 Welcome to this week's session. We're going to look at the questions related to study unit 6 and 7, which is study unit 6 was the normal distribution, and study unit 7 is your sampling distribution. In terms of both of these two study units, always in my unit, you know the properties of one of them, of normal distribution, whether the properties of normal distributions that for the data to be standardized, it will have the mean of zero and the standard deviation of one, and to standardize the data, we use the z-score and the z-score for normal distribution, we use the x-observation minus the population mean divided by the standard deviation, and that will just standardize the data that you have into a normally distributed data with each distributed with the mean of zero and the standard deviation of one, and also you need to remember the distribution in terms of the standard deviation when it's one standard deviation, when it is taller or flatter, all those things that you need to remember. If your standard deviation is bigger, does that mean your normal distribution is narrower or it's flatter? Things like that, and when you change the value of your mean, because remember the mean, it's on here, when the mean of zero is on here, if you increase or decrease the mean of your data, you know that the graph will shift from left to right. Increasing and decreasing your standard deviation tells you whether your standard deviation, the bigger it is, the flatter your graph, the smaller it is, the taller your graph will be, because when your standard deviation is smaller, therefore it means your graph will be closer to the middle and it will be taller, so you just need to know all those. So the smaller your standard deviation, it means your distribution will be taller, the bigger the standard deviation, your distribution will be flat, things like that. You need to know those properties. Also, you're not only knowing how to standardize your data, but to find the probability of that standardize data. And remember, when we find the probabilities, and this will apply also in the study unit 7 to find the probabilities, if we calculate the standardized value of z, if the value of z is less than, so we need to find the probability that the value of z is less than, then we're going to find that probability on the table. Remember, we use the table value. If it is the probability of greater than a value, then we're going to say one minus the value we find on the table, because then we're talking about the bigger values. And remember, your normal distribution table has the negative and the positive side. Both of them are for the probability of less than a value. And if you need to find the probability that z lies between two values, a and b, then you will have to go and find the probability of the second part, which is the probability of b minus the probability of a. So you will go to the table and find the probability of b first, and then subtract the probability of a second. And this is what we're going to apply just now when we look both. The questions, you must just remember that. Okay, so let's start answering the questions. Which one of the following statement is incorrect? So we're looking for the incorrect statement with regards to the normal probability distribution. A, the standard normal distribution has the mean of zero and the standard deviation of one. B, the area to the right of the mean of the standard normal distribution is one. And the area to the left of the mean of the standard normal distribution is also one. C, the score of the mean of a normal distribution is zero. G, 95.4% of the values of eight. Random variable are within two standard deviation of the mean. And E, the smaller the value of your standard deviation, the narrower or the steeper the calf. Let me go back to that because there is something that I didn't explain. When we talk about the probability, we're talking about the area underneath the calf. So we know that if we're talking about the area underneath the calf, and if in the middle it splits the calf into 50% each way. On this side is your left and this side is your right. And we're talking about the probability. We know that the probability will be the area underneath the calf. So we also know the properties of probabilities. It says the sum of all probabilities should always be equals to one. And therefore, if it means the area underneath the calf is equals to one, then it means the probability of the area of the calf is equals to one. If you look at the left-hand side, therefore, the probability on this side, your p-value on this side, your probability value on this side will be 0.5, and the probability on this side will be 0.5, because 0.5 plus 0.5, they both make up your probability of one. Also, you must remember the empirical rule. What does one standard deviation of the mean mean? One standard deviation away from the mean mean, does it mean a 95%? Two standard deviation, what does that mean? Three standard deviation, what does that mean? So you must remember those things. So remember that 68% refers to one standard deviation away from the mean. And 95% is two standard deviation, and 99% is all, or what do you call, three standard deviation away from the mean. You need to remember those empirical rules as well when we deal with normal distribution. Other than that, we should be able to answer all the questions on there based on the information that I just shared with you. So let's go. Which one of the statements is incorrect? A, B, C, D, O, E. B. I say B or D. B for Bravo. B for Bravo, B. B is the incorrect statement, because all the other statements are two based on what I just explained to you. Consider the standard normal distribution, Z. Which of the following probabilities is incorrect? We are looking for the incorrect question, or the incorrect one option on here. So A, it tells you that Z lies between two values, and remember, these are already standardized distributions. So you don't have to go and calculate the Z value because they have calculated them. We just apply the rule that we learned. Remember, those are the rules that we have learned. We just apply them to answer these questions. It should be easy. So I'm going to go to the table. We'll do each statement together using the table. And then the next one I will ask you which is your own. So the first one, it says the probability of Z lies between minus 2.8 and 0. So it means we need to go to the table and look for the probability of Z less than 0.00 on the table, minus the probability of Z less than minus 2.80. The reason why I'm keeping the two decimals is because on the table, we're going to look for, remember, the table looks like this, we're going to look for the two values, the one before the comma and the, let's say it's 0.0 on your left-hand side. And at the top, you're going to look at the last digit at the top. So in order for us to get it right, we are going to do that. So in terms of the 0.00, so it means the last digit at the top here will be 0.0000, right? So let's go to the table. We just need to go to the table. You cannot answer these questions without having the tables. And remember, the tables has the positive and the negative. So you have the negative side and you also have the positive side. So our question in terms of what we're looking for, there is a positive 0.00. So go to the positive side and we look for 0.00. So we look for 0.00 on the left and the last digit at the top and where they both meet. That is the value that we are looking for. So our first value is 0.5000. So therefore, it's 0.500 subtract. We need to go to the negative side and look for minus 2.80. So we go to the negative side and we're going to look for minus 2.80. So it means we're going to look for negative 2.8 on the left. And at the top, we're looking for 0 where they both meet. That is the value that we are looking for, which is 0.0026 minus 0.0026. Then you do your calculations. The answer will be 0.005000 minus 0.0026, which is equals to 0.4974, which is correct. We're looking for the incorrect one. Then we continue doing the same. I'm not going to do it with you. You're going to do it and you tell me the values. So you're going to look for the probability. We start with the second one first. z less than 2.10 minus the probability that z is less than 0.00. So go to the positive side of the table and look for 2.10 and tell me what is the value. So I expect everyone to have the tables because I did share the table with everyone. And the tables are part of the node. The table is under statistical tables. You should have downloaded them and have them ready on your PC somewhere where you are able to reference them. If you have facilities where you can print them, it's even easier and the best way is to have them printed and have them with you. So what is the probability of z less than 2.10? Nobody. I'm thinking it's 0.98. We can double check that. So we're looking for 2.10. So you go to the positive side of the probability table and you look for 2.1 and you look for 0.00 at the top. And the probability is 0.9821. 0.9821 minus the probability of 0.00. We did find it with 0.00, 0.50, 0.00. So calculate. Does it give you 0.4221? And the answer is 0.4821. So this is the incorrect one. I'm just going to continue so that you can practice. So let's look at C. You see on your own. See if you can get it right. Sorry, Lizzie, I have a question. If the probability of z is smaller than 2.1 and we find 2.1 on the z value, where do we find? I'm assuming we use minus 2.8 as the column value. I'm not sure. What? Are you doing it the same way as we have been doing it? I've been trying to use. Because the first part is 2.1, right? We already have that probability. The second part is minus 2.8. We did find it in the first. We did go find the probability in the first a question. This is the same. So you just use the values. But you need to write the formula correctly so that you are able to get the right answer. Are you winning? I got the correct answer. OK, so the first thing you do is the probability of z less than 2.10 minus the probability of z less than minus 2.80. What was the probability of z less than 2.10? 0.1821. 0.1821 minus the probability of minus 2.80. 0.500. No, it's 0.0026. 0026. Subdirect one from the other. And the answer should be 0.97. 0.795. 0.95, because that is correct. Now, when we get to d, d says find the probability that z is greater than or equals to minus 2.8, which is the same as the probability of 1 minus the probability of z greater than 2.8. This one, they want you to validate that the left-hand side is the same as the right-hand side. So let's check. What is the left-hand side? What is the probability of z less than or z greater than? The probability of z greater than or equals to minus 2.8, and I'm going to put 0 there, is the same as 1 minus, because the rule says, what does the rule say? If it's the greater than, then we're going to find the value by subtracting the value on the table from 1. So we do that. 1 minus the table value, and the table value of minus 2.80. What is the table value for minus 2.80? It was 0 comma 0, 026, because that's the value that we've been getting all along. And the answer here is 1 minus that. What does it give you? 1 minus 0.0026 is equals to 0 comma 997, 0 comma 9974. That is this part. Let's check if it's the same as the second part. So the second part says we need to find the probability. So the second part says 1 minus. Don't get confused with this. 1 minus the probability that z is greater than 2.8, and I'm going to put 0 at the end. Now, because whether we put greater than or greater than or equals to 4, the normal distribution and sampling distribution, they mean one and the same thing. They will not change how you find the data. So because this says it's greater than, then it means we need to say 1 minus the table value. So this will be 1 minus. We need to change the second part to 1. I'm going to put it into bracket 1 minus the table value. But this is the table value of 2.80. We don't have that because previously we worked with minus 2.8. So we need to go to the table value on the positive side. We look for 2.80, which is 0 comma 9974. So we go there. And we say 0 comma 9974. We're not done yet. 1 minus, what is the answer? 1 minus 0 comma 74 is 0 comma 0026. And 1 minus 0 comma 0026 is the same as 0 comma 9974. Are the two answers correct? The same. So the left-hand side is the left-hand side equals the right-hand side. If they are correct, if they are equal, therefore that statement is correct. That's how you check. Let's look at number E. E says the probability of Z less than minus 2.1 is equals to the probability of Z greater than or equals to 2.1. You also need to validate those two statements. Let's do that. Let's check the left-hand side. What is the answer for the left-hand side? The left-hand side says the probability of Z less than minus 2.10. So it means we need to go to the table because it's a less than, right? Less than or equals to is the less than. The rule says if it's a less than, the value you find on the table is your probability. So let's go to minus 2.1. Go to the negative side of the table. Look for 2.10. The answer is 0.0179. Right? 0.0179. That is the left-hand side. Let's check the right-hand side. The right-hand side says we need to find the probability that Z is greater than or equals to 2.10. It's on the positive side, but it says greater than. So we need to find 1 minus the value we find on the table. So let's go to the table and look for 2.10. We've been finding it. We know what that is. It's 0.9821. Remember, 2.10. We did find it. It's 0.9821. So we go and find 0.9821. And do the maths. What is the answer? 0.0179. So that's your left-hand side equals your right-hand side. They do because they are both 0.179 and this side is 0.0179. So that is correct. And that's how you're going to validate your answers or check whether the statements are correct or incorrect. Are you lost? Do you? Or is it here? Lizzie, I have one question. When we're looking for, for instance, minus 0.28, right at the top, the first one, how do we know that the column from the top is 0 and not 0.01 or 0.02? OK, because you need to keep two decimals, right? If I have 1,1, is the same as I can write it as 1,1 and 0 at the end? OK. If I have two decimals, if they give you two decimals, right? If they give you two decimals, it will be 1,11. And you will know that the last digit is 1. You need to go and find 1. So if they only gave you 1 decimals, you just put a 0 because 1,10 is the same as 1,1 if we drop the 0. Oh, yeah, yeah, OK. Yeah. So only they give us another decimal. Do we use the across the columns? Otherwise, the square is the 0, OK. Yes, if they give you another value, so let's say they give you 2.28 as the z value. They gave you two decimals, right? So we know that these are the values you're going to find on the left-hand side. And the last digit is the one that you're going to find at the top. So you're just going to go, yeah, there is only two decimals and the last digit will be. So always remember that for z, you only need two decimals. Even when you are calculating it yourself, the z score, remember to always keep it to two decimals. OK. OK, so moving on to the next question. That's one more question, Lizzie. Yes. So every time the number that you go from the table is minus sign, you must minus by one positive one. Oh, remember, when you apply this rule, if the question says less than, it got less of whether the answer of A is minus or positive. It's negative for positive. The sign less than tells you the value you see on the table. That's the value you are looking for. If it's greater than, this is the sign that you will look for, if the question says greater than, then you're going to take the value from the table and subtract it from one. If it is between, if it has two of them, it says between whether there is an equal sign or not equal sign underneath. But as long as it says it's between two values, remember to find the table value of the second one first, then the table value of the second or of the first one and subtract the second or the first one from the second like that. And that is what this is the basic thing that you can do, the shortcut way you can use to answer the questions, especially from normal distribution and sampling distribution. Always remember this. This should be somewhere where you can always remember and use it. OK, so let's look at question three. It says given that Z is a standard normal distribution, what is the value of Z such that the area between your negative Z value and your positive Z value is 0,2358. That is the probability of Z lying between two values. Let's assume those two values, take Z, negative Z as A and positive Z as B, take it as such. If you think about it in that way, you won't get anything wrong because it says the probability that Z is between A and B is 0,2358. What is that probability? Or what is the Z value? That is what we are looking for. We need to find that Z value, the small Z value. What is the value of Z such that the area between the Z negative side and the Z between the positive side is 0,23. So ask yourself this question. If they say that therefore it means we need to find how they got the 0,238 is by saying the probability of Z less than the positive Z minus the probability of Z less than the negative Z. All what we want to know is what is the value of this Z? Such that that value of Z, when we go find them on the table, what is that value of Z? That when we find that value on the table and we subtract this value on the positive side and the value on the negative side, we get 0, what do we get? 0,2358, which is very, very difficult to find because your negative and positive Z values should be almost similar to one another because one should be positive and the other one should be negative. So let's check. So we can use these values that we have here to check if those gives us the right value. So the first step that we can check is if our Z positive, the site, is 0,12. This site should be minus 0,13. So let's go check that when we subtract one from the other, we get 0,2358. Let's go. What are we looking for? 0,12. So we go here. We look for 0, or not 1, 0,1. Let's remove all the ink that we have. Sorry, Lizzie, where did you get 0,12? From the options. We're going to validate each object that is here to see if we get that answer there. So we take all these options because they say the value of Z is either one of these values. So let's check if it's one of those values. So 0,12, 0,1, and then 2 at the top is 0,5478. 0,5478 minus, and we go to the negative site. And we look for 0,1, 0,12. And that will be because I know that at the top there, it's 2. 0,4522. 0,4522. So let's take our calculator and calculate and see if we get that answer. 0,5478 minus 0,4522 equals, nope. That is not the correct one. Moving on to the next one. So that is not the answer we are looking for. We're going to change that to the next one. Let's go to the next one. 0,58. 0,58 minus 0,58. Go to the positive side. Look for the probability. 0,58 is the second last column. 0,7190. 0,7190 minus, we go to the negative side. We look for 0,58. Negative 0,5 second last column, which is 0,2810. 0,2810. Take our calculator. Let's calculate. 0,7190 minus, 0,2810. 0,43, nope. That is not the answer. We're looking for 0,2358. So that is not the answer. Moving on to that, let's go to the next one. The next one is 0,82. 0,82 minus 0,82. We need to validate each one of them to see if it gives us the answer. OK. Go into the positive side. Remove all the inks that we have here. 0,82 is 0,7939. 0,7939 minus, we go to the negative side. So remove the inks. 0,82. 0,012. 0,2061. 0,2061. Take our calculator. Let's calculate. 0,7939 minus, 0,2061. Let's hope this is the answer we get. No, it's not. So we move on to the next one. 0,62. This. And replace our Z value with 0,62. 0,62. 0,62. Going to the positive side of the table, looking for 0,62. Which is 0,26. I'm on the wrong side. We need to start this side first. 0,62 is 0,7324. 0,7324. 0,7324 minus, going to the negative side of the table, looking for 0,276. 0,2676. And let's calculate. 0,7324 minus 0,2676. Equals 0,4648. It is not correct. The last one. I hope this was the correct options that they gave you. Because the last one, if it's not correct, then it means they didn't give us the correct option. Okay, so let's see. The 0,30. And minus 0,30. So we go to the positive side and look for 0,30. Which is 0,6179. 0,6179. 0,6179. Did I write it right? 7169. 7169. Minus. We go to the negative side. We look for 0,300. 0,3821. 0,3821. Let's take out our calculator and calculate. 0,6179 minus 0,3821. And equals. Yay! The right answer is 0,30. How is that value? 0,30. Happiness? And Lizzie, the answer is 0,2358. How did we round it off to 0,3? What do you mean 2,3? The answer is 0,2358. Which is the same as the answer that we have there. Because they say the probability that Z lies between negative 0,300. So this says the probability that Z lies between negative 0,300 and 0,300 is 0,235. Thank you. It's 0,2358. That's what we proved. And that's what we will prove it. You need to also think outside of the box. Sometimes they might not give you the probability of that, but they might give you Z. They might say, find the probability on the right or on the left. If the left is this much, what would be the probability on the right? So you just need to also make sure that you think outside of the box by remembering to apply this. Because the probability to the left, which is less than, is to the left. The probability to the right, which is greater than, how do you find it? And the probability of Z, how did you find it? You need to think outside of the box. Especially when it comes to this kind of questions. Okay. Consider a normal random variable with the mean of 3,000 and the standard deviation of 1,000. So they have given you the mean, and they have given you the standard deviation. Calculate the probability that the random variable is at most. What is at most? Less than equal to. It's less than equals to 3,800. So because it's less than or equals to, so we calculate in the probability, then it means that we're going to use the formula, the probability of Z less than 8, the value we find on the table will be our answer. Choose the correct answer from the list of options below. So let's go and calculate the probability that X is less than or equals to 3,800. And because this is your data, we need to standardize that data by using the formula. Our Z distribution will be Z less than or equals to, we're going to distribute or standardize our X value by using our Z normal distribution formula. So let's standardize that. The probability that Z is less than or equals to our X is what we are given. It's 3,800 minus your population mean. I just gave you. It's 3,000 divided by the standard deviation. It's 1,200. Probability that Z is less than or equals to. Do the calculation. What do you get? 0,66. 0,66. Let's double check. 0,0 minus 3,000 divided by 1,200 equals 0,667. You need to make sure that you round off correctly as well. Right? So the answer will be in two decibels. 0,67 because the number to the right of way we went to the round of two is bigger than 5, which is 6. So we need to add 1,26. So it will be 0,67. What do we need to do now? We need to go to the table. We go to the table. Positive or negative? We go into the positive side. We go into look for 0,6 and 7 at the top. And where they meet, that is the answer we are looking for. 0,7486. If here they said at least, we would have taken this value and subtract it from 1 because it's at most. You just go to the table value and the value we find, that is the answer. Okay. Moving on. Unless there is a question. No questions. Let's move on to question five. The department of basic education found that Lenas travel time from home to school at one of the remote rural schools is normally distributed with the mean of 114 and the standard deviation of 72. What is the probability that the Lenas travel time from school is between 120 and 108? So now you need to think because it says it's between, so you're going to apply the between rule. The between rule which says the value I find on the table for B minus the value I find on the table for A. So you do the same. So let's standardize this. It says it's between 120 and 108. So if we need to standardize this, our formula will be X minus the mean divided by the standard deviation. X minus the mean divided by the standard deviation. So we do for the 121st, which is 120 minus the mean is 114 divided by the standard deviation of 72 and 180 minus 114 divided by 72. You need to calculate and give you the answers. The first one on the other side is 120 minus 114 divided by 72. What is the answer? Those who are using a sharp calculator because if it does not have the fraction like I have it, always say 120 minus 114 equal divided by the 72 equal. Don't just put 120 minus 114 divided by 72. What it will do is it will apply the botmas rule. It will do botmas and botmas says division before addition and subtraction. What it will do? It will take 114 and divide it by 72, which is not what you want. You want the answer of 120 minus 114 to be divided by 72. So always use the equal side. The top part equal divided by the bottom part equal. Those who are using the sharp calculator and you have the fraction functionality use that which makes it easy. What is the answer for the first one? 120 minus 114 divided by 72. 0.083. 0.08. Remember to keep it to 2 decibels. So it will be 0,08. The number to the right of 8 is 3. So we don't have to do anything to 8. So you will have 0,08. And do the same on the other side. It's 180. What do you get? 0.9. And that will be 0.92. Round off correctly. The smallest mistake you make in the rounding off you will not get the answer because you will be using a different number. So make sure that you know how to round off. So the number to the right of 1 is more than 5. So because it's bigger than or equals to 5 we add 1 to 1. So we will have 9, 2. So the answer is 0.92. You need to also make sure that you know how to round off. So now let's apply the rule that we know. So we need to find the probability that z is less than 0,92 and subtract the probability that z is less than 0,08. So let's go to the table and look for 0,92 on the positive side. They both on the positive side. So I'm just going to write both of the numbers here and you look for the numbers, especially those with no statistical tables. 0,92 and 0,08. Those are the two values of z that we are looking for. Write them down and then you will give them to me when we go to the next page to the presentation. Do you have them? Are you good? Yes. The first one is 0,9 and 2 which means it is this value. The second one is 0,008 which is that value. So you should have two values written down. 0,8212 and 0,5319. So the first one is 0,88212 minus 0,5319. Do the subtraction, one from the other. What is the answer that you get? 0,44 0,44 0,2893 0,2893 which is option B. So that person will get 0,44. I don't know where you get it or whether you substituted the numbers wrong some way. Make sure that you also check your numbers when you write them down. All right. Lizzie, I'm getting confused on I think it's step three where you swap the 0,9 to around with 0,08. Okay, what does they say? The probability of Z smaller than B minus the probability of Z smaller than A. So if we take the step number three and you apply that. So is the front one always A? So 0,08 is A and 0,92 B and then when I apply the formula I take B. Yes, so you will take the second one first and then the first one. Yes, I remember that A and B because it's the exact same sum between A and B. I don't know what the difference is. We want 20 minus 1,08, we want 18 minus 1,08. Okay, I see. So the first number literally is A and the second and the bigger one is B. Yes, so this is your A and this is your B. Yes, thanks. Consider again, blah, blah, blah, the same story, the mean and the standard deviation and education consultant has recommended no more than a certain minutes of learners travel to schools. If the department would like to ensure that 18.67% of the learners at year to the recommendation what is the recommended travel time? So yeah, they gave you the probability but you also need to be very careful about this because it says no more than. What does more than mean? More than means greater than. A no more than will mean a less than or equal, right? Because it's the opposite of more than. So if they say no more than then it means less than the values, right? Are we agree? Yes, we agree. Okay, so then it means this probability that they gave you is the probability that Z of a value that we are looking for of a less than or equal because it says no more than of 0,1867. So it means if we want to know what is this value of a, right? That is what we want to know. In order for us to know what is this value of a, we need to go to the table and go to the table. We know what do we know about the less than values? So if the rule says for any value less than that's the table value and we know that this is the less than value and it was 0,1867. It's a table value. We can use that to go and get the Z value. So let's go get the Z value. There's nothing we need to do. We do not have to even subtract from one or do anything. You just take the value as you see it go to the table because it's the probability of a less than because the sign says less than. So we can go and find what the value of a is. So what is the value of a? We go to because the probability is small. So we'll go to the negative side and inside this we need to look for 0, 0,1867 inside here. Look for 0,1867 Did you find it? Did you find it? I already found it but I'm asking if you guys have found it. It's under negative 0,8 negative 0,8 on the last column. The last column is 0,9. Here is the value that we are looking for. So write out what is the Z value that corresponds with this value or read out first read this value here and read that value there. What is the answer that you get? It's minus 0,8 because the last column is 9. Don't forget to write the last digit. You take those two digits plus the last digit and that is your Z which is minus 0,89. Now, because we now know this value of 8 we can now calculate because we know that Z let me write it up here Z is equals to X minus the mean divided by the standard deviation. We know that the Z value is minus 0,89 equals because what we are looking for we're looking for the X-Vid which is our travel type minus the mean of 114 divided by the standard deviation of 72. Let's apply math, multiply you're going to take it up here you're going to multiply minus 0,89 multiplied by 0, not 0, multiplied by 72 is equals to X minus 114 and X will be equals to minus 0,89 times 72 plus 114 If I apply math, right? We multiply and we take 114 to the other side and the answer we get negative 0.89 multiplied by 0.72 equals plus 114 equals The recommended travel type is 113 Did I calculate right? Let's make sure that we calculate right negative 0.89 times 72 equals plus 114 49.2 If we estimate it when we round it up to exact minutes X is equals to 50 So the approximate or the recommended travel type is 50 The recommended travel type is 50 Okay Do you understand the process? Is it clearer? Yes It's easier Moving on Two questions Unless if there is a question you are not with us mute Thank you Okay So moving on Two questions seven The emotional quotient score which is EQ of high school learners is normally distributed with the mean of 80 and the standard deviation of 20 If they were 4601 learners which is N with the high score higher than with the score higher than 91 higher than means greater than 91 How many students took the first? Okay So that is not N because these are students who got higher than So we're going to call the students X So I always tackle questions like this in a way that if I know how many students which is N which is what we need to be calculating it should be equals to X plus Y because X will be those who got higher than 91 and Y will be those who got less than 91 So if I can find out how many students scored if I know that X is 4601 Right? What I don't know is those ones the Y and I don't know how many of them those are the two things that I don't know But what I know is I can use this information to calculate the proportions because what do we know So that is one thing that we know we know that X plus Y should give us N but we also know that the sum of all probabilities should be equals to one So one should be the same as the probability of those who scored greater than 91 plus the probability of those who scored less than 91 should give me one So if I know those proportion of those students then I can calculate how many students took the test because I would know the proportion of what proportion is 4601 which will give me the rest of the other proportion because one minus the proportion of the other will give me if I know the proportion of those students who scored greater than 91 which will be one minus the proportion of those students who scored less than 91 then I can calculate the rest of the students I can know how many students are there in total right it should give me that okay so now let's do that let's first calculate and find the proportion of students who scored more than 91 so I'm going to write it here from this side those who scored greater than 91 they are how they will be one minus the proportion of one minus is yet I should not be applying one minus I must just continue and answer the question so let's standardize it Z of greater than X minus proportion our X 91 this X that I'm using here is not the same as this X I could have used P and Z by the way minus our mean set the mean is 80 divided by the standard deviation of 20 is that greater than 91 minus 80 divided by 20 1 minus 80 divided by 20 the answer is 0.55 0.55 now let's go for sorry Lizzie can you show us how you do the divide on the calculator on the KCI I'm trying to press shift but I'm not quite getting it no you can press shift is this function you just press the function oh you have this function on your calculator yes yes we just press that button okay thank you so we need to say 1 minus we need to go to the table value and find 0.55 you go to the positive side and look for 0.55 0.5 5 where they both meet 0.7088 0. 7088 0.88 which is equals to 0.29 0.29 so we know that roughly almost 29% of the students made up the entire student body so now what we must do is the following if we go back to what we had before before I confused everything else in a way we know that 1 is equals to 0.29 if I if I may use the same logic 29 12 plus the alternative will be to make up 100 will be 1 minus 0.29 which is 0.7088 which is the same thing that we have here that is what we have here right 0.2912 which is the probability of z greater than 129 is the same as 1 minus the probability of 0.78 which is this part or I can rewrite it here in that manner instead of writing it the way I'm writing it so that then it makes sense to so what we said is this is 0.2912 which is the same as 1 minus 0.7088 that's what we calculated here on this one and the same thing can you see that it brought me back to the equation that I wrote earlier so which makes it easier because I know that 29% makes up 46,000 so what will the rest what will be the rest what makes up the rest of the student body so we've got two ways of calculating this so we can say what is if I if I know let's go back to this side of things I'm gonna confuse you even more with all the formulas that I'm gonna write right now because there are many ways of answering this but I'm trying to get to a point way so if x is equals to 4601 actually our x we know what x is sorry my bad x is 4601 if we know that that is the same as 0, 2912 and we need to know what y is because x we have x right but we don't have y we know that y the proportion of y is 0,70 8,8 so let's find y so in order for us to find y we're going to cross multiply and we're going to divide so this will multiply that and this will multiply that so you will end up having 4601 multiply by 0, 7 0,8,8 is equals to 0,2912 multiply by y but we need to divide by 0, 2912 on both side as well 0, 2912 so that this and this will cancel and you will be left with y let us calculate what the value of y will be so y will be equals to use the formula on the right on the left sorry which is 4601 times 0,708 8 divide by 0,2912 equals that gives us the value of y which is 11,000 99 I'm just going to also use the decimals or we can write it off through so that we can leave it as that because if these are people people cannot be in decimal format so we can keep it as 100 and 11,900 11,199 so that is our y so if we know what our y is we can go back to our formula plus remember we can just add 11199 add them together that will give you the value of n so n is 4601 is equals to 1580 15,800 that will be the number of students so from year to year to year like I said I'm going to write so many equations that is going to confuse you okay so let's recap on what we did in the beginning to explain where we're going I needed to give you the formula that's what I wrote I said the total number of students will be those who got higher than 91 and those who got less than 91 and I went on and I wrote it in terms of probabilities because I know that that is what we're going to be using to calculate I said in order for us to get this we're going to have eventually the proportions and for us to get the proportions we know that those who the proportion of those who got 91% and those who got less than 91% will always give us one because in a way that's how probabilities weigh right and I said then it means the proportion of those who got higher than 90% will be one minus the probability of those who got less than 91% right and that is the basic thing that I started with explaining then I said okay now because on the statement itself they gave us enough information for us to calculate the proportion of those who got higher than 91% and that's what we calculated we said let's go back and calculate the proportion of those who got higher than 91% and we calculated that proportion and we found that it is 0.2912 right these are those who got more than 91% and we use this formula and I said let's reflect on this formula based on our original statement that we had and we went back to the original statement because in a way when you calculate the proportion of those who got more than 91% you will end up having to go to the table where you have to find the probability on the table and minus it from one which will give you the proportion of those who got more than 91% and I said if I rewrite this formula only this bit which gave us the answer that we were looking for in terms of the proportion it answers the question that we started with from this statement here because we know that the proportion of those who got 91% it's 0.29 and we know that the complement of it is one minus the proportion of those who got less than and which is the same as what we got here and I just wrote it there as a formula but that is not the question that they are asking they want to know how many student took the test so if we know that X amount of student wrote the test and they got more than 91 and they told us those X amount are 4600 we already know those ones and we know their proportion so 4601 students is equals to 0.2912 because those are the proportion the number is equivalent to the proportion what we don't have is those who got less than 91% but we know the proportion of them we know that the proportion of those who got less than 91% is 0.78 because there is the formula we wrote there it tells us that the probability of X less than 91% is 0.72 regardless of this one minus because one minus that use as the probability of those who got more than so since we know that 0.7088 was 0.7088% of the learners or the students received less than 91% we can calculate what the value of Y is and we went and we said if we don't know if Y is unknown but we know the proportion and we can write the formula as such because these are those who got more than 91 and the bottom are those who got more than 91% and then we apply the meds meds says in order for us to solve for Y we cross multiply and when we cross multiply it means 0.7088 will multiply will be equals to the cross multiplication of Y times 0.2912 and that's what we did and in order for us to get rid of 0.2912 we need to divide both sides whatever you do on the left you must do on the right so that we are left with Y on its own so on the right hand side 0.2912 will cancel out and on the other side we just simplify and do the calculation 4.601 times 0.7088 divided by 0.3912 is equals to 11.199 and this are the number of students who received less than 91 if we add those who received because they told us 4.601 who received more than if we add them together they will give us the number of students who wrote the test it's a long calculation but you can you get it you can get it there might be another easy way of answering the same question but in a nutshell this are the simplest for me this is the simplest way of answering this question okay so moving on to question 8 a random sample of size 120 when they start talking about sample size so we're talking about N is drawn from a normally distributed population with the mean of 160 so you just need to know that that is the mean and the standard deviation and that is your standard deviation determine the standard error and that is another thing that you need to always remember the standard error is one of those measures that you calculate from the sampling distribution and here they're asking you to calculate the standard error of the mean and choose the correct answer from the list below so we know that the standard error which is the sampling the sampling standard deviation of sampling mean of sample means is given by the population standard deviation divided by the square root of N you need to know the formulas so calculate standard deviation is 50 the square root of N N is 110 to it what is the answer anyone? it's 4,56 and the answer is 4,56 which is option C easy right right forward and easy you must always remember that standard error is the same as standard deviation of sampling this distribution of sampled means of proportions because it can be for the proportion as well that's the standard error another meaning of standard error ok now consider a normally distributed population with the mean of 990 and the standard deviation of 120 with the sample size of 50 and is drawn from the population what is the probability that a sample mean is 220 at most so there are a couple of things that we need to identify from here they have given you the mean they have given you the standard deviation they have given you the sample size and they have given you the sampled mean and they are asking you at most what is at most we have been doing at most the whole thing I'm just going to write it it's less than equal they are asking you to calculate the probability that the sample mean is less than or equals to 220 so because this is sampling distribution question you need to standardize the sampling distribution by using z of less than the sample mean minus by the standard error which is the sampling distribution over the standard deviation over the square root of n we go and calculate z less than or equals to our sampled mean is in the question which is 220 minus the mean of 190 divided by the standard deviation of 120 divided by the square root of 50 to do the calculation 220 minus 190 divided by 120 divided by the square root of 50 do we have an answer 1,77 and the answer is 1,77 and what do we need to do we'll find the answer we are looking for we're looking for the probability that's the question what's the next step this is similar to what we have been doing the whole session what's the next step how do we find the probability it doesn't mean because we are now using a different formula than everything we learned in the last 1 hour 30 minutes we forget about that so where do we go to the table on the positive side of the table and look for 1,77 1,7 1,7 and 7 where they both meet that's the answer that we are looking for which is 0,96 1,6 because also we're looking at the probability of z less than a which then it means the value we find on the table that's the value we are looking for Lizzie can you help me with it some on my calculator the KCO the 120 minus 190 over 120 over square root of 50 so you go and you first press your first fraction button and you put 220 sorry Lizzie that's why I'm getting stuck I'm pressing the function but it's not it's not doing what you're don't wait you press the first function that's what I'm trying to get to get you to be able to do the same thing so press your first function no just the button the fraction button just press the fraction button and then press 220 minus 190 and you should have the block at the bottom still a block at the bottom no it's nothing did you press the fraction button you need to have okay look at my screen I'm pressing the fraction button your fraction button should look like this on your screen but it's not showing up at the top press the mode function okay mode and then press one one and your calculator should be back to state mode to met mode and then press the fraction button press the yes and then press mode and one okay so press 220 220 minus 190 yes and then use your arrow to go down the down arrow and when you get to the block at the bottom press again your fraction button then it will create another fraction okay and then we press 120 and then you press the down arrow and then press the square root function which is next to the fraction button and then press 50 okay and once you are done then press equal side and you will see that your calculator will still have the set press the s and d the s with an arrow and a d button and it will change to decimals okay there is 1.76 you said the last one the last 30 30 pressing what is it? the s s arrow and d button that will change your 5 times the square root of 2 divide by 4 to a decimal thank you no problem I see here we have 5 minutes and there are a couple of questions we can continue with them tomorrow tomorrow is Monday yes tomorrow we will do question 9 question 11 question 12 18 and then we will be done with this and then we start with revision of study unit 8 and 9 are there any questions? no there are no questions then we can stop right here and say goodbye I will stop the recording have a lovely evening and see you tomorrow thanks Lizzie have a lovely birthday further thank you have a nice day boy bye