 Welcome to session 11 of your online tutorial sessions. Today, we're going to start a new study unit. We're doing normal distribution. We're still going to use some of the concept that we have learned when we dealt with the basic probability, that we need to know and remember that your probabilities are between zero and one, and also the sum of all your probabilities should be equals to one. Then we're going to deviate a little bit from what you have learned from discrete probabilities in terms of the sign, and I will explain why I'm saying that. When we look at normal distribution, the sign for greater than and greater than and equal, they will be exactly the same. They will be answering the same question. Whether you're doing greater than or greater than or equal, we can just use the greater than, and it's things like that, so that you don't get confused and start asking questions. But you said greater than or equal is this and this and this and this. No. In normal distribution, our greater than and greater than and equal, will always be the same. We will not be working with the equal, probability of an equal because we are using cumulative probabilities in normal distribution. But we will explain all that at a later stage. For now, do you have any comments, query questions before we start with today's session? Anything you want to ask? Speak now or forever, forever hold your peace. Nothing. If there are no questions or comments, then we can start with our normal distribution questions or understanding our normal distribution. By the end of today's session, you should be able to know the basic concepts of normal distribution. You should be able to know how to calculate the probabilities from a normal distribution using the probability of a z less than a, probability of z greater than a, or probability of z standardized normal value lies between the two point A and B. You should be able to use probabilities to find the x value or any other parameter that they would have asked. We should be able to use the normal distribution table. We are going to use cumulative standardized normal distribution, and I think it's table E2. I will explain that table to you just now. What is normal distribution? Normal distribution, we use a variable that comes from a measuring process. It uses a continuous variable, and we know that a continuous variable is a variable that is been measured. It's like we use the temperature variables, we use like your height, weight, your age, everything that is continuous variable, we can use or we can find the normal distribution of that variable. Sorry about that. We're going to learn how we calculate the probability of this continuous variable. A normal distribution is a calf, which is shaped in a belly shaped calf. If you look on the right, you can see that the calf looks like that. Normally, when we talk about graphs, we use the line or we use a bar chart that will look like that, or a histogram, and so forth. If you take all of them, if a histogram creates a belly shaped pattern, then we call that a symmetrical graph, or we call it a normally distributed information or data. We know that asymmetrical data would mean the median and the mode are equal, and that would make you make a normal distribution calf. In terms of the locality of the data, we can determine that by using the mean, because the mean is one of the measures of central location or central tendency. In terms of the spread of the data, we use the standard deviation, because the standard deviation will tell you how dispersed your data is from the mean, and if this is our mean, it will tell us whether the calf is there, it will be flat or it will be tall. The range of your data or your random variables, they will be from positive infinity to a negative infinity to a positive infinity. When you draw a belly calf graph, your graph will never touch the x-axis. If this side is your x-axis, this one fx is your y-axis, and your belly calf will never ever touch the x-axis anyway. That is a normal distribution. Some of the properties for a normal distribution is when we change the value of your mean, then your graph will shift from left to right. If the value of your mean becomes there, therefore it means your belly shape calf will move to the left, and if the mean is there, it will move to the right. Changing the value of your mean shifts your distribution from left to right. Changing the value of your standard deviation will either flatten your calf or make it taller. When you change the value of your standard deviation, it will either increase the spread, and when it increases the spread, then it means because then the values of your x-values will be far away from the mean, then your calf will be flat, because then they will be far away. But when you change, when the standard deviation decreases, then it means this calf becomes closer to the mean, then it becomes narrow. It will be like that, and so you can see that the distance between the belly calf and the mean will be closer to one another. So when you increase the standard deviation, your calf becomes flat. When you decrease, or when the value of your standard deviation decreases, it means your belly calf will be narrower. And those are the properties that you need to learn about normal distribution. The other thing about the normal distribution curve is, since we're using the x-units, when we calculate the probability, we want to convert those x-units into a normal distribution because sometimes your x-units are not normally distributed, but you want to convert them into a normal distribution so that they fit well into the normal distribution curve. And to do that, we convert all those by using what we call a Z-transformation score or a Z-score, which is just a Z-distribution, or we call it a Z-score, or we call it a normal distribution Z-value. So the Z-score is given by your observation, which is your x-value minus the population mean divided by the population standard deviation. And once we have standardized the x-score by applying this formula, then we would have created what we call a normal distributed value. And with a Z-distribution, it will always have the mean of zero, and that is the property of a Z-score. It will have the mean of zero and the standard deviation. So it will have the mean of zero and the standard deviation of one. And when your Z-distribution has the mean of zero or has the standard deviation of one, we say that distribution is a normally distributed distribution. Okay, so how do we then calculate the Z-score? If x is distributed normally with the mean of 100 and the standard deviation of 50, the value of, we need to calculate the value of Z, or we need to calculate the Z-distribution for x is equals to 200. We know the formula is z is equals to x minus the population mean divided by the standard deviation. We are told that x is distributed normally with the mean of 100. So therefore, this is our population mean and our standard deviation of 50. Then therefore, that is our standard deviation sigma. And that 50 will be our sigma. And in the question, they will give us the x value. And there is our x value, which is 200. Then we can come and substitute the values. Our x is 200, our mean is 100, and our standard deviation is 50, right? And we calculate 200 minus 100 is 100 divided by 50, which will be equals to two. Now, when we work with normal distribution, we always need to remember to leave our, I need to also change my slides, we need to always leave your answer at two decimal, always. And I will tell you why we need to leave our z score at two decimals. Always it's because when we go and find the probability on the table, then we need to use the two decimals. Okay, so always your z score, you can leave it with, when you get to the answer, leave it at two decimal. And interpreting the formula or the equation that we just calculated, we say that where x is 200, we know that that is two standard deviation above the mean of 100, or two increments of 50 rank above the mean of 100. In the exam, you will not be expected to know how to interpret, but you will be expected to know how to do the calculation of your z score. So you need to know that you need to be able to identify what you are given in the statement and substitute correctly into your formula and answer the question. Okay, so that is calculating the z distribution. Now I have an exercise. After me saying all that, let's refresh our mind by recapping on what we just said by answering this question. Which one of the following statement is incorrect with regards to the standard normal distribution? Number one, the standard normal distribution has the median of zero. So the eight says it has the median of zero. Number two, the standard normal distribution is symmetric around the mean of zero. Number three, the standard normal distribution has a standard deviation of one. Number four, the standard normal distribution has a variance of one. That is another one, variance of one. Number five, the area to the right side of the mean is one, and the area to the left side of the mean is also one. So let's go back to our question, our statement that we had. So remember what we said. We said a normal distribution, a normal distribution, the mean, the median and the mode are the same. It's symmetrical and is distributed with the mean of zero and the standard deviation of one. Remember that. Let's go and answer the question. The mean, the median are the same. We know that normal distribution is distributed with the mean of zero. So therefore it means it's symmetric, mean of zero, correct. Then if the mean is zero and we know that it's a normal distribution then also the median should be zero because the mean and the median are equal. We also know that it is distributed with the standard deviation of one. Therefore that is also correct. It is distributed with the standard deviation of one. What is the standard deviation? Remember the standard deviation is the square root of the variance. So if the standard deviation is one, oh, sorry. If the standard deviation is one, therefore it means our variance is equals to one because the variance of one will also give us the standard deviation of one. So therefore it means that is correct and which leaves us with one final answer which is number five, which says the area to the right of the mean is one and the area to the left is one. So this question would have been, oh, it's not would have been, it is incorrect because why? Remember if the area underneath the calf, if this is where we calculate the probability, therefore it means the sum of all the values underneath the calf of the normal distribution should be equals to one. So the sum of all probabilities should be equals to one. So if this is half of the graph to the left, this should be 50%, this should be 50% because 50 plus 50 is equals to one. So this should be 0.5 and this should be 0.5 because half of one is 50%. So reading the statement. It has almost a basic question order. So this would have been incorrect and that is what we are looking for. Please make sure that you are muted unless if you want to ask a question. And now I'm giving you a chance to ask a question, any question based on what we just, I just explained. Okay, so if there are no questions, then we can move on. The next exercise I want you to do, scores of high school students on a national calculus exams were normally distributed with the mean of 86. The variance of 16 calculate the score of the students that will have a score of eight. So what they want you to do is to calculate the Z score. I will write the formula for you. Z is equals to X minus the mean divide by the standard deviation. Based on the information given, what are you given? You are given the mean, the mean of 86. You are given the variance, remember that is a variance is sigma squared of 16. You need to take the variance and calculate your standard deviation, which is the square root of your variance. You are given also the X value. So calculate the Z value. Please ignore the, I don't think those are the right answers. Let's just see, let's calculate and see if, yeah. Ignore those that are not for this answer. Are we winning? I got minus 1.5. Yeah, negative 1.5. Same here. So we have, we are given the variance of 16, so the square root of 16 is equals to four. So substituting all the values that given to us, our X value, we find it in the question. Calculate the Z score, which is 80 minus the mean of 86. The standard deviation of four and this gives us 80 minus 86 will be minus six over four, which is equals to minus 1.5. And you can put a zero at the end. You can say it is 1.50. Remember to leave your answers at two decimals, always. Two decimals. And when you round off, round off correctly as well. Okay, and that's how you will calculate the Z score. Now let's look at how we calculate the probabilities. So finding the probabilities using the standard normal distribution Z score value, we need to remember the following. Probabilities for a normal distribution that we're going to use, we're going to use a table with cumulative standardized normal distribution. There is a table called standard normal distribution table, which has the probabilities of equal, which has the Z score relating to the probability of an equal of the exact. So, but the table that we are using looks at the area underneath the careful, it's what the probabilities are that we are looking for. So where you see the rate, that's where we calculate the probabilities. So the table that we're using, it uses the cumulative, because if you look at where A is in the red, you can see that if we need to calculate the area underneath this calf where the red shaded area is, this are accumulated probabilities. If we only interested in that white part area as well, this will be the accumulated probabilities that we will have to find there. It's not going to be a probability at the point there. So we calculate all of them that are in the white area or all of them that are in there underneath the calf. So you need to also know the signs because if they say calculate the probability of at least you need to know that it is the probability of greater than or equal, right? And if they say probability of less than, you need to know that it will be the less than. Now, I also said to you, you're going to ignore when you see, when they say calculate the probability of at most, at most X is less than A, we're going to represent it as such, but with normal distribution, we can just say it is the same as the probability that X is less than A. They will mean one and the same thing because we're working with cumulative probabilities, not the probability of equal or the probability of exact. So since we're using cumulative probabilities, so the probability of at most or the probability of less than, they will mean one and the same thing. We are going to use, actually we are going to always constantly use the less than. We don't have to say less than or equal every now and then. We can just use the less than sign when we calculate these probabilities. So let's learn how do we find these probabilities? So because we're using the standardized normal distribution table, you cannot calculate the probability just by using the formula of the Z. We use the Z formula to standardize the X formula. So that we can use the Z score, which is your Z formula or the Z score value that you found, we use it to go find the probability on the table. So when we're looking for the probability of a less than, the table that we're going to look at contains the probability of a less than. The table has two sides. I need to make this clear. The table has the positive side of Z and the negative side of Z. Both the negative side and the positive side are probabilities of less than. Whether you are finding it on the less than side table, the less than side, oh, sorry, the negative side of the table or on the positive side of the table, every time you see those probabilities on the table, always remember that all of them are probabilities of a less than. What does that mean? Then it means once you have calculated your Z value, when you go to the table and you were asked to calculate the probability of X less than a value and you calculate the Z value, you're going to go to the table and look for the probability that corresponds with those two Z or with the Z value that you have calculated. And we're going to look at the table and you know that that probability that you find on the table answers the question that you were asked. So for the probability of Z less than a value, we use the value on the table to find that probability or that value on the table is your probability. So let's look at an example of the table. We're going to go to the actual table just now. I just wanted to indicate this. So the table, by looking at this, you can see there it's on the negative side of things. So from the negative side of things, it's the shaded area, the negative Z values has the probabilities inside. So all these values inside the table are your probabilities. All of them are your probabilities. These values here and those values there are your Z values. So these values on your left and the values at the top, we need to combine them and create a Z value. Or when we have the Z value, we need to split it so that we can go and find the values. We're gonna look at the table just now, just wanted to show you. Then we also have the same table, table E2 has the positive side of things. So this one has the negative, this one has the positive. If you look at this, starts from the positive side and it goes to the less than. And that is why I was saying where we find the probability of less than a, the value we find on the table is your probability. So this is your probability. This is your probability. So what does all of this mean in terms of the Z value? So I'm just gonna go out of this presentation and go to the table itself. So we're going to use the example that we have. Remember we calculated the Z value. So let's say this is the question that we were asked and they say we need to calculate the probability that Z, not Z, X is less than, X is less than eight. And I'm going to assume that we already did the calculation and we find that once we standardized the X value, we found that our Z value is less than minus 1.5. So I'm interested in this. Where Z is less than minus 1.5 zero. Remember that. So let's go out of this. I'm gonna discard. Then we go into the table. You need to go and look for table E2. It should be always your first table if you're looking at your past exam papers. But if you're looking on your study guide, you should go to your study unit six and you will find your normal distribution table. And it should say cumulative standardized normal distribution. If it doesn't say cumulative standardized normal distribution, you must know that you're waking on the wrong one. It always have to say cumulative standardized normal distribution. And since we're looking for the negative side, so I'm gonna go to the negative side of the table and remember we're looking for the probability. I'm just gonna reduce it again a little bit. So we're looking for the probability that Z is less than minus 1.5 zero. Remember the reason why I said always keep your table to two decimals. It's because of this. When we work with the normal distribution table, I'm gonna give you a chance just now to do one as well so that you can learn how to use the table. When we work with this, the first two digit, the one digit before the comma and the one digit after the comma, we're going to find that on the left. The one digit with the negative side, the one digit before the comma and the one digit after the comma, we're going to find it on the left. The last digit we're going to always find it at the top. The last digit, we always going to look for it at the top. So that is why you need two decimals. So minus 1.5, we need to go to minus 1.5, that is minus 1.5, and we need to go to zero at the top. So you will notice that also at the top that it's only populated with the last second digit, all of them. So we just look for the last digit. We look at what the last digit is. If the last digit is zero, so we just go to zero and where they meet, that will be where we will find the probability. So this will be our probability. So the probability of Z less than minus 1.5 zero, let's remove this, is equals to, just give me a sec. Just want to see the values is equals to, so that probability will be equals to zero comma 0668. Now, I want you to find the probability where Z is less than minus 2.35. Find the probability that Z is less than, I can make the table bigger so that for those who don't have it in front of them, 2.35, what is that probability? 0.0094. 0.0094, so we go to minus 2.3 so the two digits on your left and we go and look for five at the top and where they both meet, that will be the probability we're looking for, right? So you know how to use the table. Is anyone still lost? Okay, if you are still lost, let's do another exercise. Let's find the probability of Z less than 1.37, 1.47, the probability of Z less than 1.47. So it means we need to go to the positive side of the table, so we go into the positive side and we're going to look for 1.47. So the two digits, 1.4, and we need to scroll to seven and that probability will be 0.9292 and that's when we will find the probability. So let's do more exercise, let's continue so that we can learn how to use this table. 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 of our X value, we know that our mean is 18, our X value, it says it is less than, so it's all the red values less than 18.6. We were told what the mean is, it's 18, we were told what the standard deviation is, it's five. Now we need to find the probability. Finding the probability means we need to standardize this X value and to standardize it, we use the Z score and we just substitute into the formula so Z X minus the mean divided by the standard deviation, our X is in the question, it's 18.6, our mean is 18, we were given, our standard deviation of five, we are given and we just calculate 18.6 minus 18 is 0.6 divided by five gives us 0.12, this is just the Z value. This is just the Z value. So we need to go and find the probability that the Z value is less than 0.12, that's what we need because that's what the question said, we need to go find the probability that X is less than 18.6 and because we're not using the X value, we need to first standardize the value, then use the standardized value to go find the probability and that's what we're doing. We standardize the X value too and we found that the standardized X value is 0.12, we use the 0.12 to go find the probability on the table. So we know that our Z value was that, we need to go to the table. So this is just to demonstrate or illustrate the standardized value. So our mean remember, when we have a normal standard distribution, the mean is zero with the standard deviation of one. So we have just standardized it, we just need to go and find the probability on the table. I'm gonna go to the table, the actual table. So we need to go find the probability and where do we find that probability of 0.12, positive or negative side? Positive side. On the positive side table. So we just go to the positive side of the table, we need to look for the probability that Z is less than 0.12. So we go 0.1 on the left, 0.1 on the left, and then we go and look for the last digit is two and the answer is 0.5478. See how easy it is, very easy to find. 5478, and that's how you find the probabilities, easy. Okay, going back, this was just to demonstrate but we already did that. So this will just show you that the area underneath the kef for the red shaded area is 0.5478. That is what we just found. Okay, there is your exercise. The owner of an appliance though uses a normal distribution with the mean of 10 and the variance of nine to model the weekly net sale. Calculate the probability that X is less than 3.5. So remember the less than, so it means you need to calculate the probability that X is less than 3.5. And calculating that, I'm going to do it for you because I just want to get to the next section. I will give you an opportunity on the next one to do the calculation, 3.5. So because I also want to demonstrate something, how we get the Z-value. So now we need to compare or standardize our X value by using the Z-score. And we know the Z-score formula is X minus the mean, right by the standard deviation. Please mute your microphone. She does not come out for you, forgive her father. So now we just need to substitute the values. So what are we given? The mean of 10. The mean of 10. We are given the variance of nine. Therefore, our standard deviation will be the square root of nine, which is equals to three. And we are told what the X value is because the X value is in the question. So our X is 3.5 minus our mean is 10 divided by our standard deviation of three. Probability of Z less than if we calculate this, what do we get? Do the calculation from your side as well. It's 3.5 minus 10 equals minus 6.5 divided by three equals minus 2.17. Minus 2.17. So don't look at this and say, oh, yeah, it's my answer. That is not your answer. This is not the correct answer. This is the Z value. So now since we have the Z value, we need to go to the table. Go out to the table, keep table. We're looking for, oh, let's go back. Minus 2.17, where do we go? To there, positive negative side. Negative side. Yes, we go to the negative side of the table. We're looking for 2.17. Okay, so we're looking for 2.1 probability of Z minus 2.17. So it's 2.1 and we need to go to the end of the table. Six, the probability we're looking for is 0 comma, 0, 1, 5, 0, 1, 5, 0. That's the probability that we're looking for and that is the answer that we are looking for. So this will be 0 comma, 0, 1, 5, 0. So the probability that X is less than 3.5 is 0 comma, 0, 1, 5. That is how you find the probability of Z less than a value, right? To look for how, or we're going to do now, how to find the probability when we are given the greater than. So also, if they say, what is the probability that Z, oh, X is, let's use X. The probability that X is at least, which is greater than or equals to a value, or, oh, not all. So if we need to find that probability of X greater than or equals to a value, it will be the same as finding the probability of Z greater than a value, one and the same thing. So the greater than or equal and the greater than, they will mean the same because we're working with cumulative values. But now, since we know that when we find the probabilities on the table, the table contains the probability of Z less than a. So that is the table. The table has the probability of Z less than a value. Then it means if we have to find the probability of Z greater than a value, then it means we need to be doing one minus the probability of Z less than a value because we need to subtract one from the probability that we have on the table. So in order for us to find these probabilities of a greater than, which is the shaded area, oh, sorry. The probability of the shaded area rate, we need to subtract the probability that we find on the table, which is the probability of a less than, which is the unshaded area, the white area. And that is why we say one minus the probability we find on the table. And that is for all where we are asked to find the probability of at least or the probability of a less than. When you are asked, find the probability of X greater than the value you are going to go to the table, the Z table, the value you find on the Z table, you're going to subtract it from one. Let's look at an example. Suppose X is normally distributed with the mean of 18 and the standard deviation of 50, oh, sorry, of five, find the probability that X is greater than 18.6. Therefore, we are told we need to find the probability of X greater than 18.6. Then we need to find the standardized value of that probability, which then means we need to find the probability of greater than, we substitute the values. Our mean is 18, our X is 18.6 minus 18 divided by our standard deviation of five. We did do this calculation, so I already know the answer because we used it as an example previously. So this will be 0.12. So I cannot find this probability of greater than 0.12 because the table only has the probability of Z less than. Therefore, in order for me to find this probability of a greater than, I'm going to say one minus the probability of Z less than 0.12 because that one, I can find it on the table. And it's a complement of the one probability that we're looking for. One minus, then I must go to the table, you go to the table, keep, we go to the table, we go look for 0.12, which we will find on the positive side of the table, 0.12, which is 0.5478. 0.12 is 0.5478. So we're going to go to back here, 0.5478. And therefore that probability will be equals to 0.4522. One minus 0.5478 gives us 0.4522. Is it being, why we doing that? Why we saying one minus that? It's because we know that the area underneath the curve is equals to one. And since we are only looking for that area, the red shaded area, and we are able to find this blue area on the table, we can use the complement event, which is one minus the probability of the event that is not part of that one that we are looking for. So that will be one minus 0.5478, which is one minus the probability of the value we find on the table, which is equals to 0.4522. Any questions? If there are no questions, then I can give you an exercise just to give you a feel of how we do the probability of greater than. Same question that we used previously. I guess you still have the Z values. So we know what the mean is. Mean is 86. We know that the standard deviation is the square root of your variance and our variance is the square root of 16, which is equals to four. And we are looking for the probability of X greater than, greater than eight. Calculate your Z score. And when you are done, if you want me to go to the table, let me know and say, please go to the table, but you need to tell me which side of the table you are looking for. Actually, we can, yeah. Are we winning? Do you want me to go to the table or do you have a table? Go to the table. Positive or negative side of the table? Negative side. Okay. You must tell me where to stop or I can make it smaller because I don't have the values. So you know what to work with. Okay, now. Yes, you can stop. Okay, now. Okay, so can I make it smaller in case I have missed some of the values? It looks fine. Okay. I'm going to move this. Are we done? Yes. Okay, so let's see. So you're saying your probability is 0.9332. Yes. Let's wake it out together. Okay. So we know that we're looking for the probability of Z less than X minus the mean divided by the standard deviation. Probability Z less than what is our X value? It's always given in the question, which is 80 minus the mean. 86. Of 86, divided by the standard deviation. 44. 44. And the probability of Z less than what is 80 minus 86 is minus 86 divided by four. You got minus 1.50. 1.50. So we need to go to the table and go find one minus the probability of Z less than minus 1.50. So let's go to the table. We go to the table. We look for minus 1.50, which is this, because I know that this is zero at the top, which will be the first color, which is 0.0668. So therefore this, take it back. One minus 0.0668, which is equals to 0.09932. There's an error there. There's an error? Yes. Should be? On number two, it should be zero, nine, double, three, two. Double, three. Yes. Okay. I will fix that as well. We'll make you guys the right slides as well, including the first one. Okay. Right. Any question before we move? So that is clear, right? You know that for the probability of Z less than A, the value find on the table, that will be your probability. For the probability of Z greater than A, we say one minus the table value, right? That's how far we've gone. You will remember this. You will always remember that for the probability of less than, you will find the value on the table. For the probability of Z greater than, you will say it is one minus the value you find on the table. Agree? Happy? Yes. Yes. Yes. So now let's move to how we find the probability of between. Now, if we need to find the probability of between, we only interested at the orange section, not the white shaded area. Sorry, not the non shaded area, the white non shaded area. We only interested in the orange shaded area. This is the between. We need to be able to find this probability. So now, when we find this probability, it's easy. We're going to find the value on the table, but the value that we find on the table, we first going to find the probability of Z less than B. So we're going to take this side first, minus the value we find on the table for the probability of Z less than A. So it means we're going to take the table value for B side of things. We're going to subtract the table value of the value we find for A. So we're just going to take the table value there and the table value there. And that will give us the probability of between. So we say that minus that side. Let's look at an example. Suppose that X is normal with the mean of 18 and the standard deviation of five, find the probability that X lies between 18 and 18.6. So we can calculate the Z value. Our X is 18. Our mean is 18. Our standard deviation is five. 18 minus 18 is zero, divided by five will give us zero comma zero zero. We do the second part, which is 18.6 calculate the Z value. X minus the mean divided by the standard deviation. Our X is 18.6. Our mean is 18. Our standard deviation is five. And we know that that is zero comma one two. Then we need to go and find the probability that Z lies between zero comma one two and zero. Then it means we need to go and find the probability that Z is less than zero comma one two, minus the probability that Z is less than zero comma zero zero. That's what we need to do. I'm not gonna go out. I'm gonna use this table. So we know that we calculated our Z value. We found that that one side is zero comma zero zero and the other side is zero comma one two. So we're going to do, this is the step, our Z of zero comma one two minus our Z of zero comma zero zero. So zero comma one two, we go find zero comma one and two at the end. And we know that that is zero comma five four seven eight. We need to go find zero comma zero zero, zero comma zero zero and zero at the end. And that is zero comma five zero zero zero. So we subtract one from the other and we find that the probability of that between is zero comma zero four seven eight. That's what we do. We take the probability that from the red and the shaded all this is zero comma five four seven eight. Remember, it would have been the red shaded all over which is zero comma five. And we just need to subtract there. We just need to subtract this shaded, the non-shaded area so that we can remove only, be left with only the red shaded side of this. And that's how you find the probability of between. So what we have learned so far, probability of z less than a is the value you find on the table. Probability of z greater than a is one minus the probability you find on the table. The probability of z lies between two value a and b is given by the probability that z is less than b minus the probability that z is less than a, which is the table value for b minus the table value for a. And that's all what you need to remember in terms of the normal distribution table. And this is what you're going to also have to remember until we get to finish off and do hypothesis testing. So when we do the confidence interval, you need to remember this. When we do hypothesis testing, you need to remember this. So just always remember to calculate the probabilities like this. Okay, so I think we have another exercise. Okay, this is just another example. Suppose x is normal with the mean of 18 and the standard deviation of five. Of five, find the probability that x lies between 17.4 and 18, sorry. So we need to go and find the probability that z lies between x minus the mean divided by the standard deviation, x minus the mean divided by the standard deviation, which then substituting, we know that the mean and the standard deviation, so we're going to substitute. So we do, we first start with the 17. So it will be 17.4 minus our mean of 18 divided by our standard deviation of five. And on the other side, 18 minus the mean of 18 divided by the standard deviation of five. 17, calculate this for me quickly. 17.4 minus 18 divided by five, what do we get? Minus 0.12. Minus 0.12 is less than, and 18 minus 18 is zero divided by five will just give us zero, comma, zero, zero. Now, we need to go find the probability. So it means we need to go find the probability that z is less than zero, comma, zero, zero minus the probability z is less than minus zero, comma, one, two. Going to the table to go find the probability of zero, comma, zero, zero. We know that that is zero, comma, five, zero. Zero, comma, five, zero, zero. We just did this, we did find it. It was zero, comma, zero, zero. We just need to go find the probability of a negative one, comma, negative 1.2. Is it 1.12 or 1.2? Zero point, one, two, one, one, one, two. Zero point, so it's not that, it's zero point. Let's go first to the top, oh wait, sorry. Two is this column. We just need zero point, we need zero. Minus zero point, one, two. So it's the third column, zero point, one, two is zero point, four, five, two, two. So minus zero point, four, five, two, two. Am I writing it right? Four, five, two, two, so just do the calculation. Zero point, five, zero, zero, minus zero point, four, five, two, two. Zero point, zero, four, seven, eight. Zero point, zero, four, seven, eight. And that's the probability of between. Any questions? If there are no questions, I want you to find the probability that z lies between the two values. Now, with this question, they are not asking you to go calculate the z value, they've calculated it, they've given you the z value. So you just need to use the z value to go find the probability, that's your exercise. So you need to do the probability of z lies between minus one point, one, zero, and one point, nine, zero. Do you want me to go to the tables? Let me know just now. So remember that you will need to go find the probability of z less than 1.9 and you need to go find the probability of z less than minus 1.1. Do you want me to go to the table? Let me just go to the tables. We'll start with the positive side. We're looking for 1.9. So that is zero point, nine, seven, one, three. And we're looking for minus 1.14. So we need to go to the negative side. 1.1, which is zero point, one, three, five, seven. Minus zero point, one, three, five, seven. Do the calculations and if you are lost, ask, do we have the answer? Number three. It is zero comma. Zero point, eight, three, five, six. Eight, three, five, six. And that's how you find the probabilities. Easy, isn't it? Straight forward, easy, easy. Okay, so you're not going to always get easy questions like this. Sometimes they might give you the probability and they might ask you to find the value of x or the value of the standard deviation. You should be able to work backwards. You should be able to know how you found this probability or that probability or the probability of the less than. You should be able to know how you did that in order for you to be able to go back. Like, for example, if you are given the probability and you are asked to find the value of x, then it means you need to use that probability. What I mean by using the probability? It means you will be given this probability, let's say zero comma zero seven three. That is the probability that they would have given you. You need to be able to know that this probability of zero comma zero seven three, you found it by using the z value of minus two point four and the one at the top. Don't forget the one at the top, this one. You will need to know that zero comma zero zero seven three corresponds to the z value of minus two point four four. Zero comma five, zero comma zero five nine four corresponds to those probability, to the z value of minus one point five six. Now, you need to be able to work outside. And this would have been for the z value of less than. So what if, so everything that I just did as an example now is the value of z less than a value. So we'll find it there. What if then they said find x where the probability is greater than. So if this one, then they say it is greater than, greater than a value is the probability of z greater than a value of zero comma zero seven three. If that was the probability that they give you, find the value of a such that the probability which is your z value, where the probability of that z value is zero comma zero seven three. Do not come here and look for zero comma zero seven three because then if you say that is the z, the value of a is minus two point four four, you will be wrong. Because this probability of zero comma zero seven three, they found it by using one minus the probability of z less than a. You must remember that the probability of z greater than a of zero comma zero seven three is not corresponding to the z value of minus two point four four because that zero comma zero seven three they found it. The answer to that question was one minus the probability of the value they found on the table. So what do you need to do in order to find the actual z value which is the value of a, you will have to say in order for you to find the probability of z greater than if you need to find the z value a of zero comma zero seven, what was that? Zero comma zero zero seven three. You will have to say one minus zero comma zero zero seven three should give you the actual value of a, which is equals to zero comma nine two seven. Zero comma nine two, zero comma zero nine two seven. Zero comma nine two seven, yes. Zero comma nine two seven. Oh, seven zero, okay. Oh yeah, seven zero. Yes, so then it means you need to go inside this probability table and go find this value which you will find in the negative side all the probabilities are small like you will notice that all the probabilities that with zero comma zero zero zero some number and goes to zero comma zero four six, so they are small. So when you get a probability which is bigger than five. Sorry, Lizzie. Zero comma five two, zero comma five one, you need to go to the positive side of the table. Sorry, Lizzie. Yes. The answer is zero comma nine nine two seven, right? Zero comma? Nine nine two seven. Zero comma nine nine two seven. I'm still looking for an eraser. Just give me a sec. So the answer here is zero comma nine nine two seven. Two seven, okay. So we need to go find this number, zero comma nine nine two seven. So we go find it on the negative side, zero comma nine nine two seven. Zero comma nine nine can go up a little. Two seven, zero comma nine nine two seven. And then I can go out and go look for that probability. Two point four four, probably. So then our A value will be two point our A value will be two point four four. That is the Z value. And that's how you will remember all this. So when it's greater than, remember that the answer that the probability that you're looking at is this one minus. It was found by using one minus that value. Okay. So let's look at this example and this will be our last section. So since we know that our Z value you will not be given this equation. So we know our Z value is X minus the mean divided by the standard deviation. If we are not given the mean, then we must make X the subject of the formula and making X the subject of the formula. It means we multiply Z by sigma is equals to X minus the mean and we take the mean to the other side it becomes positive. So it will be mean plus the Z of sigma is equals to X. You just need to know how to change the subject of the formula. If we were asked to calculate the standard deviation in that instance, so this one we were calculating X. So if we need to find the standard deviation so we know that Z is equals to X minus the mean divided by the standard deviation we multiply the standard deviation. So we'll have standard deviation and we multiply by Z X minus the mean and we need to divide the side by Z divide that side by Z Z and Z cancels out. The standard deviation will be X minus the mean divide by Z. So this Z will be your Z values on the table. So let's look at an example. To apply this. Let X represent the time it takes in seconds to download an image from the internet. Suppose X is normal with the mean of eight and the standard deviation of five. Find X, so we need to be calculating X such that the download, such that 20% of the download times are less than X. So the other important thing you need to remember is the psi. So when they say it's less than X, so therefore it means when we found the probability that X is less than an X value that we were looking for which is that, that probability was zero comma two zero. I'm going to put four zeros because on the table we work with 40 decimals, it's 20%. So if, oh sorry, if we know this, therefore it means this actually should be Z for X value. So if we know this, that this is the probability, but we also know that that probability we found it by using less than, so therefore it means 20% we will find it on the table and where the values correspond that will be our Z value. So we go to the table, let's keep everything, go to the table, let's go look for zero comma two, zero comma two, zero, zero, zero. We won't find it in the positive side, we need to go to the negative side of the table. Zero comma, remember we're looking for zero comma two, zero, zero, zero. So let's go find zero comma one, zero comma two. So it's somewhere in between those two. Okay, this is far away and this is close by zero comma two. If I look at this, even if we run it off, this will be the closest one. So it corresponds with, we're going to use minus, minus zero comma eight and then we need to go to the top. And at the top it corresponds to four. So that will be minus zero comma eight, four. So going back, so we know what the value of the small x of ours is, so this will be the probability that Z is less than minus zero comma eight, four. But that is not what we're looking for. Now, all I need to do is expand my Z, remove the probability because I no longer need that. So I can say my Z is equals to x minus the mean divide by the standard deviation. I know that my Z is minus zero comma eight, four because I did find that. I don't know what my x value is. My mean is eight. My standard deviation is five. Solve this. It will be minus zero comma eight, four, multiply by five because we multiply with what is underneath will be equals to x minus eight. And we can take eight to the other side, minus zero comma eight, four times five minus O plus eight is equals to x since I don't have a calculator with me. I will rely on you to do the calculation. So multiply zero comma eight, four times five, minus zero comma eight, four times five equals minus point four two plus eight is equals to three point eight, three point eight, three point eight, three point eight. Let's see if we got it right. So to recap, we use 20% we go find on the table and we go look for the Z value and once we have the Z value, we can substitute that into the formula and calculate and the answer is three point eight. So 20% of the value from the distribution with the mean of eight and the standard deviation of five is less than three point eight. And that's how you work back once with that. Any questions? Because it concludes today's session. On Saturday, we will do the activities that relate to this so we can practice more. Any questions before I continue? Can we do one more exercise when we're working backward? We can. Let's say I'm going to ask you now since I'm not going, I don't have any other questions but I can formulate a question just now. Let's see what time is it? Okay, we still have a lot. We have plenty of time. Sorry to come back here. I need to go out of this cat because I need to get rid of this. So we're going to use the same question. I'm just going to change a couple of things here for you to work out. So in state of less than, I'm changing it. Same question, greater than. That is your question. Find X such that 20% of the downloads are greater than X. So the first thing you need to do is to go find the probability that Z, let's use A as our example. Sorry, we're not finding Z less than A. We're finding Z greater than A. Remember that? Z greater than A of 0,000. Now, we know that they found it by using 1 minus 0,000 which is 0,800. I'm just going to use that. So it means we need to go to the table and find the Z value that corresponds with this. So let's go to the Z value table. We're looking for 0,8. We are not in the positive side. We need to go to the positive side and look for 8, 0.8. 0.823. So that will be very far. This will be very close by. So the answer will be 0,844. So going back, our PZ greater than 0,844 would have given us 20% of the time because we would have used 0,8 to go find that. So since we have that, we no longer want the probability that we can replace Z equals X minus the mean divided by the standard deviation. We replace the Z value with 0,84. We're looking for X minus our mean of 8, our standard deviation of 5. 0,84 times 5 is equals to X minus 8. Take 8 to the other side. So let's just calculate this. 0.84. Oh, sorry. Yeah, I forgot to put 4. 0.84. Multiply by 5 gives us 4.2. So this will be 4.2 plus 8 is equals to X. Therefore our X will be equals to 12.2. And that's how you will get this. I will find more exercises for Saturday so that we can practice even more. So sometimes you will find where it says exceeding, which means it's greater than or it's less. So you will see if I can find all those exercises with different questions of which ones. Hi Lizzie. Yes. So can I just ask while you're still there. So the minus 8, the X minus 8, when it jumped to the other side, does it become positive? Because I see there you're saying 4.2 plus 8. Yes. So when it moves over, it changes the side. So if it was positive 8. Minus. Was it going to be minus? Yes. Cool. Thank you so much. No worries. Any other questions? If there are no questions, we are done with normal distribution. We have learned the basic concepts of normal distribution. We've learned how to compute the probabilities of a normal distribution. You know that the probability of Z less than A is the value you find on the table. The probability of Z greater than A will be the 1 minus the value you find on the table. The probability of Z lying between the two values will be the probability of the second value minus the probability of the first value. We also learned how to use the table to find the probability. We've also learned how to use the table to go find the Z value outside. And with that, it concludes today's session. I will see you on Saturday. Next week, Wednesday, we will be doing sampling distribution. The same concepts we just dealt with today, we are going to do them again next week, Wednesday. So by the end of next week Saturday, you should be able to do your assignment three. So chapter four and chapter five. Chapter six, which is study unit six and study unit seven that we're going to do next week. They make up your assignment three. So please, those who are way behind with their assignment one and assignment two, make sure that you complete them so that you cut up with everything in order for you to be able to do your assignment three. And with that, concludes today's session. Any comment, any query, any question, any feedback, you can unmute and we are done. Thank you.