 Good morning, everyone. Good morning, Cecilise and the group. Please remember to complete the register. I have posted it on the chat. You should be able to access it now. How do I access it when I'm using the cell phone in the chat, Cecilise? Here. On your phone, probably you need to hover and then it will show some menu and then look at the top if they don't have something like a chat or a bubble, a speech bubble that looks almost like a, what's called, WhatsApp, WhatsApp bubble. You see something like I on the top there. I don't know if it is, but it doesn't open. I will share that on the WhatsApp and then you can. If maybe with the same. I don't know because I haven't looked up now. When I try to open the laptop, would I be able to access this link? Yes, you should be able to. Let me try it because I have never used it. I'm new on this technological issues. I also have a flu. Yeah. Let me close this and try to open the laptop. Yeah. I will request my son to help me. Okay. Alright, so let's start with today's session. Good morning and welcome to your 8th session of our acrylic looking at the basic statistical literacy. Today we are going to discuss how we tackle questions relating to continuous normal distribution. So looking at the schedule for June, knowing also for sure that you guys are some of you have some semester modules. You might be writing exams, which we can dedicate the two hour session every morning to do stats as well. So looking at June, we're going to look at sampling distribution the following week, and then we're going to look at confidence interval and then the last session will look at hypothesis testing and I think I can give you some break so that you can relax and then we will come back probably in August or September after the semester two registrations and then we can start again with the acrylic. I'm not sure, but if we feel there is a need to continue, we can continue. I have no problem. It's because you guys are writing exams and all that. And yeah, but because statistics is a year module, so there's nothing stopping us from continuing with the work with me helping you understand your work better. Okay. Do you have any questions, comments or queries before we start with today's session? And I hope you do have your tables, your statistical tables, because we're going to be using the statistical table and today at least we're using similar tables. So whether you're doing 16, 10, 15, 10 or 15 or 1, we will be using the same table or a similar table. Okay. So there are no questions, right? To close most of the things I've opened. I'm going to stop sharing again and start sharing again. I want to share my entire screen and to toggle between the two of my PowerPoint slides and also the statistical tables. Okay. I hope you are able to see my screen. So we can start with today's session. So like I said today, we're going to be looking at continuous normal distribution and the requirements for you to be able to do this successfully. We need statistical tables. We're going to use some formulas and we also need a calculator. So you should have all those three things ready. By the end of the session today, you should be able to learn the basic concept of normal distribution. You should be able to know how to compute probabilities from a normal distribution, especially when we're looking at different types of probabilities that we calculate in where the z value is less than or it's greater than or it is between. We should be able to find the probability using formulas or even tables. What is a continuous probability distribution? It comes from that distribution or the probability we will calculate from a continuous variable. And we know from chapter one that you would have studied in your modules and they actually introduce you to the types of variables. And I think as part of Akalit, we also did do the types of variables as part of our session. So you should by now know what continuous variables are, which are the variables that can assume any value in a continuum, which is a value that can either be measured or counted or continuous will be those that are measured. And it can take any value and that is a value that is not counted. So sorry, I said continuous is any value that can be counted. Continuous are values that can be measured, not counted. The counted one are the discrete, and we dealt with the discrete variables the past week as well. So examples of continuous variables can be like finding the thickness of an item, looking at the width, looking at the length of an item or the height of an item or a building, temperature of a solution, time required to complete an assignment or a task. As long as it's something that you are measuring, you can call that a continuous variable. And this can potentially take on any value depending on the ability to precisely and accurately measure those items. For example, measuring a building, you know that you will need a tape to measure how big the square meters and all those things. And you need to make sure that they are as accurate as possible. So when we talk about distribution, we know from the discrete distribution, we saw how the discrete distribution looks like. So in terms of normal distribution, it can be a very normal distribution and we can visualize the data by using this type of a chart, which looks like a belly type chart. And there can be different types of normal distribution. And those different charts, you can find them by varying the values of your mean and your standard deviation. And remember the mean and the standard deviation, we also know what those are because we spoke about them in the first few chapters of your study module, where it's part of your measures of central location and measures of variation. So by varying those two variables, you can look at different types of your normal distribution. And what do we mean by that? So we know that with a normal distribution, it takes this type of a belly type of a chart. And when we move or when we shift or increase the number of variables that we have, or not the number of variables, the number of the data that we have and we calculate the mean. If we increase or if our mean increases or our mean becomes larger, then the graph will shift to the right. If it's smaller, the graph will shift to the left because to the left are your smaller values. Right are your bigger values. In terms of your standard deviation, when you calculate the standard deviation, it will tell you whether your graph will be flat or tall or it will move up or down because the standard deviation tells you how far apart your values are from the mean. So we know that if in the middle it is our mean, which is the middle part, if the standard deviation increases and goes, therefore the chart will also flatten out because this line will become like bigger on the side. So this standard deviation of the first original chart, you can see that it is small and this standard deviation, yeah, it will be huge. And similar, if it is narrow, let's assume that this will be, I don't know how to draw. So bear with me. I didn't like drawing even at school. So this will happen. No, man. I don't want it to be the same size as the original one. Let's change it again. If you look at this standard deviation, you can see that it is even smaller. So then the taller your chart, the narrower your standard deviation will be. That tells you the spread of your data as well. So that is how you can interrogate or look at the different types of your normal distribution shape as well. So if your normal distribution moves from where it was to this side, therefore it means our mean has increased. And if it moves this way, therefore it means the mean value has decreased or it's small. And in terms of this standard deviation, we talk about being narrow or being flat. When it is narrow, it means we have a small standard deviation. And when it is flat, we have a large standard deviation. All right. So when we have continuous variables, sometimes your data can be skewed. Remember, we also spoke about the skewness of your data, whether it can be positively skewed or negatively skewed. So when your data is skewed, you cannot infer your results to the population if you're doing some analysis and insights and you want to talk about the population and all that. So in order for us to use that data, which is negatively skewed or positively skewed, we can standardize the information so that we are able to infer back the results when we're doing inferential statistic infer back the results to the population. Sometimes in order for us to understand the outliers and the extreme values, we can also standardize the values to understand the impact of the outliers on the data that we have. So to do that, we use a z-score or z-standardized normal distribution. In some cases, they call it a z-score. In some cases, we call it standardized normal distribution. So you will see that you might see some way it is used interchangeably to say this is a z-score, this is standard. In this purpose, we're just going to call it the z-score, right? We're continuing. So what we do with the z-score is we take your original values, we transform them into a z-standardized value or unit. And this standardized unit always has the mean of zero and the standard deviation of one. And that is the property of a normal distribution. This does not mean that you will not get questions where they have given you the mean and the standard deviation, but those are not standardized normal distribution values. We need to standardize them. We're going to use the z-score formula in order to convert our mean and standard deviation into standardized normal distribution in order to transform our values that we get from our analysis into a standardized form, which will take a form of a zero mean and a one standard deviation. So far, what we have spoken about is that normal distribution is a belly-cave shape graph. It is symmetrical. It means the mean, the median, and the mode are the same. But normally we don't also use the mode, but we can always say the mean and the median are equal, are the same. So the mean and the median of the data should be the same for it to be normally distributed. Or we can say it is symmetrical. So we know that the location is to be determined by the mean because the mean is one of the measure of central location or central tendency. It tells you the location of your data and the spread is calculated by means of using the standard deviation. Sometimes when you answer the question, they might not give you the standard deviation, but they might give you the variance. And we all know that the variance is the square root of your standard deviation. Or your standard deviation is the square root of your variance and your variance is the square of your standard deviation. So if in the question they didn't give you the standard deviation, but they give you the variance, you first need to calculate the standard deviation standard deviation before you can answer the question as well and we also know that it should be a random variable that has an infinite range which will be from negative infinity to positive infinity because it goes goes goes goes goes and with normal distribution as well you need to always remember that the values of your distribution or the belly calf will never touch your x-axis at any point so therefore there is always those small margin of error in between the belly calf shape and your x-axis so it will always go go go go up to negative infinity and go go go up to positive infinity and it will never pass through your x-axis at any point or it will never touch your x-axis at any point and that is why you can see from this graph that your belly calf shape does not touch the x-axis at any point in time. So how do we then standardize the values? We use this formula which is the z-score or the z-distribution by subtracting your observation by subtracting your mean from the observation and dividing by the standard deviation. So the formula is x minus the mean divided by the standard deviation and that is your z-score. Let's look at an example. If x is distributed normally with the mean of 100 and the standard deviation of 50 the z value for x will be so now we need to be calculating the z value. What have they given us? They have given us the mean which is 100, the standard deviation which is 50, 50 right? Then they also told us in the question that the z value for x so they gave us the x value of 200. So we go and we substitute into the formula because we know the formula is z is equals to x minus the mean divided by the standard deviation. Our x is 200, our mean is 100 and our standard deviation is 50. 200 minus 100 is 100 divided by 50 is 2 because 50 goes two times into 100. Therefore our z value where x is 200 is 2. Then it also states that it's the same as if I have to interpret this that the with the x value of 200 it is two standard deviations away from the mean or above from the mean. If it was negative 2.0 we would say it is two standard deviations below the mean. If it's positive we say it is above below when it is negative and above when it is positive. Usually they don't ask you to interpret the value of the z score unless sometimes some way they might ask you to do that but you should be expected to know how to do some calculations in terms of finding the z value because it's very important to know how to calculate the z value because when we later on when we want to calculate the probability of a standardized normal distribution value we need to use the z score to calculate our z value and then use the z value to go find the probability on the table but we're going to do that later on. So this is very important you need to know how to calculate the z score. This is your exercise for a particular group of scores the population mean and the standard deviations are 25 and 5 respectively so therefore it means 25 is population mean and 5 is your standard deviation. The z score of the raw score of 19 is so here they're giving us our x is equals to 19 so they're asking you to calculate z since we're still learning I'm going to give you the formula and you do the calculation so we know that this is our mean and this is our standard deviation should be easy and quick to calculate so should we type in a chat yes you can type in in the checked box while others are still calculating you can give you can give your responses in the chat I will keep an eye on the chat let me look in with my phone as well so I can see the chat okay I think now we've got a couple of answers so what is our x x is 19 and our mean is 25 and our standard deviation is 5 so 19 minus 25 minus six it's minus six minus six divided by five negative one comma two negative one comma two so the raw the raw score of 19 is 1.2 below the mean of 25 and that is option three and that's how easy it will be to answer questions if they ask you just to calculate the z score so you also need to pay attention to the type of questions that they are asking you and also you can look at the options and see if it's zero point some number zero point if it has very high numbers with negative numbers sometimes then it means you need to be calculating the z score so but you need to read the question carefully other questions that they can ask you will be general questions including also whether do you know the properties of a normal distribution so this we can do that together because I just told you just now and some of you might not have gone through the normal distribution content as yet which one of the following statement is incorrect with regards to the normal probability probability distribution statement number one says the z score of a mean of a normal distribution is one is that correct what did we know what did we learn we learned that the normal distribution is distributed with the mean of zero and the standard deviation of one so here they're asking is this the mean of one it's incorrect because we know that the z score is distributed or the the normal distribution normal probability distribution is distributed with the mean of zero so that would have been the one that is incorrect but let's continue and look at the other steps because that will talk to the properties of the normal distribution the smaller the value of the standard deviation the narrow and the steeper the curve will be is that correct we did this remember the graph I showed where we had the normal distribution like that and I drew another one and I said it will look like this and we looked at if all of them have the same mean we looked at the the standard deviation this is what they are asking if the standard deviation is smaller if this value is smaller the curve will be narrow and steeper can you see that it's narrow and it is steeper so it's taller steeper or taller let's call it that way that would be the correct one the mean of a normal distribution can be any numerical value in the negative zero or the positive so remember we said it can go up to negative infinity and it can also go into positive infinity and we know that normal distribution has the mean of zero there at that point but it can also remember the graph can also move can shift do you still remember we spoke about this the mean can also shift from being at that zero it can move it can move there it can move there it can move there it can move I told you that I don't know how to draw so mind my calves the way they look but I've tried so it can move even if here is your zero so we can have the mean there the mean there the mean there the mean there as well because the bigger the or the smaller the mean value so the mean value can take any value because it can take any value in the negative infinity to the positive infinity so that is also correct so we spoke about this as well that you can it can shift to the left or to the right remember that it's part of the introduction what we didn't speak about it's what we're going to talk about later on where it talks about the area to the right of the mean so we know that with normal distribution you know that this is the mean right so it means with normal distribution it splits the graph into two parts and if it splits the graph into two parts the other thing that we know about probabilities from from what we we have learned the probabilities the sum of all probabilities are equals to one so if everything underneath the belly calf if I'm going to tell you later on that everything underneath the belly calf is a probability therefore it tells me that everything underneath the belly calf if I add them all up I should get the sum of one then if I split this belly calf into two I am splitting it into two parts so it means I'm dividing this into two I'm dividing that one into two this is the question that they're asking you the area to the right of the mean of the standard deviation of the normal distribution will be 0.5 so it tells me if I'm splitting one into two I'm creating 0.5 this site and I'm also creating 0.5 because 0.5 plus 0.5 is equals to one and the area to the left so this is to the left and this is to the right it says the area to the right of the mean is 0.5 and the area to the left is 0.5 which means this is correct so I'm also covering some of the things that I was not going to cover in the slides but yeah I'm introducing them and this is the nice way of introducing the probability later on. The last one which is five this is based on what you also know from the empirical rule that we did with probability or not with probabilities when we were looking at measures of variability I think we also included in that discussion empirical rules where we looked at whether do we have one standard deviation two standard deviation away from the mean and three standard deviation away from the mean because with normal distribution we still use the mean and the standard deviation we can also still use the empirical rule because in terms of the empirical rule it is a normal distribution because it tells you how far apart your values are away from the mean is it one standard deviation is it two standard deviation is a three standard deviation because of the empirical rule we can also include it in the numer in the normal distribution chapter so a question can be asked 95% of the values of a normal distribution are two standard deviation away from the mean what do you still recall on the empirical rule that we did when we were doing the measures of variable no variability we said a one standard deviation is 68% 95 two standard deviation 99 three standard deviation you still remember all those and this is the correct statement because 95% of the values will be two standard deviations away from the mean okay so now let's talk about how do we find these probabilities that I have just introduced so finding the probabilities we're going to use the z score to go find or locate the probability so let's assume that I need to find the probability of a less than the thing that you need to understand is in your module we always going to use the standardized normal distribution we always going to calculate the value of the less than or the value of the greater than we will never or you are not going to be asked to calculate the value of an exact probability where it is equal unless if you are doing mathematical statistics a pure statistic um and I think those who are doing 1501 I think you are expected to know that no because I looked the only table you use is the cumulative standardized normal distribution so you are also not expected that this at your level to know how to calculate the probability of an exact especially when we calculate in the normal distribution probabilities the probability of an exact will be the probability at the point and that probability is different to the probability of a cumulative probability which is the probability of a greater than or a less than there is a table that contains those probabilities but usually the probability of an exact sometimes it can also be equals to zero but we're not going to touch on that because the session for today is to introduce you to this type of probabilities which are your cumulative probabilities so in order for us to calculate the probabilities which are the areas underneath the calf which is like for example the shaded red area if I need to calculate the probability of a less than I need to use the table now the table that you have both for 1501 or 1510 and 1510 and 1610 your tables contains only the probability of z less than a value there are two sides two sides of the table there is the positive side and the negative side both of them contains the probability of a less than in the positive side it contains the probability of the bigger side right in the positive it contains the probability of the bigger side which means it contains bigger probabilities it contains large probabilities as you can see with this large shaded area this probability will be big as compared to that probability of the non-shaded area the shaded area probability will be big the negative side of the table we're going to get there just now so here I'm referring to the tables right the table positive side of the tables where we talk about the z values the positive side of the table contains the bigger side of the probabilities the negative side of the table if I didn't I don't have a graph that shows I will draw one this is your normal distribution which is your mean there and I'm just going to shade this patch the negative side of the table will contain this smaller side of the probability which is the smaller shading this patch so it will only contain this if you are asked to find the probability of a less than value easy to find on the table you go to the table you find the probability if it's the z value is negative you go to the negative side you find the probability that's the value you are looking for if the z value it's positive you go to the z table you find the value of your positive probability which will be the bigger value right how does the table look this is the type of a table that you're going to be using it has the negative side of the z and the positive side as you can see there negative positive but your table also has the top patch now when we answered a question just now we calculated the z value and we said the z value was minus 1.2 all what I asked you is edge a zero always keep your z value to two decimals therefore it means round off correctly so that you can have your two decimal so don't drop decimal points before you get to your final answers so round off correctly so if I have a z value of minus 1.2 how do I read the probability on the table on the side which is your left on your left you will have two digits which are the first two digits at the top you will always have the last digit so if I read the z table I will read minus 1.2 on this left and one zero at the top so I must look for minus 1.2 so it will be somewhere there minus 1.2 I don't have minus 1.2 here but it will be somewhere there and the value for minus 1.2 is just give me a second 0.8400 so it will correspond with somewhere there it will be 0 comma 1151 I'm gonna keep it two decimal because I will read so on this side of the table what the value looks like so to find the probability let's say I need to find this probability of z less than because I'm not gonna look for z equal I'm looking for z less than minus 1.2 zero and that probability I'll find minus 1.1 and zero at the top and that will be my probability 0 comma 1151 that's how you will find the probability if I need to find the probability of z less than 1.20 I will go to the positive side because my z value is positive I'll go to the positive side and I will look for 1.2 and I will look for 0 at the end and that will give me 0.8849 now if I take this probability 0.1151 plus 0.08849 they will give me 1 because the sum of both probability should be equals to 1 and that is how you're going to read the table right easy easy easy any questions any questions yes yes I don't understand here I see these two tables now this one which is been shaded and this one which is not shaded so I don't understand where do you how do I get you say I can look at 1.2 to get a z negative if maybe I'm missing here okay so let's go back to our question so we went and we calculated let's say the question was asking us let's calculate the find the probability let's let's call it find because the question will say find the probability that x is less than 19 that's what we calculated right and we went and we calculated and we found that our z value was minus 1.2 and then I say 0 to the end because it needs to be two decimals now we need to go to the table it got less of the shaded areas ignore the shaded the most important thing is the value whether is it a negative or a positive and that you got if it's negative like this minus 1.2 so we come to the negative side and we look for minus 1.2 on the side of the table and there we have found it then we go to the top to look for the last digit you only looking at the last digit on the table and our last digit is zero so therefore it's the first column where they both meet that is the probability and that is what I maybe I did even remove it I did remove that and that is what example that I have done with you right here so then I turned around then I said what if what if when we calculated this let's assume that we calculated this and we found not negative but we found positive 1.2 zero as our answer so we go we found it as a positive value so we can't come to the negative side of the table we need to go to the positive side where there are no negatives so the positive we look for 1.2 and then you look for a zero at the top and that is the probability so let's look at more examples so that you can have a feel and be able to know how to navigate this table for these questions okay so let x represent the time it takes to download an image file from the internet suppose x is normal with the mean of 18 seconds and the standard deviation of five seconds find the probability that x is less than 18.6 the key words here is this less than that is very important it will tell us how we're going to find the probability on the table so for now I'm introducing the less than so how do we do that we need to find the probability that z is less than and we write the formula of z which is x minus the mean divide by our standard deviation we know what the standard deviation is we know what the mean is and we were given the x in the formula so p times the z of less than our x is 18.6 minus the mean of 18 divide by the standard deviation of five and that gives us p z less than 18.6 minus 18 equals divide by five equals 0 comma one two zero comma one two so now I need to go find this probability on the table you can just imagine I'm not putting anything in there because I'm finding the probability this tells me I must go find this probability so we go to the positive side so I can see that the answer here is positive it's zero comma one two so I'm gonna find this zero comma one on the positive side of the table zero comma one on the left and at the top I'm going to look for point two right let's go to the table you can come to the table we still remember what we do we're looking for zero comma one two so we're looking for zero comma one there right and the last digit comma five yes and that will be zero comma five four seven eight there we go yes yes on this table how do I identify the positive and negative side on the left hand side you can see this one that we are on right yes it's positive because there is no negative there is no minus in front of zero comma eight right when you go to the top table this is the other table you can see there is the negative the minus okay is your negative and where there is nothing is your positive yes that's how you identify you look at the answer you get on the z if it's negative you go to the negative side of the table if it's positive you go to the positive side of the table okay thank you oh sorry I need to clear the ink okay so that is what we just calculated right now we standardized 18 to a zero we standardized standard deviation of five to a one and we found that our z score was zero comma one two and that is how you would identify the question okay so that is the probability of the last done so I'm not going to show you again this because we've done this already how do we then find the probability of a greater than now we spoke about the probability of a less than now remember this picture this what I just showed you the first example I said the table contains the probability of the less than so all these values that we see here you're going to get the answer if the question is asking you to find the probability of a less than always if the question says less than you just come to the table you find the probability for greater than since the table that we are using to find the probability has the less than to find the probability of a greater than we're going to subtract the probability we see on the table from one so we're going to always say one minus the table value for all questions that are asking you to find the probability of greater than a value go into subtract the value you see on the table from one suppose x is normally distributed with the mean of 18 the standard deviation of five of five find the probability of z greater than 18 we've done this it's the same question that we just did when we started with the exercise now on probabilities so I'm not going to go into details because we calculated that and we know that it was zero comma one eight one two sorry zero comma one two we calculated this probability and we found that it's zero comma one two but now the question is it is greater than so because it's greater than we need to go to the table and we know that we did find it it was zero comma five four seven eight we need to take this value and subtract it from one so we take one minus zero comma five four seven eight and we find that the probability of x greater than 18.6 is zero comma four five two two so what we do is we go and take the probability of a less than and we subtract that from one so that we can find the complementary probability and that is how you're going to find the probability of greater than always subtract the table value from one let's look at more example oh no I don't have an example but we're going to get to the examples now let's look at how we find the probability of between finding the probability of between it's a little bit different to finding the probability of greater than remember greater than one minus the value and it's also different from finding the probability of equal of less than because with less than we know that the value we see on the table is the value that we are looking for for the between we will use the same method as the less than the value we see on the table for the b value or the second value we're going to subtract it from uh we're going to subtract the value of the first value on the table so we say the probability of z lies between a and b so we're going to find the value of b ignore the the what I'm saying is for the between just ignore the signs ignore whether it's less than less than or equal or less than they mean one and the same thing ignore the sign all what you need to do is make sure that the probability of this second part which will be which will mean we go to the table we go find this first so you first find this part on the table you look at this and then you go and find the value of this second table value then you subtract you say the first table value minus the second table value that's what we say with the less than that's the easy way to remember how to answer the question with the less than so with the less than don't go and complicate your life by trying to figure out why because we said it's less than and we do this and this other side it's a less than don't worry about that all what you need to always remember is write the value correctly because the smaller one needs to be first and the bigger one needs to be last so we're going to go find the value of the bigger one first minus the value of the smaller one second right how do we do this suppose x is normally distributed with the mean of 18 and the standard deviation of 5 find the probability that x lies between two values 18 and 18.6 you can see that 18 comes first because it's small and 18.6 comes second because it's big we go and find the probability so we calculate z score of 18 and we go and calculate z score of 18.6 and those are the z score so we can come back here and substitute and create a formula that looks like this the probability of a z lies between 0 because for 18 it was 0 and for 18.6 it was 0.12 so we have the probability of between here how do we then go and find that let me do it outside of this i'm gonna come here to the table to this so i could see that both of those probabilities that we're talking about they are positive right they are all positive like this but let's go there they are both positive it's 0 and 0 comma 1 2 so they are both positive so i'm going to work on the positive side of the table so i know that i'm looking for the probability that z lies between 0 and 0.12 like i said don't worry about the equal sign it doesn't create any difference whether you use the less than or the less than or equal one okay so how do we then do this remember we said we're going to find the probability of the first one first so we can write it as the probability of z less than 0 comma 1 2 minus the probability of the other side also do not worry about the sign we're just going to find the probability of the value that we find on the table which we know that the probability that we find on the table is of less than 0 right and because it's 0 and we want two decimals so i'm going to change my 0 to 0 comma 0 0 because i need to leave it to two decimals all right so now i need to go and find the probability of z of 0 comma 1 2 remember the second one first so i'm going to find the second one first 0 comma 1 2 0 comma 1 2 that is my probability which is 0 comma 5 4 7 8 and this side is 0 comma 0 0 so it's just the one on top 0 comma 0 0 which is 0 comma 5 0 minus 0 comma 5 0 0 0 and you go and calculate that and that is what can go back to our power so we went and we found 0 comma 5 4 7 8 minus 0 comma 5 0 0 and the answer we get is the probability of between is just those that are between 0 and 1 2 and those are 0 comma 0 4 7 8 so in order for us to find the probability of x that lies between 18 and 18.6 and that probability would be equals to 0 comma 0 4 7 8 and that's how you find the probabilities we're going to go into some exercises up until after 10 but before that sometimes you are expected as also to find not only the probability of between or the probability of the less than or equal or they might give you that probability and then you need to be able to calculate it so i'm not sure if we will be able to look at when you are given the probability and then you need to go find the z value because you want to reverse engineer everything so that you go back and find the z value so you can calculate your x probably we will have one of the examples just to show you so let's look at more examples in terms of finding the probability of between suppose x is normally distributed with the mean of 18 and the standard deviation of 5 now find the probability that x lies between 17.4 and 18 okay let's go and find that probability finding the probability i always like to write it inside so i know that i'm answering that question finding the probability so for 17 i'm going to find the probability that it lies because now i'm going to be calculating z and i'm going to do also for 18 and and the probability now we can substitute our x is 17.4 minus our mean of 18 divide by the standard deviation of 5 that lies between 18 minus 18 divide by 5 therefore i need to find the probability what is 17 minus 18 divide by 5 17.4 minus 18 is 0.6 divide by 5 is minus 0.12 minus 0.12 z less than 18 minus 18 is 0 so i don't even have to go and find divide by 5 because it's still gonna be 0 so now i have the probability of between and what i know is in order for me to answer the question i will have to find the probability of the first one of the second one first which is z less than 0.00 minus of the second one the probability of z less than minus 0.12 so we need to go to the table so we have 0 and 0.00 and we have what do we have minus 0.12 so we have the positive and the negative so the first one we're going to find it in the positive side it's all right and we we did find that which was 0.05 00 right so i can write that 0.500 minus the second one we need to go find it in the negative side so we go to the negative side of the table and we're looking for 0.12 which is the same as what we have been using uh no not the same it's 0. so we're looking for 0.12 right so before it disappears let's go to the last digit and go down and look for 0.45 002 0.45 002 0.12 is 0.314 0.13114 am i writing is 0.1814 0.1314 which then will be equals to 0.4 you said it's 0.36 86 sorry Lizzie you said 0 you said negative 0.12 oh negative yes and then you are in negative 1.2 oh sorry we went to the negative one we need to go to negative yeah and it's negative 0.1 ah yes i see yeah oh thank you for that which is 0.5 002 yes oh thank you for for 5 002 for picking that one up okay and that is and that is what you will do there will be a lot of mistakes because of these numbers there are so many of them so pay attention and double check your work as well um that will be 0.0478 as our answer of the between and that is what you would have learned on how to use the normal distribution table how to calculate the probabilities of between and how to calculate the probability of less than or the probability of a greater than remember the probability of a less than the value you find on the table that is the probability you are looking for the probability of greater than will be 1 minus the value you find on the table the probability of between will be the probability of z less than b minus the probability of z less than a which means the value you find on the table for b minus the value you find on the table for a right and that's how you do that um maybe there will be one of one or two questions that we can look at where you are given the probability but you need to go find the you need to reverse so you are given the probability of z less than a value which is 0 comma 1212 let's assume that and they're asking you find the value of x what is the value of x so we need to be able to move backwards as well to say if I know what the probability is therefore it means I must come inside the table look for 0 comma 1212 0 comma I hope there is one two one two one two one two one two one two oh we don't have one two one two and let's assume that that was the question that they gave us so we need to be able to go out and find the z value there and go up and find the last digit of the z value and say that is I'm going to change my question which is minus 1.17 so let's say that is 1.10 so we went and we found that and we know that now our z value which is the value of a a is minus 1.1 so if we know that a is 1.1 therefore we know that our z is minus 1.1 which will be less than and we can remove the less than and use equal x minus the mean divided by the standard deviation and hopefully they would have given you those two and you just substitute the values and substitute the values and solve for the value of x um let's hope that there will be one question that we can look at that looks like that so let's go into the question mode or exercise mode so these are the type of questions that we might get remember you're doing a multiple choice question and every statement with the multiple choice question it means you need to evaluate that question in order to find the correct answer so all this question one to five you need to go and find the answers for lucky enough with questions like this they already have calculated your z value so you don't have to go and find z minus the mean divided by the standard deviation they already calculated that and they're telling you that the answer to that is those values like zero zero minus one and one and they're asking you what is the probability so let's start with number one we can do this one together so you are going to use the tables and give me the answers like i said probability of finding the exact should always be equals to zero for your cumulative normal distribution because that right so i'm gonna give you the first one uh so the probability of z equals to zero will be equals to zero let's ignore the first one right go to the second one the second one says we need to find the probability of greater than zero so because we want to find the probability that z is greater than zero therefore it means the value we find on the table we need to sub direct it because on the table we're going to find the value of z less than zero right that's what we're going to be finding so and remember because we're working with two decimal tables so we can put two decimals at the end so let's go to the table and go find zero comma zero zero it's zero comma zero five all right zero comma five which is correct can you see that because that will be one minus zero comma zero comma five zero zero which is the same as zero comma five zero zero zero which to two decimals it will be zero comma five number three find the probability of z less than zero is that correct or incorrect that's correct that's correct because that's what we just did here is the same value that we've just it that is correct find the probability of z less than minus one the probability of a z less than a value it's always going to be found by looking at the value or finding the value on the table that is correct that is correct okay okay now the last question says find the probability that x is the problem is that z is greater than one and and they say is the same as one minus the probability of z minus one so this is incorrect correct you should be one minus the probability of z less than or equal to one is less than or equal because of this it will make that one incorrect but before you before you jump to that conclusion that it is incorrect what I will also suggest that you do is go and find the probability of z less than z greater than one so finding the probability of z greater than greater than one will be one minus the probability of z less than one right you'll go to the table you'll go find one which is zero comma zero zero which is zero comma eight one eight four one three right zero comma eight four one three eight four one three one minus zero comma eight four one three and you solve that what is the answer one minus point eight four one three which is zero comma one five eight seven and then you come to this side as well you go find the probability of one minus because this said the answer to this should be the same as the answer to that so one minus what is the probability of less than one comma zero minus so we go to the negative side we're looking for minus one and zero zero which is zero comma one five eight seven right one minus one minus zero comma one five eight seven which is equals to and probably it is equals to zero comma eight four one three and those two are not the same you need to be very careful when you answer the questions like this as well just go and double check that you are able to evaluate each one of them and make sure that they are correct so don't take any shortcuts because actually we we shouldn't even have went to the table because we did find that was zero comma one five eight so if they could have removed you can see that right if they would have removed this minus if they had to remove this one minus the value there and they ask you if it's correct can you see that this will be the same because the answer to this is zero comma one five eight seven therefore the probability of z greater than or equals to one is the same as zero comma one five eight seven you need to be very careful especially when it comes to multiple choice questions please make sure that you evaluate each statement and not make any assumptions and you can see the answers would have been correct and it would have been the correct one as well but we know that we know that that is so the incorrect one will be that one so let's look at the next question consider the standard normal distribution of z which one of the following probabilities is incorrect i'm gonna give you some few minutes just find the incorrect one you can write your answer in the chat and then we will come back and do some feedback uh zolika yes you can use it and that is also you can use it for the probability of the exact sometimes it can also be useful for that if they ask you the probability that x is equals to the value because that is the standardized it's not the cumulative probabilities but the standardized probabilities but we need to know how to use it properly otherwise you must just rely on the negative and the positive table are we winning not specifically i don't have a table i will need to have a table first so i will not win for now don't you have your study guide next to you justice no it is not with me now okay yes actually they didn't send it i need to type it to print it but it's online you can download it online okay cislis on my unisa yes okay i will do so you must always remember to bring your tables every week from now on until until until june ends until all the sessions always we're going to be using introducing new tables it's going to be i will come easier if you bring your tables with you so that you are able to work through them i will do so cislis okay are we winning or are we still calculating no response we are still calculating man guys you need to respond we are still calculating all right i'll give you more extra time are we done let's see do we have any response on the chat nada next are you guys struggling are you winning are you not sure option five so many essays option five let's let's see let's see so option number one remember like i said very tricky questions you need to evaluate each one of them and make sure that they are correct so to answer this first one we need to go find one minus the value we find on the table because it's greater than so we're going to minus 2.80 right so we come to the negative side of the table delete all this negative 2.8 negative 2.8 and zero it's at the top and that is zero comma zero zero two six i agree zero comma zero zero two six and the answer here will be one minus one minus point zero zero two six is equals to zero comma nine zero comma nine nine seven zero comma nine nine seven four right that is that one so this one is a little bit tricky as well because it says one minus the probability of greater than two positive 2.8 right so this we treat it as one minus and we convert this because we don't have any value on the table that is greater than so we say one minus the probability we're going to find on the table which is of less than 2.8 so it means we go to the positive side of the table and we go find one minus one minus because that is always there one minus one minus is that patch that is the one minus and i just expand this into one minus the probability of a less than so we go and find on the positive side 2.8 so we go to the positive side of the table so we'll use this column zero comma nine nine seven four right zero comma zero comma nine nine seven four right and that is one minus zero comma zero zero two six which is equals to zero comma nine nine seven four so therefore this is correct see how tricky it is that they would have tricked you by looking at this and you are in the exam you would panic and you'll say oh yes that is the incorrect one and you stop waking throughout the whole session on other options because you would have assumed that that is not correct so that is correct number two the probability that z is less than negative 2.11 we can go and find this value on the table and finding the value we go to the negative side of the table so we go negative side of the table and remember what we're looking for forgot minus 2.100 at the top so it means we're going to always use the first column so the first column is way it is zero 2.1 we're looking for 2.1 all right negative 2.1 you see I have to double check because the last time I made it an error because I didn't double check my work which is zero comma zero comma zero one seven nine right and then we know that this is zero comma zero one seven nine this side it says it is greater than so with greater than we know that it's one minus the probability of z less than 2.1 so we need to go to the positive side and look for 2.1 so go to the positive side and we know that we're looking at the first column so 2.1 is zero comma nine eight two one so we go there we say one minus zero comma nine eight two one which is equals to if we calculate that it will be one minus point nine eight two one which is equals to zero comma zero one one seven nine which is the same so the left hand side is the same as the right hand side okay the next one is the between so between we know in order for us to find between we're going to find this first z less than zero comma zero zero and minus the probability of z less than minus 2.80 and that is equals to let's put them on the line like that I'm going to do it like this okay so that I am able to calculate correctly so we're going to find the on the positive and on the negative so let's first start on the positive because it's zero comma zero zero which is zero comma five zero zero so that will be zero comma five zero zero minus and on the negative 2.8 did we find that I don't have to go to the table because some of these things we went and we found them which was that value of because we're going to the negative side when we found this value we set it to as one minus the probability of z less than minus 2.8 that's what we did and which is the same as this one so I'm just going to use that value which is zero comma zero zero two six which is equals to so you just say point five zero zero zero minus point zero zero two six is equals to zero comma four nine seven four which is the same which is the same which is correct then we go to number four also number four says the probability of between so we're going to find the probability of the second one first z of less than 2.1 minus the probability of z less than minus 2.8 so we did find the probability of z 2.8 we did find the probability of z less than 2.1 which we did here remember and we found that was zero comma 9821 so I don't have to go to the table because some of these values as you go along you have already found them so because this was less than 2.1 and it's positive that is less than 2.1 which was positive we did find that was zero comma 9821 otherwise you can go to the table and go find 0.221 and go there and find that is zero comma 9821 oh you can just rely on the information you used previously minus and we did find this I'm not going to go to the table we said it was zero comma 0026 and that is equals to 0.9821 minus 0.0026 equals 0.9795 that is correct that's why I don't like I don't enjoy doing multiple choice questions because you went very hard five times as much because now I should have been on question number five of my exam paper I'm still on question two this is a struggle so number five we need to find the probability of the second one which is z less than 2.1 minus of the first one which is the probability of z less than zero like I said you just ignore those greater than and less than and for the exercises you don't need to add too much about them all you just need to remember is take the second one minus the first and that's it you will know how to answer these questions and that is the base your e2 tower will teach you how to do or read these things normally like a normal statistician me I'm giving you the skills I'm giving you the skills allowed to answer your questions that's it okay if it means showing new shortcuts I do that because I want to see you succeed okay so let's see 2.1 we did find that it was 0.9 0.9821 minus and for zero we didn't oh we did find it it was 0.05 remember we did find the probability of less than zero and it is uh zero comma five zero zero five zero zero zero equals so nine eight 0.9821 minus 0.500 it equals to 0.4821 which makes number five incorrect happiness yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes I can see that most of you know how to do these things so it's gonna be I'm gonna skip this one because you can take a screenshot of it um because you you already know how to answer some of these questions like this I want to get to something different so this is another question you can take a screenshot of it um or otherwise you can come back to the recording to look at exercise number three and if you're still not getting derived you can talk to me on whatsapp as well here is another question that would surprise you so here they say the shaded area underneath the calf following a standardized normal distribution calf is equals two and they give you the shaded area so what they did here is they give you the z values right because the shaded area you always need to remember that the shaded area is the same as probability so when they talk about shaded area or area underneath the calf and all those things think about it as the probabilities so looking at the values here that we have since they didn't give us the mean the standard deviation or they didn't tell us it's between this and that but we can read from this that this is our z lies between those two values and we can find the probability of that that's as easy as it is you can see that I've just implemented the question on top of what they just gave us to answer because they just want to know what is the probability oh come on what is the probability of the value between two values right so what we do we go to the probability of z less than 1.25 minus the probability of z less than 0 comma 0 this site I don't even have to now look at the table because we've been using it now I know that is 0 comma 0 5 now this other site we need to go to the table so let's go to the table minus 1 let's go back I must remember 1 comma 25 1 comma 25 it's on the positive side we're looking for 1 comma 2 and then at the top we're looking for 5 I always do it that way because we're working online when I scroll it hides all the other values and the answer is 0 comma 8 9 0 comma 8 9 4 4 and that is equals to 0 comma 8 9 4 4 and then we answer the question is it right 0.89 4 4 minus 0.5 0 0 0 equals 0.39 4 4 which is option number 4 that is how you will answer the question because at the end of the day you go in to look at this question and panic because they didn't ask you what is the probability they are not giving you the z values they just give you the graph and they expect you to know what to do with this kind of a question so you just need to quickly visualize it in your mind to say oh but then they gave us the z values here because if they are asking us to find the area underneath the calf it means they're asking us to find the probabilities and this value should be the z values because they didn't give me the mean and the standard deviation so it means they are not expecting me to calculate the mean and the standard deviation to answer this question to find the z because they gave me my z values okay and you just say oh it's it's the shaded area is between two values so you just put the probability of between two values and answer the question like is the probability of between so this is another question do I want to work through this we've done more questions relating to probabilities that look like this but let's look at this suppose x is normally distributed with the mean of 100 and the standard deviation of 20 the probability that x is greater than 145 is so they've given you the mean they've given you the standard deviation they've given you your x of greater than and they're asking you to find the probability that x is greater than 145 and to answer this question we know that it will be one minus the probability that z like oh sorry the probability that z is greater than 145 not 145 but z is greater than our x minus our mean divided by the standard deviation which we also know that it will be one minus the probability of z less than our x minus the mean divided by the standard deviation why I'm doing all these things it's because it's greater than and we know with the greater than we'll eventually have the answer like this so we can also substitute from there 145 if you want or you can substitute into the next question otherwise you can give it as the question as it is as you practice alone so one minus the probability that z is less than 145 minus the mean of 100 divide by the standard deviation of 20 and you go and calculate one minus the probability of z less than what is 145 minus 100 is 45 divide by 20 it is 2 comma 2 5 2 comma 2 5 so we're gonna go to the table on the positive side and look for 2 comma 2 and then at the top we look for 5 where they both meet which is 0 comma 9 8 7 8 and we come to that one minus 0 comma 9 8 7 8 and the answer is 1 minus 0.9878 equal 0 comma 0 1222 1222 and that's how you will answer questions this I like for multiple choice questions because then it's one question I need to choose which one is correct based on the answer and I had to calculate it once not like the others okay I'm not oh yes this is one of those questions that I was referring to to say they give you the probability but they don't give you all the things that you need and they ask you to find either x or the standard deviation or the mean or any of those values so they say here the probability that x lie or x is less than a is given by that which is 0 comma 1515 and it is the area underneath the curve of a random variable x with the mean of 30 and the variance of 16 the value taken by is equals to so what is our x that's what they are asking us yeah all right so what else have they given us they've given us the probability so we can use this probability to go find the z value because we know our z formula is x minus the mean divided by the standard deviation so we're looking for the x we have the probability so it means we can find the value so we know that the probability of z less than a is 0 comma 1515 I'm using this scenario because they told me that that is the probability of that because I know that this is the probability of x less than that x less than a which is the same when I go to the table I'm using z not x I use the z and I know that to find the value of a I use this formula this formula finds me the value of a right that's all what I know for now so since I know that a gives me this therefore it means I need to find what this a value is that gives me this 0 comma 180 so let's reverse engineer let's go find this value here let's go find this a by using 0 comma 1 and this is the value inside the table right that is the value inside not outside inside so we go in to the negative side because we know that on the positive side of the table these are big probabilities we're looking for one zero comma one five so it cannot be on the positive side it will be on the negative side so we're looking for zero comma one let me write it here zero comma one five one five that's what we're looking for inside the table any of these values should be one five one zero comma one one one one three one five one five one there is our probability that we were given now we need to go out and collect our first two digits which is minus zero comma one I'm going to go and write them here so now we have our value so we know that z of less than minus one comma what did we get one comma zero one comma zero but it's not the end I need to go out up to get my last digit always remember that right that the table has all the z value has three digits oh sorry two two digits after the comma so that is three and that is three is the same as zero comma one five one five so it means my z value I have my z let's substitute into the formula our z is minus one comma zero three which is equals to x is what we are looking for minus the mean they told us that the mean is steady they also gave us the standard deviation nope they gave us the variance how do we find the standard deviation standard deviation is the square root of the variance and the variance is 16 therefore the standard deviation is four and then it is four and we use mate multiply because it's dividing so we can multiply four times minus one comma zero three is equals to x minus thirty therefore x will be equals to what is four times point one oh three which is negative it's negative zero comma negative zero comma four two zero and this is minus so when it moves this side it becomes positive so that will be 30 minus oh minus 0.412 plus 30 will be will be equals to uh i'm calculating something wrong me so um what have i done wrong pardon 25 comma eight eight why is my thing so wrong is there something that i did wrong let's see point one yes i see that i did something wrong that's why i am so it is negative four point one two oh and the answer is 25.88 and that's how you will answer the question so we left with three minutes i just want to scroll through some of the questions that i've included in the notes notes are downloadable from the my unisa site where you access the the recordings um some of the question might look like this here they gave you the z value as well if you look at this let's look at this one as well so it says given that z is normal or is standardized normal distribution what is the value of z so yeah we're looking for the z value such that the area to the right of z is 0 comma 261 so we must also be mindful of weights like that the area to the right it means the area greater than let me not put the equal sign the area to the right it means the area greater than the area to the left it means less than so since they say the area to the right greater than so to the right the area to the right which means it's greater than they are telling us that this area here is is 0 comma 2061 all what we know is in order for us to find this 0 comma 021 we had to do pre minus the probability of z less than a and that would have given us 0 comma 2061 that is this area here based on that formula remember for the greater than we say 1 minus the value on the table and that gives us the probability so in order for us to know what that value of a is we need to use this probability that we have but we need to go and subtract it from 0 so that sorry from 1 so that we can find the electron value of z if we use this let's go it's 0 comma 2061 so we go and look for 0 comma 2 let's delete this 0 comma 2061 there if we take this and go out which is 0 comma 8 2 minus 0 comma 8 2 if we say the answer is minus 0 comma 8 2 if we say it like this we will get it all wrong because we're going to assume that the value we find on this we found it by using minus 0 comma 8 and then we went and found this probability this will be incorrect what should be correct should be we need to go back one way you need to substitute that into this so we need to go find this value so we know that this is what we have but we know that this we found it by finding 1 minus 0 probability of z less than a is equals to 0 comma 2061 we found it by using that and that is the value we found on the table so in order for us to find this on the table we're going to say 1 minus 0 comma 2061 is equals to the probability of z less than a because that is the value we used on the table and that will be the probability that we found on the table was 1 minus 2061 which is 1 minus 0.2061 which is equals to 0 comma 79 3 39 that is the value that we used so we need to go to the table and find this 0 comma 79 so we go there on the positive side because it's 0 comma 7 it's a bigger probability so we come here 0 comma 7 9 7 6 7 8 7 9 now I forgot what else was there 7 9 3 9 7 let's write it down 0 comma 7 9 3 9 so that we know exactly what we're looking for 7 9 3 9 which is that way so we go out 0 comma 8 go out 2 so the answer we have our a which is what we're looking for the value of z so our z of greater than 0 comma a 2 so the answer we're looking for it's actually number b and that is how you will answer the questions okay any questions any questions before or before I get to the questions because I can see that we are two minutes over but it's fine there's our other questions you need to also calculate your x you are given the z value so now here you just substitute into the formula you will just substitute x minus the mean divided by the standard deviation you have the z value you just substitute the z value you have your standard deviation and you have your mean you just substitute and find the value of x which is straightforward the other question you need to calculate the probability of x but here you are given the variance so you need to calculate the probability that z is less than x minus the mean divided by the standard deviation and remember this is the variance you're going to first find the square root of your variance and it is less than the value you find on the table that will be the value you find on the table z of less than a it's the same as the value you find on the table question number 10 was going to ask you to calculate the probability of student failing so they gave you the standard deviation and the mean don't get confused even if they give you in percentage form you can use the percentage or you can convert this to a decimal this is 0 comma 5 6 and this is 0 comma 0 4 what is a percentage of a fail a fail is anyone who get less than 50 which is 49 and below that is a fail or you can use 50 i'm not sure which one now what is the probability of a pass pass is more than 50 fail is less than 50 or maybe you can say the probability of less than a 50 and see if you get any of those questions there which is very tricky now if you use a 50 for a fail or you use a 49 you can check both and see if any of those probabilities you do get and then the last one also is asking you to calculate the probability of at most at most is less than remember at most at least at most is less than or equal but you can just use the less than and then calculate the probability and that is everything you needed to know about continuous normal distribution probabilities or probability distributions are there any questions please remember to complete the register those who haven't completed okay i'm gonna repost it because i think something went with my link i'm about to complete the register those who can access the register let me know as i indicated this is easy i couldn't use the laptop so i'm struggling to access the register here okay i'll put it now you want to say guest here okay sorry you want since you are a guest here you need to login with your