deviation away from the mean. So that is what in a nutshell practical uh that means uh let's look at an exercise so that you can practice. For a particular group of scores, the population mean and the standard deviation are 25 and five respectively. So it means this is your mean and this is your standard deviation. Find the Z score for this row score. So it means they're saying use the formula X minus the mean by the standard deviation and find what the Z score is where the row score is 19. So this is your X. Is it A, B, C, D, or E? It isative 1.2. >> It's 1.2 2 because we know that our X is 19 minus our population mean of 25ide by our standard deviation of five which is 1.2. So when we are moving forward into the probabilities there are a couple of things I need to remind you. So when we calculate the Z score and we find that the value of a Z score is equals to a negative 1.2 we keep it as is but we also going to uh say your Z score will be1 2. we always keep it to two decimal and I'm going to explain this shortly just now when we uh when we deal with the tables and finding the probabilities. So having a negative number it's very important because on the tables we are going to use the the normal distribution tables has the negative and the positive side of the table. Uh so that is very important but also the Z tables has two decimal um at the top. So we always have to keep the values into two decimals but for this exercise uh we look at what the answer would be and we just keep it at that. So it means very important you need to know how to round off properly. So remember, if I'm rounding off to two decimal, I must look at the number from the two decimal to the left of it. So sorry to the right of it. If it's greater than or equals to 5, I must add one. If it's not, then I must leave it as is. So you need those principles needs to remain. Uh because the minute you add another number, you're increasing the probability numbers. If you don't include or if you don't round off properly, you are missing out on one of those probabilities because you will get the wrong answer. So, it's very important that you know how to round off properly. Um, okay. So, this is one of those questions that I said or one of those things that I said. You also need to know and understand the basic properties that make up a normal distribution. It's very important. Knowing how to calculate is also very important. But also knowing the basic properties of a distribution can also be a breaking point for you to pass or fail a module. So you need to know this. So which one of the following statement is incorrect with regards to the normal distribution? So we're going to go through uh option by option and eliminate do a an elimination pro process. So we need to find the one that is incorrect. So I'm going to read pro uh option one. The z score of a mean of a normal distribution is one. The z score of a mean of a normal distribution is one. Number two, the smaller the value of a standard deviation, the narrower and the steeper the cf. Number three, the mean of a normal distribution can be a numerical value that is a negative, a zero or a positive. The area to the right of the mean of a standard deviation is 0.5 and the area to the left of the mean of the normal standard deviation is 0.5. 95.4% of the values of a normal distribution variable are within 2 plus or minus two standard deviation of the mean. What did I not do to you? Is the following. I have never explained that. So it means I expect you to say you know how to get to to that. I have never explained that. So it means you need to know where in the uh what do you call in the study guide where to find that information. Uh what else didn't I explain? I didn't explain that. So, as you can see, most of these things I I didn't even touch in my explanation of the basic probabilities, and they expect you to know how to respond to those. What else didn't I not touch? The only thing I touched in our um conversations is that one. So if we need to find the incorrect statement, we then need to do a process of an elimination based on what what we also know. I don't know if you want to give it a try. before you also give it a try. What I didn't explain around this 94 which is two standard deviation is the following. There is what we call an empirical rule. Rule which state that you need to calculate from both side the mean plus or minus the three standard deviation and it will give you whether where it is. Um and and that it is always 99.7%. The mean plus or minus three stand uh two standard deviation is 90. I'm just going to give you the answers for this one. It's always 95.4%. And oh, let me write it at the top. Um then the mean + one standard deviation is always 68%. So it means in this instant number um [snorts and clears throat] number five is correct. What I didn't also explain is number four. What does number four mean? So we said now I'm drawing a skewed if we if think about this as uh something that is perfectly drawn is not skewed to one side uh since I don't know how to draw properly. So if we split if we split this into two this side should be 50%. To this side why what did we say about the probabilities? We said the sum of all probabilities we did this at uh in study unit 4. The sum of all probabilities should always be equals to one. And what does that mean? It is always equals to 100%. So if I split my normal distribution into two parts, what's that that mean? It means I'm dividing the 100% into two. So it means what? 50/50. So the area to the left will be 50%, and the area to the right will be 50%. That's what number four says. So done. We covered that part. We know that it is correct. Moving on to the next one which is um what are we covering in the next one? It says oh the mean the mean of a normal distribution can be any numerical value whether it's a negative a zero or a positive value. So if I'm talking about a normal distribution and I say my mean here represent a zero. So my mean there is zero. Therefore it means if my mean comes to this side it will be equals to a negative because this side there are positive. So what does that mean? That does mean my statement number three is correct. Definitely because if my mean is zero there any value any mean value on the on the left of that zero will be a negative mean any value on the right of that uh zero will be a positive value. So therefore it means statement number three stands correct then leaves me with the two What are the two? Number two, or do we go with number one? Oh, let's do number two because we covered number two just now, not so long ago. Ah, what did we say? We said uh if I go back two three slides back. There we go. We said if we change the mean it becomes there if the uh what do you call that now? We said the standard deviation increases the spread or actually we can use this. This is the most important one. We can use this. The question there said it's steeper when the standard deviation is smaller or it's flatter when the standard deviation is bigger. So let's go back. What does that mean in this statement? The smaller the value of your standard deviation, the narrower and steeper your curve will be. What does that mean? So if I have the mean here, so it means my standard deviation will be smaller. So if I have my mean here and I know it cannot be bigger like that will have to be like that. So the mean is there and my standard deviation is there. So it means the larger the standard deviation it becomes wider. So large how do we write it? Large standard deviation it's wider and flatter whereas smaller will be narrow and steeper. So what leaves are what leaves uh or what is left for us to evaluate and make sure that it's correct is the the first one. The zed score of a mean of a normal distribution is one. It's not because let's go there. We calculated the zcore of a normal distribution. Did we find it as one? Nope. Did we find it? It can take any range. It can take any form of a value whether negative value or a positive value. So it is not one and also one other thing what does the the mean of a set distribution it's always zero. So if the distribution can take any value but the mean of that Z distribution is always equals to zero. That's what we said right only the standard deviation is equals to one. Go back the Z score mean because I only used that in my explanation here because that's the Z score. It can take any value. But the question say the mean of a normal distribution of that. We know that that mean the mean is always equals to zero because we said it in the um properties. The mean is always equals to zero and the standard deviation is equals to one. So in this instance, one is the one that is incorrect. So one is incorrect. So you need to know and remember all of Those um what also I didn't cover is uh let me see. So I've covered the empirical rule which I forgot to show you. Uh what I will do on on um uh on WhatsApp I will share with you a summary fact sheet that I created. You will see all these from 0.1 until 5. I'm covering all of them. So the zed score mean of a normal distribution is equals to zero and the [clears throat] z is equals to one standard deviation above from the mean and a negative one standard deviation below the mean and so on and so forth. I think that will help with that explanation. The other thing that uh you will get will be the uh the empirical rule. Now here is the the most important thing that we always need to remember with the empirical rule as well. Sometimes they might ask you to calculate them. They might ask you to calculate two standard deviation, three standard deviations and so on and so forth. You should be able to just substitute if they give you the mean and the standard deviation, you should be able to just substitute and calculate them as well. Right? So that is that. Now let's move on to the next part. The most exciting part which is how do we then find the probabilities themselves of a normal distribution. What does that mean? We want to find the area underneath the curve. So if I need to find the probability of a Z value. So now we standardized our X values and we created this X uh unit. So let's assume that this line at the bottom represents the values of our Z [snorts] uh unit. So if I'm saying uh what is the probability that a a zed value will be less than a value. So I'm going to go and select where that zed value is. So if for example that zed value is uh 1.2 so therefore it means this is 1.2. So I want to know everything below. So everything from there going there. What is the probability of that area underneath? Now, here is the catch with everything. If you look at this, there is this small white unshaded area. We ignore that one unshaded area and we use the less than probability. Now the table that we are going to be using which is the standardized normal distribution table contains only and only like I'm stressing it right the probability of a Z score value less than a value. If you need to find the probability of a greater than I will tell you and show you how to find that. So the table only contains this probabilities. So it means every time I calculate the probability of a less than a value I will find that probability on the table. That's why I said how we calculate the probability will be different to how we calculate them using the binomial and the poison table. So we need to know how to use this table. So what is the table and how does it look? So let's first understand the table. So that will be the first table which is table E2. So what does that mean? So it means table E2 contains all the probabilities that a Z value is less than a value. We need to remember that. So it means any question we get find the probability that Z score is less than three is less than 3.00 0 0 is less than -3 mama mama. It doesn't matter as long as less than we will find that probability here. So then do we use this table? So the table contains the positives table. If your Z value was negative, you go to the negative side of the table, right? But our table has values at the top and the values on the side. How then does it? So remember when we calculated the Z score, I said to you, you need to always leave it to two decimal. So, we're going to use our example of Z or now I'm going to change it to a less than because I need to use this table of 1.2, right? And I said put a zero at the end so that it is two decimals. Okay. Now, we're going to use this to find the probability. I need to go and find that probability of that z score. How then do I read it on the table? Easy. The first two numbers, the one before the comma and the one after the comma, you will find it on the side of the table. So it means on the side I must look for negative 1.2 two on the side of the table. I'm going to find the value before the comma and the value after the comma. So 1.2 I'll find it the last digit which is my second digit. I will find it at the top. So you can just ignore all the other numbers like 0.0 and only look at the last digit on this. So here our last digit is zero. So therefore it means our last digit will be on this one because you can see that it carries only the last digit and there is populated even when you go to the positive side you will see that it's only the last digit that is populated at the top. So since our last digit here is zero so it means I'm going to use the 0 comma 0. Then I'm need to come here and look for 1 comma 2 and there. So the answer here is 0a 1 1 51. Easy. That's how you find the probability of a less than value. If the uh z score that we found was let me just make up a number. So if we need to find the probability of Z less than 3.12, let me complicate things and put random numbers there. Find the probability that Z is less than 3.2. The first thing I need to check is this is positive. So it means I must go to the positive side table. And I know that I'm looking for the probability that Z is less than 3.12. So I write it there. And I need to find 3.1 because it's 3.1 2 right [clears throat] the value before comma and the value after comma I will find it here. And at the top I must look for two and there and it's column number three. Therefore, it means that is the probability 0.999 and focus your angle. That's how easy it is. Okay. So, let's go back to our presentation. I've explained this. So, your table has the positive and the negative side. Um let's find the probability that let X represent the time it takes in seconds to download an image file from the internet. Suppose X is normally distributed with the mean of 18 seconds and the standard deviation of 5 seconds. Find the probability that X is uh less than 18.6. So we know now we need to go and find uh this because we need to standardize it. So we know normally this is if the mean is here which is 80 our standard deviation is.5. So it means distance from here to there is five and they told us we need to find less than 18.6. So 18.6 will be somewhere here. So we then shade the entire area. So we need to find this area, the red area because that's what we need. But we cannot just go and find it on the table. We need to standardize it. So it means we need to go and calculate the Z value for this. So we go and standardize. We use the Z formula which then it's finding the Z of less than X minus the mean by the standard deviation which is what we have there. I know that I'm using an equal sign because I'm just using the formula as is for the ease of calculation. But that is the same way. So therefore it means we're going to find if after I've calculated and substituted all the values I find the Z value of less than 0 comma 1 2 that is what I have because I have to round it off to two decimal now since I have this I go to the Z table it's positive so it means I must be on the positive side of the table right so we standardized our units to a normal distribution, a standardized normal distribution. Uh, for some reason my laptop is not charging. Just give me a second. Fix it. Otherwise, it's going to Hey. Oh, pen. The pen. It's the pen, not the for now, I'm going to explain without using the pen because it says the pen is running low, not the laptop. Okay. Okay. So [snorts] now we know that we standardized this, right? We standardized it and we now know that our Z value is uh on the Z line. We need to be finding this area. Where do we find this Z area value? On the Z table. So we go to the Z table. So here is my snippet of it. I have positive. I went to the positive side. I need 0.1 on the side. 0.1 at the top. I need only two. And where they meet, that's where my standardized value would be. And this area from here to there, it's only 55%. which is 0 4 uh 0a 5478 and that is the probability of x less than 18.6 six easy let's do I don't have an exercise now I said I'm going to also explain what then happens if I need to find the probability of a greater than value because then the table does not have that the table has the less than values so now if I have to find the probability of a Z standardized score of greater than a value. It means since the table has the greater than and we know the sum of all probabilities should always be equals to one. Therefore, it means I can subtract the less than probability and get the greater probability. Because at the end of the day, in a nutshell, if you think about it, probabilities are easy to do. [clears throat] [cough] We know that the probab probability of Z less than a value plus the probability of Z greater than or equals to a value. At this point I'm just going to use the both side will be equals to one because this does not include a this includes a right. So both of them if I add them together they will give me one. They should always give me one needs to be collectively exhaustive. So it means and mutually exclusive. So it means if I add all of them they should always give me a value of one. So I know that this value I can find on the table but I don't have this value. So how do I find this value? Then to find the probability and because we don't use the probability at the point we just convert our probability to a greater than a value even though we know that it might not cover all of them but uh with normal distribution we assume that we are covering all of them. Minus we take this on the other side. we will find the probability of Z less than A. So we've got our collectively exhaustive probability covered. So that is why every time [clears throat] you will have to find the probability of a greater than you just say one minus the probability of a value you see on the table. Right? Easy. For every time you need to calculate the probability of a greater than it will be 1 minus the value you see on the table. Okay. How then do we do it? So we going to use the same exercise we did. So we know that we need to standardize this x value of 18.6. Our mean is 18. Our standard deviation is five. So we go and calculate the standard deviation. We did calculate the Z score, right? We know that it was 0, 1,2. We calculated it previously. So we just say 1 minus the probability of the value we're going to find on the table. And we know that the probability of the value we found on the table, we did find it because I'm using the same data. Didn't change anything. I just changed the sign. So we found it. It was 0 5478 and that's what I'm using there. You just say 1 minus that da da da and it gives you the remainder cuz this shaded area in blue is what we calculated previously on the probability of a less than and the red is the probability of a greater than. and the right or the left figure. I'm just showing you that both the blue and the red when you combine them they give you one. So if I add 0a 5478 and 0a 54 22 it will give me one. That's that. That's the probability of a normal distribution. It's easy as that. Do you need to know and remember the graphs I'm drawing? No, not necessarily. The only thing that matters is remembering the formulas. Remembering that when it's greater than, it's 1 minus the table value. When it's less than, it is the table value. That's it. That's how easy it is. simple, straightforward, easy to find the probability of a value especially for a normal distribution. Now the other thing is a between which looks a little bit tricky. Not it is not. Think about it this way. [clears throat and cough] You have this side and that side and you just need this middle part. You don't need all of them. If if you take remember if this whole thing is equals to one. If I take this less than here and I take this one here and I subtract both of them from one, what do I what do I get? I get the orange. If I subtract this wide area here and this wide area here, which this wide area side is called A and it's called A and this white area is called B. If I subtract A and B, I subtract them from one. So I will get this shade. But I also don't have to do it that way. Right? Easy, easy, easy. If I treat this B area as a standalone less than side and I subtract it from the A area, I will still be left with the B area. Why? This is easy. What do we need to do? Easy for cheesy straightforward. We just need to go and subtract the value on the table for a minus the sorry for b because I'm going to use this portion instead of this one. I'm going to use this whole portion and subtract this. So this becomes my b and this becomes my a. So B includes the smaller portion. This one this that's B. It includes that and it also includes A. B is made up of orange plus A. And if you want to remember this, hence we're going to subtract a so that we are left with b. So we're going to subtract a. So we're going to take away a again so that we can only report on how much is orange from from that. So it means we are going to say the probability of Z less than B minus the probability of Z less than A. So that we can only be left with that portion this portion this this this this because our B starts from here to there. It includes this small one. So if this is B and this is A what we need to subtract is just to be left with that take away A and we will left with we will be left with this S plot. That's what we're going to be doing. How do we do that? Easy, straightforward. I'm going to use the same information so that then it doesn't become complex. So we know that the probability of X lies between 18 and 18.5 with a mean of 18 and the standard deviation of five. What we didn't calculate is the probability of 18 or sorry the zed score of 18. So quick I do calculate it. uh the zed score of 18 will be 18 minus 18 because our mean is 18 divide by the standard deviation of 5 and the answer is 0 and remember it is 0 comma 0 0 don't always just write the zero always put remember to do it in two decimal so that you don't forget this one we did calculate it before it was 0a 1 2 and we know that the z score lies between 0 comma 0 at 012. Based on our definition, we said we're going to take the B value and subtract the B table value subtract the A table value. So we go and do that the B table value as 0a 54 78 and we subtract the A table value which is 0 0 which is 0A 5 0 and we get this small portion here. It is just 0 comma 0478. This red shaded area. Easy. Any questions? Am I moving too fast? Everything is >> Hi, ma'am. Hi. >> Um, I joined the class late. Can um will will I later get like the the full class? Maybe a recording. Um, the recordings are always shared. I shared the link, but they take their sweet time with it with uploading. I see that they only had two recordings so far. Uh did you join based on the link I shared on the WhatsApp group? >> No ma'am I'm not on the group. So that's why I was wondering if we like it uh this recording later on. Oh can I leave like share my number so that I can be added on the group. >> Let me send you let me add the WhatsApp group link. Uh >> okay thank you so much. Now you see you're reading all my let me do it later so that because then >> yeah I [clears throat] will I I will have to stop the sharing and then do the WhatsApp link don't leave as yet and then I will share it. Okay so let's do exercises now. Uh I I want to skip all this so that we can now practice. We still have time. The session ends at 11:00. We have how long do we have? Yeah, we still have 30 minutes. So, in a nutshell, what I'm I was trying to say, um, we've covered everything I I said we will do, right? So, we looked at the basic concepts. We've covered some of them, not all of them, but some of them in detail. Some of them you still have to go and read more about the normal distribution. We I've showed you that this values probabilities you find on the table. [snorts] This ones you say one minus the table value. So it means the value you find on the table you subtract them from one. This one you're going to use the table value of B subtract the table value of A. Right? So table value and table value. Always remember B the second one minus the first one. Second one minus the first one. You always need to remember that you need to standardize your x values by using the formula. The formula stays the same for standardizing. Next week when we do the sampling distribution, this formula will change a bit little bit slightly. The concept will still remain the same. So if you can master it this week, you can remember and know how to do it this week. Next week is going to be easy, easy, easy, easy, easy because we're going to use the same same. Nothing changes actually. We just going to add one or two things, but nothing stays the same. The principle stays the same. Just a reminder, please for any technical issue like the recording and all that, please send an email to CTN T. I I will remind them to to uh upload last week's session and this week's session. If you are struggling with your module content, you still unsure whether it's things that we've covered or you are way ahead and I'm still behind and you want some assistance, feel free to reach out to me. You have one-on-one consultation, 1 hour, free consultations with me that enables us to discuss where you are stuck, what problems you need me to help you with, and so on and so forth. Do not hesitate to to book those because I am available. I'm free, free, free, available anytime this time around. I don't have anything that is holding me back. uh because I only have this classes. Uh now please do not forget to complete the register before you leave and also at the end of the session when you leave remember to do the evaluation form. The link to both forms is pinned on the chat. I think you can even if you join late or not you will always find that links there. Please make sure you do that. My name is Elizabeth Boy. Okay. So, next week we will be covering sampling distribution which is similar to what we're doing. It's also a normal distribution um session as well. [clears throat] Oh, I must still fix my slide. Uh this is the exam prep on the 12th of September. Then we will be doing exam preparation and I think there might be a couple of more sessions as well depending on when the exam starts at UNISA. We might have some more sessions. So now let's look at exercises for this week. Which one of the following statement is incorrect? So it means you need to one have your tables open uh you can find the table on your textbooks in your study guide. I think in the study guide they also do have it. You just need to know [snorts] uh where in the study guide this table is on. So you need to have this table open so that you can answer these questions. So as you can see there it says the Z is equals to the Z is less than and so on and so forth. Uh but since it you it uses zero so I'm going to assume you need to change the 0 to 0 comma 0 0 and the one to -1 comma 0 0 and the one will be 1 comma 0 and I can just Put remove the ink just to help those who don't have the table readily available. Uh the problem is it only for what I'm showing you it only covers number one, number two, number three and number. Okay, number five. You don't even need the table. Um but you can continue. We are looking for the let me see the incorrect statement. So do a process of elimination. So maybe let's do it together and then number two I will let you do it by yourself. So number one is that correct? Uh the probability that Z is equals to zero. Okay, for number one it will be correct because remember we are not these are cumulative probabilities. So we don't have a probability at a snapshot but also a probability at that point will always be equals to zero. So number one is correct but we're looking for the incorrect one. Right? The problem is when I type my thing it disappears. So I'm not longer going to write on the slide. Number two, anyone who wants to take a guess whether is it correct or incorrect. Number two, the probability of Z greater than zero. >> That is correct. >> That is is it correct or incorrect? Is that what >> it is correct? Because if you look at the probability of 0 comma 0 is 0a 5 and we know that it is the greater than. So therefore it means it's 1 minus 0 comma 0 uh 0a 5 0 0 which is 0a 5. And number three, is it correct or incorrect? >> It is correct as well. >> It is correct because it's just that number that we see there on the screen. It's 0 comma 05 50. Going to number four. Oh, sorry. What did I do now? Number four. I'll go to the negative side of a table so that then we can bring number sorry negative it's at the top on this. So we're looking for 0 uh -1 comma uh1 comma 0. We know that the first column is where the zeros are. So it is also correct. Number five. Is number five correct or incorrect? And can you identify why is it incorrect? If it's incorrect. Okay, let me help. Remember when we do the less than or equal when we do that we don't change the A remains A. It cannot become B. Right? The minute the the the value here changes to another number, therefore it means we no longer talking about the same thing. So if you look at number five, it says the probability that Z is greater than one. So what should they be? It should say less than one, right? So the answer here should have been or the correct way of writing this one should have been 1 minus the probability that Z is less than or equals to what or something like that because I can see that they are using the less than or equal on this statement but it cannot change because then this and this are not the same. They have to be the same. Same same same same. So your incorrect answer is number five. So because you're also writing a multiple choice question, it means it's double the work. So as you can see that we had to go through 1 2 3 4 before we get to the correct answer there. So you are you work double you work triple for like yeah more than you supposed to. So consider the standard normal distribution set. Which one of the following probabilities is incorrect? Now I'm not going to help you here. I can already foresee. >> [cough and clears throat] >> If you have the answer, you and also post it on the chat or you can say it out loud and say the answer is option whatever the number give you some time open There. This one says it's 2.80. If I go there, 2.80. it is 0.026 and this one says it's 1 minus the P of 2.8. So if for example just to test the logic if this is 1 minus the probability of uh greater than 2.8 eight. So therefore, it means I can always test it and see if it's the same. So let's test it. So 1 minus the probability of Z greater than 2 8. What does that mean? It means it's 1us 1us the probability of z less than 2 8. You cannot just also assume. So okay here is the catch. What's the difference between what I'm doing now and this? It's because this was the less than and there was the less than. they they were the opposite and that's why I could automatically make an elimination with the number negative on this one. This is greater than it's not less than because this side is greater than it says the probability of Z greater than -2.8 is equals to. So it means it's the same as 1us the probability of z greater than uh 2.8. So I need to test that. So 1us 1 minus and I need to go and find 2.8. So don't take it because the previous one we did it that way. It means it also the logic will also work on this. It's 2.8. So I need to go to the positive side. And I must look for 2.8. And there's no other number there. It's zero. And I know that zero is always the first one. So it's the first column. Come on. So it's this first column. So that will be the value 0.9974 0.99 74. So but while I'm still at it, what did I say this value is as well? So because now I can go to my calculator 2.8 I said it is -2.8 is 026. So I said this is 0 comma 0 26. So let's see if we get the same oneus 1 - 9974. It is 1 - 0 0 26 which is 1 -26 is equals to 0.99 74 already I showed you using the process of elimination You can do the next one. Is number two correct or incorrect? You have negative 2.21 0 which is 0 comma 0179. then they say is the same as the probability of uh Z greater than 2.1. So which means the value we're going to find on the table we need to subtract it from one to always constantly write these values down 0179. So this one is 0.0179. 017 KN and this one will be 1 minus the value we find on the table and that value would be positive2.18 to 982 0.98 Q1 and what is that answer is it 0.17 in the exam I would have found my incorrect answer and moved on but because we are practicing let's practice so that in case that was correct then we need to move to the next one and see how we we answer that so That will be 1 - 98 21 and that is 0 017 9. So that is correct. Can you see >> incorrect option name is number five. There >> I think the incorrect option is number five. >> [clears throat] >> Wait. So this is 0. Uh so what is 2.1? 2.1 is 0 9 8 21 and this is 0 5 0 0. So if I subtract those two so 9. So this one will be 9 uh 0.9821 subtract 0.5 0 0 I know that the probability of 0 is 0.5 uh because we've dealt with it in the activity before and if I go here you will see that it's 50% there am I doing it's 50% there so that is what I'm doing. Remember this is in between. You take the second subtract the first and that will be and that is equals to 0.4 221. If you take that minus that, you can test it. You will see because one uh 1 - 0 is 1 and 2 - 2 is 2 and 8 am I calculating right? No. Oh, this should be 8 0.48. 4821. So, they've got two questions that are incorrect because this is incorrect as well. Oh, sorry. I did it all wrong. Number one, sorry, my bad. I didn't take that into consideration. The sign, the sign says greater than. So, this will be one minus that. So this will be also correct. Yes, you are right. Number five would be the incorrect one. I didn't take the greater than into consideration. Um and [clears throat] this one will be 0.500 0 and this is -2.8 is 0.026. You take this subtract this should give you 0.49. So you just say 0.500 0 subtract 026. We borrow 10 4 uh 9 - 2 is 7. Uh 9us is 9 and 4 - is 4. So that is correct. And on this one we know that this is 0.26 and 2.1 did we get 2.1 2.1 which is positive is 0.9821. So we just say 0.98 21 subtract 0 0 2 6 1 we borrow 11 - 6 is 5 uh 1 we need to go borrow from eight 11 - 2 is a is uh is 9. Uh and then here we have seven and nine. So that it's correct. So only number five is incorrect. You are right. And that's how you will solve. So there is a difference as you can see. So you will need to be able to to validate and test and check. Suppose that zed is normally distributed with the mean of zero and the variance of one. Which of the following statement is incorrect? So with this one you have to go and say 1 minus the probability that zed is less than a value 2.64. So it means you're going to say 1 minus the value you find on the table. What do we get on the table? 2.6. Uh, it's 2.6. But we need the four. Four is column number. Oh, sorry, my bad. [snorts] 0 1 2 3 4. It is the last one on here. 0 1 2 3 4. So 2.6 6 D 0.9954 0.99 Am I Did I write it correctly? My surface uh 0.9959 59 And that will be 12. So, I've done number one for you. Number two should be easy. It's just the value on the table. 0.87. So now with this one I need to reduce this size. We need to go and find the eight is the second last. Uh we're looking for seven. Seven is the third last. So this is negative that's seven. So it means number two also it's correct. Oh and then we can do number three. I can give you the values. So the first thing you do is you look at the second which is 1.4 4 and then you go to the positive side 1.4. Then you take that number 0.91 92. Then you go to the negative side and you do the same.40 which is 0.8 0.0808. Take those two numbers. So you will take those two numbers subtract on this side. 91 92 0.91 92 Subtract 0.0808 08 08 and see if you get the answer. 91 92 subtract 0808 which is correct. So that is correct. Then you go to number four. She's in the negative. And we're looking for 2.8 then zero which is Bob is your answer. There is your answer. Pops your angle. Oh no. Adap is incorrect name. Option four is the incorrect one. Option five gosh. Option five. Um so option five can also confuse you if you don't pay attention. As long as here they didn't put one minus. So it means you go and find the value here which is 1us the probability of z less than 0 74 and go and check if that value you get here is the same as that value which then takes us to I don't know why my table is behaving like this my PDF it says the PDF contains unapplied reduction I don't do need reduce so 1 minus 0 comma so we go to the positive side come on You go to the positive side and look for 0 74 which is 1 - 0 comma 77 1 - 0 77 04 which is 0 comma 22 96. Then you go to the negative side and see if uh 0 comma 7 I must also go up uh and they are the same 1 - 0a 7704. It is 0a 2296. So it's also the same. So this is 0 comma 22 296 and this is 0a 22 96. So only option four is incorrect. I want to skip some of the activity but you you can um uh do them. I I've shared the slides on the WhatsApp group. Those who are going to join later then I will share with you the the slides with you as well. Um don't put your your your cell phone numbers on the chat. [clears throat] Okay. So here is one exercise. It looks like this. So they're asking you to find the the area underneath the curve. So already this has standardized the value. So you take this minus that. So you go to find the table value subtract the table value from 0 0 table value and that will give you the answer you are looking for. You can take a screenshot and do it. This one they want you first calculate Z. Uh sorry it says greater than. Oh sorry it's greater than. So it's it's greater than X minus the mean divide. That's what you are calculating. So therefore it means you're going to say 1 minus the probability of Z less than X minus the mean divide by the standard deviation. I'm doing it this way so that then you just substitute into that because then when you get to the answer you will get Z less than a value and then you go and find that Z value and you subtract it from one. That's the principle. So your standard deviation is 20, your mean is 100, and your x is 145. So you just substitute there 145 minus 100ide by 20. Then you will get 1us the probability of z less than and then you get the answer and then you go to the table and find the table probability value. Then this one, what are they saying? Easy for cheesy. They're saying the probability of x less than a is this. So what you need to do is you can either go backwards uh to go and calculate because that's what they want you to calculate to go and calculate the value. I don't know what value they want you to calculate the value of a. So we work backwards because we need to go and find the Z value. So you go and look for this. I'm going to show you. You go to the table. Sorry. You go to the table. You look for 0 comma 1 515 inside the table. So it will always be on the negative side because the negative side has smaller values. 0 comma 5. Those are 1312 75. There you do two things. You go out to the left that is Z is -1. Oh, what did I see now? What did I do? See 1.0. You go up that's three. So that will be 1.03. I know what my Z is now. Easy. Remember the formula is Z X minus the mean by the standard deviation. Your X is the same as the A there. So what you do is because this was less than that's why I found the Z value easy. If it was greater than I will have said 1 minus this value so that I can get the actual Z value. So I just substitute - 1.03 = a minus our mean is 30. Pay attention variance standard deviation is the square root of your variance. So therefore it means this is the square root of 16 which is equals to 4 and 4 * -1.03 = a - 30. Therefore this is the same as whatever the answer here would be. What is 4 * now I'm taking time 1.03 * * 4 is 42 and you add 30 to it and that you will get the answer. I'm not going to do the entire equation for you. So 30 - 4 [clears throat] you will know what the answer is. So I was just showing you uh and the same as with this one it says the area to the right. So it means this area is 0 comma 2 6 or 2 2 0 6 61. That's what they say. The area to the right, choose which one is the correct answer. They just want to know what is the Z value there. The area to the right. How do we find it? It's 1us 0 2 0 61. So you will find the answer there and then you go to the table and find this answer. I'm I'm just going to call it the question mark. You're going to go and find the question mark and then go out and go out and find those two values and find what the value is. I hope you understand what I'm I'm saying cuz I'm rushing everything right now. So area to the right you find it this side. We know that that will be the greater than the greater sign. Right? If it's area to the right we know the table contains only less than. So that is why we subtracting it from one. We get the answer after we subtract the probability. this probability they gave us from one and that probability we go inside the table. We go look for it inside the table like I did with this one. You find that probability somewhere in here. You go up, you go out. It can be depending. Let me see how big is the value. So this is 0.2. So it will be on the positive side because you can see this is ne smaller numbers. So it will be 8 something or 7 something or 9 something you will find it here. Then you go out and you find the Z uh the first two values and then you go up you get the last value and then that will give you what the answer would be. I don't know the answer there because I didn't calculate it throughout. This one same same same same. They give you the Z value, right? And nothing about greater than or equal. But what they asking you is to calculate. You do the same as this. You do the same as this because they asking you to find X. So that is it. X. That's what they they need you to calculate minus the mean divide by the standard deviation. They gave you your z, they gave you your mean, they gave you your standard deviation. Substitute find x. I've showed you how to do that. This one is the normal one. They're asking you to find the probability of x is greater than 35 or 3.5. easy to find because you just need to find the probability that Z you standardize this value Z is less than your X minus your X is 3.5 minus the mean divide by the standard deviation you just substitute the values uh which is 3.5 - 10 / the variance variance. You need to find the square root of 9 which is equals to three. So divide by 3 and you find the probability of Z less than the value and you just go to the table and the answer or any one of this will be one of them. You will find the answer there. I'm not going to give you the answer. I'm just giving you the hint in terms of how you will tackle the question. I'm not going to go through all of them. You will go and sort it out uh yourself. On that note, I'm going to conclude right here and say thank you for joining the session. Remember any consultation you can book it with me. Don't forget to sign the register. Don't forget to do the evaluation. My name is Elizabeth Boy. I will see you next week when we do sampling distribution from 9 until 11. That will be on the 22nd. On that note, are there any questions? >> No question from my side. Let me share the link to the group chat. Uh how do we share this? >> Already shared it. >> Oh, you already shared it. Oh, thank you. Thank you. Thank you. Thank you. Oh, I can see you already joined as well. Uh what I will do is I will resend everything again to you. Don't worry. I will I will resend them on the group. those who received them, they will still receive them again. Um, yeah. Other than that, have a lovely, lovely Saturday weekend until next week. Bye. >> I have the rest of enjoy the rest of your week. Yeah. Thanks, man.