 That's great because I'm going to toggle between the screens. Okay, so welcome to your session eight. It feels like it has been long and today, we're going to start doing study unit six, which is continuous normal distribution and in short, I'm always going to refer to study unit six as normal probabilities. Because we don't do all continuous distribution, we're only doing one continuous distribution, which is the normal distribution. So in your module, even when you use your prescribed book and study, do not study everything else, we only do one normal distribution or normal probability distribution. That is the only section that you need to worry about. The other thing in terms of the normal distribution, because you're not doing pure stats, you are not going to worry about the probability density functions, like we didn't do probability density functions. I'm going to apologize since log shading started, my network has become unstable. So if you don't hear me, let me know so that then I don't lose anyone, or I don't just talk to myself. So we don't deal with probability density functions. So you don't cover that, don't even worry about that. If I don't mention it, it means it's not important to you, or it's not important to your module that we are busy with. So based on our session plan, you need to keep on watching the session plan so that you know when we have tutorials and when we do not F, especially the online sessions. So today is the 10th, we're having this continuous normal distribution. Next weekend on Sunday, we're going to look at sampling distribution and remember, we're working towards you being able to submit your assignment three, and session 10 will be on the 24th of July, and we will concentrate on the two chapters, the continuous normal distribution and the sampling distribution, and we'll do question and answer, and answer more activities, and do more exercises, and so on. All right, and then you just need to follow this. I will also amend the calendar, because I think people saw that I didn't change anything on the calendar. When we do not have a session, then the session is already pre-scheduled, because I just did an automatic pre-booking of all the Sundays, not thinking of there will be Sundays that we are not having any sessions, but I'm going to adjust the sessions online as well. Okay, if you have any question, or you can stop me right here and ask any question as well before we start with today's session. If there are no questions, then yes? Yes. While going through the notes this week, I came across the nominal probability table. Where can I find this table? Okay, I'm going to show you where to find the table. There are some books, they do have two tables. They do have a standardized normal distribution table. They also have a cumulative standardized normal distribution table. We do not use just a standardized normal distribution table. We use the cumulative one, and I'm going to show you during the course of the session today how to use that table, because that is the table we're going to be using from now on until you go write the exam when we do confidence intervals, or even when we do in the next session in study unit seven, sampling distribution, we're still going to be using the same table. When we do confidence intervals, we're still going to be using the same table. When we do hypothesis testing, we're still going to be using the same table. So, but I'm going to show you how to read it and which table we are using. Okay, don't worry. Okay, so in order for you to be able to do this chapter or this study unit properly, you need to have your statistical table. You need to know your formulas. The formula here, it's easy. It's the simplest formula that we're going to be using and you need a calculator to do that. By the end of the session today, and that is what I also thought, because I forgot to publish the study unit area for you to also refer to it and go through the content before we do the discussion today, I went back and I revised the whole chapter. So today we're just going to do the whole chapter, the whole study unit in this one hour, the minutes that we have. But in a nutshell, by the end of the session today, you should be able to learn how to apply the basic concept of normal distribution. You should be able to calculate the normal distribution probabilities by looking at the table, finding the probability of a z less than, finding the probability of z greater than a number, finding the probability that z lies between two numbers or finding the probability or any values using the empirical rules or not using the empirical rules depending on what you need to be finding in terms of normal distribution. And I guess you did lose me right there. Just give me a second. I need to check if I can connect via my phone because my wifi is having these difficulties. I must just apologize for that. Just give me a second. Do you still see the presentation? Probably I should just stop and re-share again. Put the journal in front of that. Okay, like I said, we should be able to do all of this by the end of today's session. Okay, what is a continuous probability distribution? It comes from a continuous random variable. So a process that comes from a continuous variable, those are the variables or it's a process where it is measured, right? Because we know that a variable type that is continuous or quantitative continuous, it's that variable that is measured. We can assume that any value on a continuum takes up that continuous variable. For example, the thickness of an item, if you take a measuring tape and measure the thickness of a nail or a bottle or something. The time required to complete the task, you need to have a clock to time or check the time or record the time. The temperature of a solution, you need a thermometer to test the temperature of that solution. If it was boiling, you will need a thermometer to be able to measure the temperature. You can't put a finger or you can watch it and say the temperature is this much. So you need an instrument to do that. The height, you need a measuring tape also to measure the height, either in inches or in square meters or in whatever the unit, the measurement unit you are using. Okay, and this can potentially take any value depending only on the ability to precisely and accurately measure that variable. And this adjust your continuous random variables. And when you have those variables or when you have measured, for example, the thickness of the nail, you measure each and every nail that you have, let's say the population nail for that factory when they produce the nails, they're about 3,000. You measure the 3,000 nails that they have produced and you can take the, you can draw a histogram of the measurement of those thickness, of the width of the nail and it can be displayed on a histogram and the pattern of that might follow a normal distribution and a normal distribution is something that looks like this. Where you might find that depending on the value of your mean or your standard deviation, we know what the mean is. It's a measure of central location. It measures how far away are your data. It describes the location of your data and we know what a standard deviation is, is the measure of your variability or dispersion or how far up at your data is from the mean, right? So when you look at the distribution of your data or the thickness of that nail and you can see if it follows any of the patterns, it can be the mean of that might be at the central point and then the standard deviation might tell you whether you're getting a top, like a top pick like the yellow one or a normal distribution sort of pattern or a flat one. So this, if I draw the mean, let me just do it that way. If I say this is the mean of this one, the mean is at this point, you can see that the distance between that mean and the standard deviation is smaller so that might be a one standard deviation from the mean and it creates that peak and this might be a two standard deviation away from the mean for the blue one and we can see that it looks like a normal, a median normal distribution curve and with this one, I can draw the mean here but if you look at the distance between the mean and the outside of the curve, you can see that this is a big distribution where the gap between the mean and the standard deviation are very huge and it creates a flat normal distribution curve. But if we also look at the difference between differing the means, you can see that the pinkish color, the mean is more than the mean of the blue and the yellow one because the blue and the yellow ones, they've got the same mean except the pink one has a different mean. So when you vary the mean, your graph will either move from left to right, we'll discuss this later on, it will move, it will shift from left to right based on the mean and your standard deviation will shift your graph up and down in terms of whether it becomes narrower, taller, shorter and normal. So like I just mentioned, when we change the value of our means, the normal distribution curve will move from left to right and when we change by increasing the value of your standard deviation like you saw, the curve will either flattened or it will become narrower and taller by just looking at the spread of the values. So when we talk about normal distribution as well because we do collect the actual data which is your observed data that when you go and you start measuring something, that is the actual data and sometimes your data might not be normally distributed. Like I showed you with the graph, it can look like this to say it's normally distributed sometimes it's not in that case when the data is not normally distributed, then we need to standardize the data. By standardizing the data, we say within any normal distribution with the mean and the standard deviation combination can be transformed into a standardized normal distribution by applying the Z, which is also called a standardized normal distribution. I know that earlier when we started looking at measures of central locations and measures of variation we did touch on some Z scores when we were talking about outliers and all that but in the nutshell in your module, the outliers we do not always refer to the Z scores when we calculate them and all that we don't refer to those ones but those who are doing pure math or pure statistics usually they do have to understand and interpret what the Z score is and how they calculate or find their outliers. In your module, we use the Z score usually for only standardizing the distribution so that it becomes normally distributed and that is where we only use the Z score. So in order for us to standardize we need to transform our X observations into the Z units and to standardize the normal distribution it will have the mean of zero it doesn't mean that when I'm saying it has the mean of zero and the standard deviation of one it means everything that you need to be working on in this module will have a mean of zero and the standard deviation of one, remember? Any object will have or any population that you would have selected or the sample that you are using will have a mean related to that but that mean might be from a sample or a population that is not normally distributed therefore when we standardize it we then say it has the mean of zero and the standard deviation of one, right? That is when it is standardized What is a normal distribution and how do we identify it? It is what we call a belly calf shape so you will see that it looks like this the graph on our right is a belly calf when a pregnant woman is pregnant they have this belly so that it looks like that and sometimes we refer to it as a symmetrical distribution or the data is symmetrical and we know when it is symmetrical in terms of measures of central location we say the mean and the median are equal so it means the mean and the median are on the same point and the location or the locality of the data which is your central location of the data is determined by the mean and the spread for the dispersion is determined by the standard deviation we use those two values the mean and the standard deviation and any value because there can be any value of x you can see that x can be from negative infinity to positive infinity so any random variable can be lying on any of the x-axis anyway on the x-axis whether on the negative infinities or on the positive infinities it can be lying between those values there so when we talk about infinity we're talking about the number that we don't know how much it is in the biggest or the smallest smallest number that we don't know what that number is it is infinite and infinite okay but what you need to also realize is that usually in your module because they cannot ask you to find an infinite mean or standard deviation or sample from an infinite thing we always are told what the population is or we are told what the population parameter looks like and then we can make assumptions from there as well so how do we standardize the data we do this by using the z-score so our z-formula which is also called the z-score it's also called the z-distribution so it doesn't matter which word you use but we can call it the z-score or we can call it the z-distribution or the z and you will see when we go to the table I'm always going to say go to the z-table so we're going to be using the z-z-z-table or the z-distribution table so the z is your observation x minus the population mean divided by the standard deviation and that is how you standardize your x values and we know that this formula or this z-distribution is always standardized by the with the mean of 0 and the standard deviation of 1 so we're not going to put here 0 and put here 1, no, no, no that's not what we mean but this whole formula is standardized with the mean of 0 and the standard deviation of 1 what do we mean? so in the middle on your z remember now I've transformed my x value so this is my y and this is where my x was my x value I've transformed it into a z then my mean will be 0 and my standard deviation will always be 1 standard deviation away from the mean and that will be where my standard deviation that will be my z-distribution in a nutshell it will look like this that is a normal standardized distribution now on the left of the mean so we know that this is the mean on the left of the mean there will be some negative negative values so this side we know that there are negative values and on the right which is going that way there will be positive values of z so it means also on the table that we're going to be using we're going to have negative values and positive values so remember this when we get to the table as well let's look at an example of how do we standardize the x values if x is distributed normally with the mean of 100 and the standard deviation of 50 the z value of x is equals to 200 is given back so we talked what the mean is so we know that it's our mu our standard deviation we know that it's our sigma and the z value they told us what x is 200 we just go to our z score formula and substitute x is 200 the mean is 100 the standard deviation is 50 and we calculate 200 minus 100 is 100 divided by 50 is 2 now I'm going to ask you later on when we go to the table we need to leave it into 2 decibels so you're going to just add another 0 there it's very very very important to remember that right so the answer will be 2.0 you are not allowed or you are not expected to interpret the z score good people there is no way in your module they're going to examine you of interpreting what the z score mean no they are not but there will be some theory questions they will ask you about the normal distribution all those things that we spoke about you will be examined on but not the interpretation so if they in case because I will never know your lecturer sometimes surprises me in case they ask you to interpret the z score then you will say with the x of 200 it means your z your x of 200 it is 2 standard deviation because the answer there was 2 it is 2 standard deviation away from the mean or above from the mean because it's positive it was negative you will say below the mean so it's 2 standard deviation above the mean because it is above because it's positive if the answer here was negative negative 2.0 then you will say below below the mean of 100 which is the same as 2 increments of the 50 rents per each that is only if you are expected to interpret the z score other than that I've never seen anyway in the past exam papers or in the assignment questions where they ask students to interpret the z score and this is done when you're doing pure steps okay so let's see if you remember everything that I just said in 2 seconds or 5 minutes ago which one of the following statement is incorrect with regards to normal distribution so this one is one of those theory questions that they can ask therefore it means you need to know your normal distribution property the basic concept of normal distributions everything that we just learned okay so number one remember we're looking for the incorrect statement number one says the z score of the mean of normal distribution is one what do we know about the mean of the z score we know what what do we know what do we know about the mean is distributed with the mean of zero and the standard deviation of one yes so they said the mean so they said one so this should have been zero so it means which one of the following statement would have said that but it is incorrect right the smaller the value of your standard deviation the narrower and the steeper the kev this question that we just dealt with so let's assume that this is 0.5 standard deviation and this is one and this is 2.5 so the question was the smaller the standard deviation the narrower or the steeper the kev is that is true right because if this is smaller than one you can see that the kev the yellow one has the smaller standard deviation that would have been correct the mean of the normal distribution can be any numerical value in the negative or positive or zero someway there and that is this the mean can be either in the negative or in the positive right or it can also be zero because we know the standardized normal distribution is the mean of zero so that one is correct right the area to the right of the mean of standard deviation is 0.5 and the area to the right of the mean of the standard normal distribution is 0.5 what didn't I tell you now is the area now going back to the graph now I can just explain this if you look at this graph I'm just gonna remove all this the area underneath the kev so everything underneath this kev yeah all these values we call it the area the area underneath the kev this is your probability values all this the area underneath the kev is your probability values now if this is the probability value therefore it means the sum of all the probabilities will be equals to 1 if I split this graph into half because it says if I split it of the mean of 0 where it is it tells me this will be 0.5 this will be the probabilities on the right of the mean will be 0.5 because if I add both of them they will give me 1 the sum of all these probabilities underneath the kev will give me 1 so that is what the question was asking you so if you get something like this asked you just need to know how to visualize it and explain it so the area to the right of the mean of the normal distribution if this is my mean the area to the right will be this side to the right and the area to the left will be this one and we know that this is negative and this is positive the area there and we have the mean 0 in the middle but the probabilities which are there the probabilities underneath or the area underneath the kev the other thing you need to be aware of in terms of the area underneath the kev that's the other thing that I didn't mention you will notice that your values of your area underneath the kev so this belly kev will never touch the X XS anyway will always be closer to the XS but it will never touch so there's always some little bit of an area or a gap between the XS and the and the belly kev so they never touch now this is something that we also have studied before but before we introduced the normal distribution we dealt with impercal rule remember that and I think they introduced the empirical rule with your we introduced the empirical rule when we were doing measures of certain locations and measures of dispersion now from what you remember with the empirical rule remember there are three areas there is the mean in the middle and there is area number one and area number two and the last area so we know that it's one standard deviation, two standard deviation and three standard deviation so what we know about one standard deviation there are 68% two standard deviation there are 95% three standard deviation there are 98% or 99% I can't even remember but there are 98% three standard deviation so two standard one standard deviation so I'm just gonna put it there one standard deviation two standard deviation and three standard deviation so the question here was asking you if 95% of the values of normal distribution are within two standard deviation of the mean we know that 95% are two standard deviation away from the mean and that's how you answer the questions so it's great for what is easy as that okay so now let's go back and look at how do we then use this information to find the probabilities because we haven't found the probabilities now we're going to start working okay finding the probabilities we find them in three ways remember we can find the probability of a less than we can find the probability of a greater than we can find the probability that it lies between so let's first start with the probability of less than so when we find the probability like I've already explained the probabilities are the area underneath the kef right the area underneath the kef a we call it a u c it's the same as the probability and it is what we're going to be finding so to find the probability we're going to use the table the cumulative standardized normal distribution table we're going to use cumulative it's table e2 if you have past exam papers or anyway they have it and it is called the cumulative standardized normal distribution and you can see also they've got a picture there and it shows you certain things now this table this cumulative standardized normal distribution table we're going to get to it just now so if you have a question asking you to find the probability of z less than a value what is a z? a z you would have already calculated your z by using the standardized formula remember the z distribution you would have calculated that to find the value of a so this value of a you find it it's the same as that so this value of a it's your x minus the means divided by the standard deviation you find a value there and your value will be in two decimals so if it was 2.00 if the answer was 2.00 you just add a zero to it and I'm going to show you or tell you just now so how are we going to use the table? the table has the positive and the negative like we said it before you need to remember that this table contains only like I said only the less than values the two tables that you have in front of you if you have them or these two tables that you're going to see they only contains data or probabilities of the less than only less than less than less than anything if the question is find the probability that x is this and y is this and the other the mean is this the standard deviation is this find the probability that x is greater than you won't find it here but you will do certain things but if they say find the probability that x is less than then the value you see on this table inside the table this are your probabilities all these values are your probabilities all these values you see here are your probabilities you come here you find it and that will be your probability of a less than let's go to the actual table I like doing demo on the actual tables so finding the probability of let's say the question first find the probability that z is less than 1.25 right that is in two decimals now practicality means we practice so let's practice that we can see that that we have 1.2 and then we've got the 5 right 1.2 the 1.2 you always going to find it on this side of the table on your left side so the first values will be on the left side the last value will be 0.05 and this will be at the top so the last value will always be the last value of the value at the top of the table which is this top of the table right like I said I have 1.25 this is a positive value this table has negative values of z positive values of z you can see there right this are positive sides so if I need to find 1.25 it means I'm going to find 1.2 on this side and I'm going to find the last digit at the top so let's go and find 1.2 let's make it bigger 1.2 is there and I go to the top I look for 0.05 because I'm looking for the last digit at the top of the table that is why you always have to leave your answer to 2 decimal so where they both meet that is the probability that I am looking for right that's the probability of a less than so the probability that z is less than 1.25 will be equals to 0.89 4 4 that's easy right let's then go to the negative side let's change this let's change this let's say we want to look for the probability let's say I want to find the probability that z is less than minus 3.18 let's say let's say 3.18 so 3.1 minus 3.1 I'm going to find it here and 0.08 I'm going to find it at the end because 8 is what I'm looking for so at the top the last digit 8 on the side 3.1 minus 3.1 where they both meet that is the probability that I'm looking for and therefore the probability is 0.00 7.4 how many zeros there are 3 now you need to also pay attention to the table that you are using my table has 4 decimals but if you start from 3.1 there are 5 decimals or not even from 3.3 from minus 3 decimals but yeah using this table that you have you should be able to answer any question so that is how we going to find the probabilities easy any questions is it clear are you happy will you be able to use it okay now I'm not even sure whether I need to do some exercise like this but let's look at this exercise let X represent the time it takes in seconds to download an image 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 less than 18.6 so we know that it's less than therefore it means the probability of Z we going to be finding it on the table there is nothing we need to do we just going to go to the table and find the probability after we have standardized this X value so standardizing the X value it means we moving from this X distribution value we going to apply the Z distribution formula where we calculate our X is 18.6 the mean of 18 standard deviation which is our sigma of 5 seconds substitute we get the Z score of 0,12 and therefore we can go and standardize it by going to the table so now we have our Z of 0,12 and 2 0,12 0,12 go back to there and it is positive right so we go to the positive side of the table just when I delete this and we are on the positive side we looking for 0,1 remember 0,12 0,1 and 2 at the end and there is where they are 0,1 on this and 2 at the top and that is 0,5478 so we standardize the mean to be 0 the standard deviation of 5 to be 1 and we standardize our X to be a Z from 18 to be 0,12 and we went and we found the probability and the probability is 0,578 now this tells me that 50 so we can multiply this by 100 and get 55% of the data is below 0,12 or it is below 85% because the probability is in decimal it's a proportion and if we want to find the percentage we just multiply the probability by 100 to get the percentage so 55% of the data is less than 18,6 lies below 18,6 that's how you use the probabilities now how do we then if I know how to find the probability of the less than then how do I find the probability of the greater than so finding the probability of a greater than it means since the table contains only the less than and the less than are complement of your greater than so you can see that the white area this white area is your complement area right the complement this is your complement area and what do we know about complement so remember the probability of a complement can be found by 1- the probability of a or the probability of a can be found we can find the probability of a by finding 1- the probability of an a complement so if this we know that this probability of a less than a then we can find the probability of a greater than a because less than a is a complement of greater than a so what do I mean by that what I mean is that if I need to find the probability of z greater than a is the same as finding 1- 1- the probability of z less than a so what it means it means that if I need to find the probability of a greater than I will go and calculate my z score which is x- the mean divide by the standard deviation and use the answer I get to go to the table but the value I find on the table I need to subtract it from 1 sometimes we get an example now find the probability that x is greater than 18.6 this is similar to what we just did remember the mean of 18 the standard deviation of 5 we were doing an example with less than 18.6 now we doing it with greater than we just swapped the side so I'm not going to go and calculate again because our z of mean divide by x- the mean divide by this we found that it was 0,12 right we did find that so now if we need to go find the probability of z greater than 0.12 we're going to say 1- the value we find on the table and we know when we went to the table we found table as 0,5478 similar remember I told you that the area underneath the calf the whole area the shaded area will be equals 2 and 1 we did find this blue area earlier on remember when we were doing this exercise we did find that it was 0,5478 now if we know what that is then we can say 1- the complement and it will give us this probability that we are looking for which is 0,54522 that is the probability of z of x greater than a value I hope it makes sense we're going to do some exercises and then it will become clearer as well now how do we then find the probability of between finding the probability of between it's a little bit different to what we just did now so remember when is the probability of z less than a that will be the value on the table when is the probability of z greater than a that will be 1- the value we find on the table now with the probability of between because it's this probability that we're looking for this area in between we need to subtract those two areas or we need to do something so since we don't know what this less than is and we don't know what this greater than is but we can on the table find this value and we can find that value we can find both of them and subtract this value from that value or subtract this value from this value so that we can get the missing part so let's do that how do we do this so when we find the probability of between we're going to find the probability of z less than or equals to be okay that's the other thing I don't even have to use the equal sign when we deal with normal distribution you don't have to worry about whether there is an equal or less equal sign or not equal sign to it so they mean one and the same thing only for normal distribution the equal and the less than or the greater than they mean one and the same thing here so you're going to find the probability of z less than b which is the second one you're going to also find the probability of z less than a which is the first part and subtract one from the other so we're going to find the probability of z less than b minus the probability of z less than a because those are the values that we can find on the table how do we then do that so x is normal with the mean of 18.0 and standard deviation of 5.0 find the probability that x lies between 18 and 18.6 so we go and find the we calculate the z distribution for 18 so we go and do that one and we go and do that one so for 18 is 18 minus 18 because our mean is 18 and our standard deviation is 5 18 minus 18 is 0 so therefore it means the answer will be 0 because any number divided by 0 0 divided by any number will just be 0 so for 18.6 minus 18 divided by 5 gives us 0.12 remember we have now the probability that the z lies between those two values so we can go and say we go into the table to find the probability that z is less than 0.12 minus the probability that z is less than 0 now with 0 you can just put 0 0 at the end then we go to the table and on the table I'm lazy to navigate between the two on the table we go into first find the probability that z this is what I just wrote on the other slide z of less than 0.12 so 0.1 and 0 0.02 at the top where they meet is 0.5478 we write it down minus the probability of z of 0.00 so if I go to 0.0 and 0.00 at the top it will be 0.05 and that will be 0.5 0.5478 minus 0.5 000 it gives us the probability of the between is 0.0478 that is just the shaded area that we are looking for another example of between if I need to find the probability that x lies between 17.4 and 18 this time you can see that we are looking for this blue shaded area the last time we were looking for the red shaded area now we we are looking for this side and we go and calculate the 17 z of 17.4 minus 18 divide by 5 and the answer will be minus 0.12 and then you go and calculate the value of z of 18 minus 18 minus 18 divide by 5 which is 0.00 now you are going to go to the table before we go to the table because then I have my 0.012 and 0 so we need to take the second which is the probability of a z of less than 0 minus the first which is minus the probability of z less than minus 0.12 so now I am going to go to the table just for this so we know we go into the positive side of the table to look for 0.00 and 0 remember 0.00 so the 0.00 and the last digit of 0 there and we can see that 0.5 and the other one was minus 0.12 so we go into the negative side which is the top table and we are going to look for negative 0.1 and at the top we are looking for 2 where is 2 is the last column so I am just going to highlight the last the third column and that is 0.4522 and that is 0.4522 and we subtract that from 0.5 and the answer we get is 0.0478 and that is the probability of between okay I am left with 2 minutes to conclude my recapping just to recap on the empirical rule because we are just going to use it just now as well we can say that about the distribution of the value around the mean for any normal distribution that it can be and values will fall 68% of the values will fall within one standard deviation remember that which is the mean plus or minus because the minus will be in this side of the mean and the plus will be on the other side of the mean so it will be one standard deviation on the left or on the right of the mean and if it is 2 standard deviation we say it is 95% and I said 98% so it is 99.7 I was short of 1% it is 99.7 for a 3 standard deviation away from the mean so sometimes let's assume that you are not or you are given the probability of the area underneath the kef and they tell you that the area underneath the kef which is the probability is 99% but we don't know what your x value is so in order for you to be able to calculate that x value you need to use the empirical rule because then if you look at the empirical rule you have the mean you have the standard deviation and all just what is missing is just finding the z value which is 3 with the z value but that is just for the empirical rule so if we look at the probability questions that we have right now if we need to find x and we are not given x but we are given the probability or we are given the z value we can be able to find any of the value whether it is the mean, the population mean or the standard deviation you should be able to if your formula is z is equals to x minus the population mean divided by so if I need to find x I will just leave x on its own multiply your standard deviation with your z and take the mean to the other side and that will be a positive mean so then you can see that this is the same so if we need to find the x value for an unknown or for a known probability let's say they gave us the probability but not the z score for the known probability we first need to find the z value and then we can go and find our x value so let's look at this let x represent the time it takes in seconds to download an image from the internet x is normal with the mean of 8 so they have given us the mean of 8 and the standard deviation which is our sigma the standard deviation of 5 find x such that 20% of the downloaded times are less than so it's also very important that they mention weights like this because then it tells us how are we going to find the z value on the table if the if the answer here because this is less than we know that we just going to go to the table if it said it is greater than then we need to know that the value this they have given us they found it by doing one minus the table value this you need to always remember so because they're telling us it's less than then we're going to assume that this probability that they have they found it by using the z value and that's how they got to 20% so we are going to do the same because we need to use x is equals to the mean plus your z times the standard deviation we are given the mean which is 8 we are given we are not given the z value but we are given the probability so it means we're going to need to find the z value we are given the standard deviation which is which is 5 so how do we find this 20% so we know that they told us that it's less than so it means is this shaded area which is less than so since it is less than then it means we need to go to the table inside the probability table we need to go find 0 let's go back 20% divide by 100 is 0.20 right and if I need to convert it I can make it to 4 decimals because the table is 4 decimals so it means I'm going to go inside the table to look for 0,2 inside this table don't go to the positive you see in the positive then values are bigger so we're going to go to the negative side table to look for 0.2 so if I go in there I have 0.20 0.20 0.29 so I must go back back back 0.250 I'm going to take that one so when I get there I need to read my z value so I'm going to read z is 0, minus 0,8 so let me write it down 8 plus minus 0,8 that's not the end I must go up up up up to look for the last digit don't forget about the last digit and the last digit is 4 times times 5 and then you solve this that will give you your x so that's what I did so you go inside the table you look for the value that is close to 0.20 which is 0.2005 on the table you go out and you go out to look for the last digit the first two digits and the last digit so 20% of the area in the lower tail is consistent with the z value of minus 0,84 so I do have my z now I can just go and substitute and calculate and find that my x value is 3.8 so 20% of the values from the distribution of the mean of 8 and the standard deviation of 5 are less than 3.8 and that's how you use the normal distribution to find your answers what you have learned in the last hour the basic concepts of normal distribution you've learned how to compute probabilities from a normal distribution finding the probability of a less than finding the probability of a greater than and finding the probability of between remember the probability of a less than is the value you find on the table the probability of a greater than is 1 minus the value you find on the table the probability of between will be finding the value on the table of the second value minus the value on the table of the first value but don't forget that Before you find the value on the table, you need to standardize your values by using the Z value. Remember that this whole formula is equivalent to A value. We've learned how to find the probabilities using the formula and using the table. That's quite today's session. In the next 30 minutes or less, let's look at some questions or exercises. I will leave some more exercises for you in the handout so that you can do it and we can have a conversation on WhatsApp or on my Unisa as well. The first question is that STA 1610 final mark shows a normal distribution with the mean of 56 and the standard deviation of 4 percent. They have given us the mean and sigma, our standard deviation. What is the probability that a random chosen student fails a module? If the student is failing the module, then it means they got less than 50, right? Because if you get less than 50, you fail the module. It means we need to find the probability that X is less than 50. So X is less than 50 will be given by the probability, because we need to find Z of X minus the mean divided by the standard deviation, and substitute the values, which is our X is what is given in the question, which is 50 minus our mean 56. Because I'm working with percentages, my bad. My bad. What I need to do is convert this to decimals. So this will be 0,5 and this will be 0,5 and 0,5 and 0,56. Divide by standard deviation is 0,04. Okay. The probability that Z is less than, let's do the calculation. Open my normal calculator from here, because this is just a simple. So there is nothing wrong with using a normal calculator. 0,5 minus 0,56 equals divide by 0,04 equals minus 1.50 minus 1.50. Now, we need to go to the table, go into the table, go into the negative side of the table. We're looking for minus 1.5, minus 1.5 and at the top is 0, so it's on the first corner. So that is 0,0068. I think the answer is 0,0668. Any questions? Any questions? Any clarity? Any comment? So I just, what's to ask? I've come across this, appears all tables that you can get of a standard normal cumulative distribution function. They go from negative 3.4 to positive 3.4, and then the preferred one we should have, it goes to four decimal units. Is that correct? Sure. Is it also called a cumulative standardized normal distribution? If the name is the same, and the table has positive and negative values, like a table with the positive z values and negative z values, does it show it like that? If so, then that is the table that you can use. As long as... The table I have, it starts off with the negative values, it goes all the way till it gets to, it goes to the negative values, until it gets to negative 0.0, and then it goes to positive 0.0 to 3.4. Yeah, it should work. If you say it ends on 3.4, therefore it means you are missing all these other values, and if it also starts at 3 on this side, therefore it means you are missing those other values. You just need to be very careful in cases where your lecturer asks you a question and the answer is negative 3.7. Right? If the answer is negative 3.78, so 3.77 or 3.0, and your table does not have all those values, you just need to make sure that you have the one that has all the values. So at least up to negative 6 z value should be enough, even if it continues, it should also continue to up to 6 positive 6. Sorry, guys. I just checked, so the book that we have, Introduction to Statistics, that's the one that they deliver to us. It goes from minus 3.0 to positive 3.0, so it's missing everything above your... It's your study guide, right? Yes, yeah. You can talk about the study guide. Do you have page numbers? Page number 120 and 121. 120 and 121. So go to your study guides, guys. You will find this table, and this table is very important, like I told you from now on until we do hypothesis testing, all the study units that we're going to be doing now on once, they use these tables. So next week, we're going to use the same table. The following week, we're going to use the same table. And next month, we're going to be using the same table. Okay, so now let me get number two, exercise two. So the other thing that you also need to pay attention to, remember we've been talking about standard deviation, the mean and the standard deviation. So this question says, the owner of an appliance stores uses a normal distribution with the mean of 10. And the variance, we know what the variance is, sigma squared of nine to model the weekly net sales, calculate the probability that x is less than 3.5, and it's less than, then it means it's easier. So because they gave you the variance here, what do we need to do? What must we do? If we given the variance, we must square root it to obtain the standard deviation. So we're going to find the square root of nine, which is three, which will give us the standard deviation. So now let's go find the probability that x is less than 3.5, which is the probability that z is less than x minus the mean divided by the standard deviation, which we'll find the probability substituting into the equation, 3.5 minus the mean of 10, the standard deviation of three. We did find that it's three. Probability z less than what is 3.5 minus 10, divide by 3, it's minus 2.17666666666, so which we can write it as minus 2.17. So we're going to go to that table to go find on the negative side of the table. Let's delete the old things. Minus, I forgot now. What are we looking for? Minus 2.17, minus 2.17, minus 2.1 and 3.7 is right here. I'm just going to highlight it. Where they both meet, it's 0.015, 0.015, 0.015, 0.015, 0.015, each is option number five. Is it right? I'm not going to, oh, let's just answer it. It's fine. I was going to say I'm not going to answer this, but I see that it's very different. So this one, it says if the z score given is minus 1.96, the distribution of x is normally distributed with the mean of 40 and the standard deviation of 5, then what is the value of x? So we can do it by just using the formula or if you already remember how to convert this, you can already automatically say the mean plus or minus z of the standard deviation. They will give you one in the same thing. So we know what our z is. It's minus 1.96. Our x is what we need to find because that's what we need to find. Our mean, we were told that it's 40. Our standard deviation is 5. And we can just multiply minus 1.96 times 5 x minus 40. And minus 1.96 times 5 is minus 9.8. Then I move 40 to the side. It will be plus 40 is equals to x. And therefore minus 9.8 plus 40 gives us 30.22, which is x. So x is equals to 30.2. Or you can come in the side and say the mean of 40 plus or minus. I don't know why I have plus or minus. We just need the plus, not plus or minus. Minus 1.96 times the standard deviation. The answer here will be 30.2. It should be the one in the same thing. And that will be answer number one. I just showed you if you're given the mean, the standard deviation, easy, you first need to calculate your z. Sometimes you are not given the mean, the standard deviation. You are given already the z value and they want you to evaluate each statement. Like for example, all these statements need you to evaluate them to choose which one is incorrect. So you don't have to go and calculate your z of this and that and no. Already your z of x minus the mean divided by the standard deviation has been calculated, which is your a value, which is 2.64, which is minus 0.8. All that all you need to do is use the value that you are given, go to the table, find the answer. Now, always remember the probability of z less than a a is the value going to find on the table. Probability of z greater than a that will be one minus the value we find on the table. The probability of z lying between two values will be the probability of z less than b minus the probability of z less than a, which is the table value for b minus the table value for a. Okay, so now in the next seven minutes, let's see if we can answer all these questions. I hope you have your tables really. You are, I'm not going to navigate, you are going to give me the answers. And like you're going to call out the value on the table. So if I get two values that are different than I can just go back and check. So we're also going to pay attention to the sign. So the first one says this should be the probability of z greater than 2.6 should be that. So let's see if that is true. We know that this we're going to find it. By using one might the value we find on the table, which is the probability of z less than 2.64. So go on to the table. And find 2.64 and tell me what is the answer? It's 0.9959. Yeah. 0.9959. Subtract one from the other one from zero. Oh, sorry. One minus 0.9959. What do we get? The calculator gives gives a 3.1 to the power of 10. Okay. Yeah. 10 to the power of negative three. Yeah. So that would be converted. Yes. So say plus one and equal and remember that you've added one to the answer. It should be one comma some number. Now, properly you are able to read it. What do you see? And when you read it, so you will have one comma zero, zero and some number. But when you read it, remember that you've added one. You just start with zero. Your calculator is giving you? My calculator is giving me 0.041. Which is number one. So your answer on your calculator, your scientific calculator, gives it to you in an exponential format. Because if there are so many numbers, zeroes, then it will make it to the negative three or something like that. So therefore it tells you that there are three zeroes, one before and two after the comma. If it's four, it will mean one zero before the comma and three after the comma. Some things like that. It's just a scientific notation of your calculator. If you get answers like that, where it gives you to the exp that you just add one, say plus one, but always remember that you've added one. Start reading from zero by removing the one. So now you know. Okay. So number one is correct. Let's go to number two. Number two says we need to find the probability that z is less than negative 0.87. So we know that when it's less than, the value we see on the table. So on the table, go and find minus 0.87 and tell me if it's equals to 0 comma 192. Yes, it is. That's correct. Now we have the probability of between. So you go into find the probability of z less than 1.40 minus the probability of z less than minus 1.40. So go to the table and find first the probability of 1.4 and give me that probability. 0.9192. And then go to the negative side and find minus 1.40 and give me the probability. 0.080. 0.8. Yeah, so 0808 at the back. Yeah. Like that. Yes. So I'll direct and tell me if it's equals to that or give me the correct answer. 0.9192 minus 0.0808. 0.8384. 0.8384. So it means this is correct. Okay. Now we have the probability of z less than 2.8. So it is less than the value you find on the table that will be the value we're looking for. So right, go to the table, find the probability of minus 2.80. You can just add a zero at the end. And tell me if it's the same. Zero minus minus 2.80. Yeah, it's 0.026. Is? Sorry. Just confirm me. Okay. So we're looking for minus 2.80. Minus 2.80, that is that. So it's 0.0026. So 0.0026 because we're looking for minus 2.80. All right. We're looking for the incorrect one. So that will be the incorrect one. So let's see this one. This is a very interesting one. It says the probability of z greater than 0.74 is the same as the probability of z less than negative 0.74. So we need to go and find, let's go and find this probability first. So yeah, we're going to say one minus the probability of z less than 0.74. So go and find this probability of z, 0.74. It's on the positive side. And we're looking for 0.74, which is 0.7704. 0.1 minus 0.7704. That's what we got, but that's not the end. So we need to subtract it from 1. 1 minus 0.7704, which is equals to 0.2296. So that is the site. Let's go and verify this site. Let's go and find the probability of 0 minus 0.74. Go into the negative side, 0.74, which is 4 at the top on this column. It's 22, 96, 0.2296, 0.2296. So they are the same. So that is correct. And that's how you will validate, not just by looking at this and say, oh, they are not the same. They should say this and you will find it that you might be wrong. So the probability of z greater than 0.74 is the same as 1 minus the probability of z less than 0.74. But it's also the same as the probability of z less than minus 0.74, because they are both equals to 0.2296. Okay, so that is the end of our session for today. You can go through this exercises. So this first exercise is looking at that we didn't cover. They say the area under the curve is equals to. So you must pay attention because here they gave you between. And these are z values. These are already z values. So it's a z value of between. So you just calculate it like normal. So you're going to find the probability that z lies between 0 and 1.25 and apply the same rule that we have when it is between say the second one minus the table value of the second one minus the table value of the first one. Okay, and exercise six, they gave you the probability. You need to go find the value of a, which is your x value. But then it means you need to go find the z value and calculate your x equals to the mean plus z alpha. Or you can use your zx minus the mean u and y, the standard deviation. This is the probability of z of a is equals to 0 comma 1515. So it means you need to go and find on the table here inside this table, 0 comma 1515 somewhere inside here. And then go find the z values and substitute because it's less than, it's easy. The less than is easy. It means the value you find on the table is the value you are looking for. But pay attention, you are given the variance as well. This is straightforward. It says at most, at most we know that it's less than or equal. So you are asked to find the probability of a less than. This is similar to what we just went through. I think it's even the same question. It is nothing. This has changed on this one. So read my back. Okay, the other questions. Here also, they are asking you to find the value to the right. Remember, if you don't know where the right is, just draw yourself a diagram and say the right is on the positive side. So if they told you about the right, therefore it means it is the greater than the value you are going to be finding is from 1 minus the probability of z less than a value. So this probability that you found, because this is the probabilities, this area underneath the curve on the right, it was found by using this. So in order for you to find the correct z value, you need to take, you cannot take this and go find the z value. So you need to take the answer that you get from. So this will be 1 minus 0.2061. And once you have that, that will give you the answer of the z value that you are looking for. And that will be the answer that you are looking for. So I'm just giving you hints as well as I go along. And that concludes today's session. Are there any questions? We are seven minutes over time. Are there any questions, comments? If none, then thank you for coming through and being part of today's session. Unfortunately, I cannot even click anywhere on this thing. I cannot stop the recording. Hello. Yeah, we can hear you. Are you able to hear me? What do you see on the screen? Just the names and stuff. Yeah, the names. Are you able to see the names? I'm unable to click anywhere. Okay. I don't know why. I'm not doing the same early. I couldn't switch on the type of interface, but it's working now. All right. I hope that's all yet. I am unable to click anywhere to stop the recording or even not even the recording. To stop sharing. Now we see your video. See you. Now you are seeing me. Yeah. Yes. Now I've switched it off. All right. Okay. If there are no questions or comments, then enjoy the rest of your evening. Cool. Thanks, YouTube. Bye. Bye. Thank you. Bye. Okay. How do I stop this now?