 Okay, so good evening and welcome to today's session, sharing, it's Risha, welcome to this week's session. So we're going to be doing normal probabilities. We're going to touch base in terms of introducing new to the concepts of normal probability as well as how we calculate the standardized values. And then we move into how we find the probability of a normal distribution. And we will do a little bit of activities or exercises. Oh, we're going to also look at the table. So remember, I'm going to be using table engine, which is cumulative standardized normal distribution table. Okay, so by the, I didn't change my slides, but by the end of the session today, you should be able to know the basic concepts of normal distribution. You should be able to compute your probability from a normal distribution, finding the probability of a standardized value less than or a standardized value greater than a value, or if the value lies between the standardized value lies between two values. And we should be able to use the formulas and also use the table to find the probabilities. The formulas we're going to need to know how to calculate the Z score, which is the standardized value. And then the tables is what we're going to use to find the probabilities. So unlike with Poisson and binomial, when we find the probabilities either by using a formula or by using a table with normal distribution, we're only going to find the probability by using the table. We use the formula to calculate the Z score. And then we will get some of the activities that we can do together. So since we're talking about normal distribution, it comes from a continuous process where your variable would have been a quantitative numeric variable, which will be continuous in nature. So since it comes from a continuous variable, that variable can take any number in the continuum. But it is a number that can also not be counted. So it cannot be counted because if it's counted, then it becomes discrete. And you know the discrete, we deal with discrete in binomial as well as in Poisson. So with normal distribution, your values would be continuous because they are measured values. Such as the thickness of an item, you cannot get an item and count how thick that is. So you'll have to take a measurement tape and measure that. The temperature, you cannot look at someone or look at the weather and say, oh, today the temperature is 26 degrees. You need to take a thermometer and measure the temperature. So it's a value that comes from a measuring process. This can potentially take any value, depending on the ability to precisely and accurately measure that. And remember also, because with continuous variable, it can take decimal. So as long as you are able to take something and measure that and get the unit out of it, that would fall part of the normal distribution. So in terms of normal distribution, we touched on the topic when we were dealing with a histogram or when we were dealing with how we summarize and view data for numerical data. And we did a histogram and we look at the distribution of the data. And part of the distribution of the data, we spoke about the data being symmetric or skewed on all that. All those form part of finding out whether your data is normally distributed because when your data is symmetrical, then we say it is normally distributed because the mean and the median and the mode are actually the same. But here I'm just showing you an illustration of different types of normal distributed normally distributed data. So you can have, oh, sorry, if I move to the next one, you can have this purple one where you can see that it's flat. And you can have this blue, the blue one, which is almost normal, the normal, normal care that we use normally, the normal, the normally normal care that we use, the green one where it is taught. All this can also be normally distributed because it also, it only depends on the population parameters, whether we move or shift the value of your mean or your standard deviation. If we increase it or decrease it, it will determine how your normal distribution can look like. So for example, in a way, if we shift or if we change the value of your mean, if we decrease it, it will shift to the left. If we increase it, it will shift to the right. So depending, so when the value of your mean changes, your normal distribution will move either to the left or to the right. When it comes to the standard deviation, the population standard deviation, when we change the value of your standard deviation, whether we increase or decrease, it changes the shape of the graph in terms of the spread, whether it becomes flatter or it becomes taller. So the bigger the standard deviation, the flatter your care will be. The smaller your standard deviation, the taller your normal distribution care will be. So if we look at one standard deviation, a two standard deviation and a three standard deviation because the standard deviation, the smaller on the first one and it's bigger on the second one and on the third one, it's even bigger because it's three standard deviations. So that's how you will be able to identify some of the properties of a normal distribution, especially with regards to the shape, how the mean and the standard deviation influences the shape of your graph. Okay, so with normal distribution, because we deal with the raw data, which is the measures that we have collected in order for us to convert, sometimes you work with this data and it does not conform to being a normal distributed data. But because we need to make decision out of it, we need to make sure that we transform that raw data into a normal distribution or into a normally distributed value that we can use to make a decision out of it. So when your data is not normally distributed, so we transform it. And the method that we use to transform that data is what we call a standardized normal distribution, which is the Z score. Usually we call it the Z score or the Z value or the Z or the standardized normal distribution value, but it will always be a Z. So in a way, we take, where am I? Or we take your actual value, we transform it into the Z score and how we do that. Remember now, I just moved too quickly, we do that in order for us to standardize your value of your mean and your standard deviation. So for your normally distributed value, it has to have a mean of zero or we say your normally distributed values would have the mean of zero and the standard deviation of one. That is a normal distribution. So it's normally distributed with the mean of zero and the standard deviation of one. I already spoke about this. As I said, we mentioned that for a distribution to be normally distributed, we say the mean and the median as well as the mode will be equal and it will take a barely shaped calf. Where the mean is your measure of central location and your standard deviation is your measure of spread or your measure of variation. So how do we transfer or translate the data or transform the X observation into a normally distributed view? We do that by using the Z score and the formula would be your X observation minus the mean divided by the standard deviation. Therefore, it means in the statement that will give you what the standard deviation will be or the population standard deviation is. They will also give you the population mean or what you need to do is subtract the mean from subtract the mean from your X observation divided by the standard deviation. And this Z score will be distributed or always distributed with the mean of zero and the standard deviation of one. That is the property of a normal distribution. Even though your mean or the population mean might be 380 and your standard deviation might be 50, but in order for us to convert that 350 and the standard deviation to convert your X observation of 252 becoming a normally distributed so that it has the mean of zero and the standard deviation of one, we use the Z score to transform that. How will we do that? So let's look at this example. If X is distributed normally with the mean of 100 and the standard deviation of 50, find the value of Z or the Z score where X is equals to 200. So we go into identify what we're given from the statement. We are given the mean of 100, the standard deviation of 50 and our X is 200. Then we go into substitute into the formula. Our X is 200, our mean population mean is 100, our population standard deviation is 50 and we substitute and we calculate and we find that the Z score is 2.0. Now I'm going to ask you that when you calculate and you get to the answer, always leave your answers to two decimal in my example. I'm not leaving it into two decimal, but always remember to leave your answer two decimals. The two decimal means two zeros after a comma. So you must make sure there are two numbers after the comma. So you will also have to make sure that you round off correctly as you leave the two decimals. When the value is greater than five to the right of where you want to round off to, you need to add one. If it's less, you leave it as is. You need to be able to know how to round off. So by calculating the Z score, we can say or we can interpret this by saying that the X of 200 is two standard deviation or two increments of 50 units above the mean of 100. You do not have to learn how to interpret, but you just need to learn how to calculate the value of the Z. So when it comes to finding the probability, you need to have calculated your Z. If you haven't calculated your Z, they would have given you the Z value. So you need to be able to know how to use that Z whether calculated or given to make a decision or to go and find the probability. So there are three ways that you can be asked about the probability of a normal distribution. If they ask you and always remember before I even get to defining what the probability, how do we get there? The probability under the normal distribution, the probability is the area underneath the kef. So it means all the values that are underneath this belly kef are what we call probabilities. So and we know that the sum of all probabilities is equals to one. So the other thing that you also need to be aware of, especially when you look at normally distributed kefs, is that your value of your kef, your belly kefs, will never touch your X observations or your X axis because if it touches the X axis, then that probability will probably be equals to zero because it will be the probability at the constant. So we use these probabilities to calculate the cumulative probability, but not a probability at a point. We calculating always, you must always remember that that we calculating cumulative probabilities, not a probability at a point. So since we calculating cumulative probabilities and we know that the area underneath the kef is what we need to be calculating when we calculate probability or find the probability, we need to remember the following. Even with the normal distribution, the sign, the mathematical sign is very important because it guides you or tells you how you're going to find that probability from the table. So yeah, you calculate your Z or you are given the Z. You are going to use that Z value that you calculated, which is true decimal value. You're going to find the value on the table and that will be the area underneath the kef, but how do we know where to find that area underneath the kef? The sign is very important. If you have to find the probability that the Z value is less than a value or the Z score is less than a value A and if value A is here, so therefore it means we are looking for all these probabilities underneath the kef, right? Because these are all the probabilities. The shaded area, which it might not be visible for you, I just realized that on this one, the area is this that we are referring to. So that is the area underneath the kef because it's less than the value of A, that is our A. So in order for us to find that value on the table, we are just going to go on the table and find that value. Later on, I'm going to explain to you because the table has two sides, but we're going to get to that. So if I need to find the probability that Z is less than a value, I'm going to use the table value and find, regardless of whether I'm on the negative side or the positive side of the table, the value I find on the table, that is my answer, or less than, always remember that. If I need to find the probability of a greater than, then I need to subtract that probability from one. Why? Because like I said previously with the probability of a less than, you need to be aware that, regardless of the positive for the negative, the Z table contains all probabilities of a less than. So if the Z table contains all the probability of a less than, therefore it means if the question asks you to find the probability of a greater than, then you need to find the complement of that, which is one minus the probability of a less than value that is on the table. So when you want to find the probability of Z greater than a value, you're going to say one minus the value that is on the table. Whether the value of A is negative or positive, regardless, you are going to say one minus. So for this side, it's going to always be one minus the value from the table. What about when it's between? So since it is between, the other thing I forgot to mention is that if I know that the probability on the table is the sum of all the probabilities is equals to one, therefore, if I split this normal distribution into half, I'm going to use this one here, if I split this halfway, this side will have 0.5 values and this half will have 0.5 values because the sum of both probabilities should be equals to one. Now, coming back to the between, when you have to calculate the probability when it lies between two values, there is a smaller value and a bigger value. Always remember that. Whether one value is negative and the other one is positive for both of them are positive or both of them are negative, regardless, there is the smallest value and the biggest value. So let us assume that A is our smallest value and B is our biggest value. So if Z lies between these two values, in order for us to find the probability, what we're going to do is we're going to go onto the table and find first the value of a bigger value, which is the probability of B. So we're going to find the value of B first on the table and we're going to subtract the value of A from the table. I hope you will remember that. For the between, we're first going to find the value of B, subtract the value of A and that will give us the probability of between. I think I've explained a lot. Let's get to the table that I've been referring to. Table, table, table. The table, which is your cumulative standardized normal distribution table. If you are using a prescribed book, please be very careful because you will find that there is also a standardized normal distribution table and that is the probability at the point. It only has the positive values there. It does not have two sides. You need to read the table carefully and use the one that says cumulative standardized normal distribution. If you are using past exam papers, it's table E2. The normal distribution table has two sides, has the positive side and the negative side. They all contains the probability of AZ less than A value. If you look at this, here is our Z. The shaded area you can see from the negative side because if this side is negative, this side is positive. On the negative side, that is the value from the negative side. On the positive side, if we start the table from the positive side, these are the values from the positive side. If you look at the probability where Z is 00, it will be 0,05 and when you go to the negative side and you look for negative 00, you will find 0,5 because this and that should give you the same. Now, we need to learn how to read this table, the negative and the positive side because this is the probability of a less than. All the values that we see here are probability of a less than. If you don't understand how to read the table right now, after this session, you need to practice because the next session and the next session and the next session, we're still going to be looking at the same table. When we're doing sampling distribution, when we're doing hypothesis testing, when we are looking at confidence interval, we are going to be looking at the same table. So you must make sure that you know and understand how to read this table. Okay, let's get down to it, since I've explained a lot. Let's find the normal probabilities. Let X represent the time it takes in seconds to download an image file from the internet. Suppose X is normal with the mean is mute. Please make sure that you are muted unless if you have a question. Okay, thank you. So let X represent the time it takes to download an image from the internet. Suppose X is normal with the mean of 18 and the standard deviation of 5. Find the probability that X is less than 18.6. And to do that, it means we're going to have to calculate the value of set. And from here, we know what our mean is. We have our standard deviation and we have our X value always given in the question. So our formula X minus the mean divide by the standard deviation. Our X is what is in the question, 18.6 minus the mean of 18.0, you can also in 20 if you want, divide by 5 comma 0. I'm going to remove this because I do have the, so we're going to use the formula and we substitute the values and then we get the answer. So our Z is 0 comma 12. That is our Z score. If they ask you to calculate the Z value, easy. You just substitute into the formula and calculate your Z score. If they ask you to find the probability because they say find the probability that X is less than. So in order for us to find the probability of a less than, then we need to go to the table because we know now that we need to find the probability of Z less than 0 comma 12 because that's what we calculated. We go to the table on the positive side because 0 comma 12 is positive. So we're going to go to the positive side on the table. We're going to look for 0 comma 1 on the left-hand side and then we're going to look at the last digit which is 2 at the top. At the top, all the values are 0 comma 0, like will be comma 0, 0 comma 0 1. So at the top we're only interested in the last digit. And on the right, oh sorry, on the left we are interested in the value before the comma and the value after the comma. That's how you read the normal distribution table. So let's look for 0 comma 1 and 2 and we find that that probability is 0 comma 5 4 7 8. Of course, it will be this portion, this smaller portion there plus the 50% makes up. So the smaller portion here is just the 0.0478. But in a way, we just say we're looking for the probability of Z less than 0 comma 1 and 2 and that will be 0 comma 5 4 7 8. That is the probability of a less than. Now, if we need to find the probability of a greater than, same information, but now the sign says greater than. In order for us to find the probability of a greater than, we did calculate this, I'm not going to calculate Z again. We did calculate Z and we found that Z was 0 comma 1 2. So our Z is greater than 0 comma 1 2. From our unit, remember our not standardized unit. So our unit was X of greater than 18.6. When we standardized this 18.6 we find that it is 0 comma 1 2. To find the probability that Z is greater than 0 comma 1 2, remember it's greater than. So we're going to say 1 minus the value we find on the table. So we're going to say 1 minus the value we're going to find on the table and remember on the table, we only have the less than values. Now, you will ask me later and say, why am I using less than and now I'm using less than or equal? It doesn't matter. We're doing accumulative and for the purpose of you understanding the concepts and you understanding the work, we're going to treat both of them for normal distribution as one. So you can use them interchangeably because we're looking at the cumulative probabilities. So even though they mean two different things in terms of mathematical side, so this one says it includes 0 comma 1 2. The other one says it doesn't include 0 comma 1 2 and and vice versa, but in a way we're going to just interchange them in this session. So 1 minus the probability of a less than because the less than is what is on the table and we know that from the table and less than was 0 comma 5 4 7 8 and the probability that X is greater than 18.6 is 0 comma 4 5 2 2. That is the probability of a less than. Looking at the between if we need to suppose that X would the mean of 18 and the standard deviation of 5 and we need to find the probability that X lies between 18 and 18.6, we calculate the Z value for 8 for 18 and we calculate the Z value for 18.6 and we can then say we have the probability that Z lies between 0 because for 18 is 0 and 0 comma 1 8 for 18.6. How do we find the probability on the table? Remember we go into take the probability on the table from the bigger one 0 is smaller than 0 comma 1 2. So 0 comma 1 2 is the biggest one. So going to take 0 comma 1 2 find the value on the table, subtract the value on the table for 0. So how do we do that? So we're going to first find the value of 0 comma 1 2. So 0 comma 1 2 is 0 comma 5 4 7 8 and the probability that X is 0 or Z is 0 and Z is 0 will be go there and 0 is the same as 0 comma 0 0. So we're going to find 0 0 and 0 at the top and that is 0 comma 0 5 and we put there and the probability of between only between is 0 comma 0 or 7 8 and that's how you will find the probabilities. Are there any questions, any confusion, anything you want to still want to find out? Now let's recap before we go and look at the the activities. We said the probability that Z is less than a value, it is the value you find on the table. The probability that Z is greater than a value is 1 minus the value you find on the table. The probability that Z lies between two values we're going to find the table value for B. I'm going to use table value table B minus table A. If you can remember this, we will be able to know how to solve normal distribution questions. Are there any questions before we move to the activities? Okay, no questions? So let's go on to the activities. Which one of the following statement is incorrect? Now then it means we need to go and find the probabilities. So in this instance they didn't ask you to calculate the Z. They have given you the Z value, all you need to do is go to the table and like I said we need to toggle between two. I'm going to stop sharing and share again but then share my entire script. I have too many things open on my screen. Let's hope it's not going to cause any chaos when I open the wrong thing. Okay, so which one of the following statement we can start from number two and then move on. I will come back to number one. Anyway, let's start with number one because then number one is fine. So number one is asking you to find the probability at the constant. What did I say about the probability at the constant? It will always be equals to zero because we always interested in the cumulative probabilities. So this we're looking for the incorrect one. So this will always be equals to zero. So it is correct. The probability that Z is greater than zero. Do you know how you will find that? So you will find one minus the table value and what is the table value is zero comma zero zero. So let's go to the table because now I'm doing it shortcut. If I'm not doing it shortcut, then I will be finding the probability of Z is less than zero comma zero zero. So it means when we go to the table, when we go to the table, we need to look for and it is positive. So that's the other thing you need to be aware of. This is negative side. So we need to go to the positive side. So you go to the positive side of the table and we look for zero and zero. And that is zero comma five zero zero. And that is zero comma five zero zero, which is the same as zero comma five. So that is correct. So I'm going to do exercise one for you. You will have to do the next exercise on your own. The next one, the probability that Z is less than zero because of the sign says less than zero. What did we say? Remember, less than zero, the table value. So it means we're going to just go to the table value. So the table value is zero comma zero five. So we go to the table value and that is zero comma zero, zero comma five zero zero. Okay, then the first one, what I did, I forgot to inform you was that one minus zero comma zero five is the same as zero comma zero five. It's my mistake, my bet. I should have alerted you to say I've already subtracted one from zero comma zero five from zero. The reason why I put that zero comma zero five thing. But that is for the probability of less than because if you say one minus zero comma five is the same as zero comma five. If it was another value, we would have had to subtract zero comma zero five from one or whatever the value from one. Number four, the probability that is less than minus one. So this one we need to treat it as minus zero minus one comma zero zero. So it means we're going to the negative side. So you go to the negative side, you look for one comma zero negative one comma zero and you look for zero at the top where they both meet. So it's the first column. The answer is zero comma one five eight seven. Zero comma one five eight seven. So that is correct. Now we get to the last one. The last one says the probability that z is greater than one is equals to one minus the probability that z is less than or equals to minus one. So let's check that out. In a way, we know that the probability that z is greater than one would be one minus the probability of z less than one. That is a given. We know that that would give us the same answer but let's find out if this gives us the same answer as this. So let's go find one minus the probability of a table one. So we need to go to one on the positive side one comma one comma. So the answer is zero comma eight four one three zero comma eight four one three and that is equals to seven. We need the calculator. It's going to open my calculator from here. My normal calculator. One minus point eight four one three equals zero comma one five eight seven zero comma one five eight seven. That is this site. The answer for that would have been zero comma one five eight seven. So let's find out if it's the same as this because they say it is equal. So what is the probability of z less than equals to negative one? We did find it there right? We don't have to go far which is zero comma one five eight seven and that will be equals to zero comma eight four one three which this side we said it is one zero comma one five eight seven. So both of them are not the same. So the incorrect answer is five. That's how you will answer questions in the exam or in your assignment as you receive them or find them. Are there any questions? Are there any comments? Nothing. Nothing. Next. Okay. So let's look at another. This you can do on your own. You can take a screenshot and do this. I'm going to show you one because we didn't do the between and then you can do the rest. So let's look at number three. I'm not saying it will be the incorrect one but we're going to get number three only. So here we have the probability that z lies between two values. So remember we first need to start with the probability of z less than zero. So we need to go to the table value of where it is zero comma zero zero. So it's zero comma zero zero. We know that it is zero comma five zero comma five zero zero zero minus and then we need to go to the negative side that look for negative two comma eight zero. So we go to the negative side then we look for negative two comma eight and zero at the top and that is zero comma zero zero two six zero comma zero zero two six and the answer would be and that's how you will do the probability of between point five zero zero zero minus point zero zero two six equals zero comma four nine seven four which is correct. So you can do the others. You can do number three, number four and number two on your own. I just want to get to another question so we can also do this one on your own. You can take a screenshot of it and go and play around with it because it's the same as what we just did previously. I want to do this one. The shaded area under the following standard normal is equals two. So now they have given you the shaded area then they want to know what the probability of this shaded area would be. They've given you two values which means what is that probability? Oh remember the z value is the area underneath the cap is the z value. So we know that z lies between two values zero and one comma two five. That's how you're going to interpret this. You can write this as z value lies between zero and one comma two five. We know we start first with one comma two five. So you're going to go to the table values and look for one comma on the positive side. You're going to go and look for one comma five zero and that is zero comma nine three three two. Oh it's one comma two five. One comma two five not one comma five so my bad. You also need to read the table carefully and correctly especially your z values because the minute you make a mistake you cannot get it right. So where they both meet zero comma eight nine four four zero comma eight nine four four and then we go to zero we've been dealing with zero. I can tell you just right now that it's zero comma five zero zero and use just subtract one from the other and that will give you the area in between point eight nine four four minus point five point five equals zero comma three nine four four which is option four. That's how you're going to answer questions as you get them in the exam or assignment. Okay so the other one there are so many of them. Okay so this one they are asking you to find the probability but they're giving it to you in weights. You need to be able to know how to interpret the weights into into the mathematical side. What is greater than? What is less than? What is between? You need to be able to read those and write it so you should be able to do this as well. So you need to be calculating the probability so you will have to first calculate your z score x minus the mean divided by the standard deviation and then go find the probability using this correct side. The last one that I'm going to do with you is this one which also looks as complex as the other ones. They telling you on this one that the probability of x less than a value is zero comma one five one five. For the fact that they gave you the less than therefore it means this value is the value they found on the table. So it's going to be easy to go and find the z value. So we're going to use this to find the z value. What else are we giving? They telling us that that is the area underneath the cap so therefore it means this is the probability which is the area underneath the cap which is normally distributed with the mean of 30 and here is the other catch the variance. This is sigma squared of 16 therefore you need to find the standard deviation which is the square root of 16 which is equals to 4. That's what we're going to be using to substitute. If they have given you standard deviation it would have been 5. The value taken up by this is equals to so they are asking you to calculate the value of x. That's what they are asking you. So the probability it means we need to find the z value so that we can find the x value. We are given the mean. We are given the standard deviation. So let's go find the z value that corresponds to that probability because it says 0 comma 1515. We need to go inside the table. It will be on the negative side. The other thing you need to always remember on the positive side the probabilities are big. On the negative side the probability is as polar. So when the probability is smaller in the question like 0 comma 1515 which is what we are looking for 0 comma 1515 they could have even told us that it is 15 percent or 15.15 percent so we will need to convert it. So we're going to go inside this table and look for 0 comma 1515 and there is 0 comma 1515 and we're going to go out and we find minus 1 comma 0 we need to find the last digit and the last digit we're going to find it at the top. So you're going to go to the top as well because that value is made up of two values one on the left and one at the top. So you're going to go to the top and when we get to the top the last digit is 3. So it's minus 0 comma or minus 1 comma 0 3. So our z is minus 1 comma 0 3. Our x is what we are looking for minus the mean is steady divide by 4. Now we're going to apply maths we can multiply because we want to get rid of 4 multiply the side by 4 multiply that side by 4. So we're going to say minus 1 comma 0 3 times 4 is equals to x minus 30 and we need to get the rid of minus 30 and bring it to the other side. Mathematically it says when it moves across it changes the side so we have 1 comma 0 3 times 4 plus 30 is equals to x. So our x is equals to and we can calculate that minus negative 1 comma 0 3 what is my negative multiply by 4 equals minus 4 comma 1 2 plus I must add 30 to it equals 25.8825.88 and that's and that's how you can also use the information to solve equations. I know that you were supposed to talk today and didn't talk to me I feel bad but we are at the end of the session and are there any questions? Based on what I just showed you you can also answer this question because it's almost so this one they just want you to go and find the value of set so based on the information that they have given you think about it very carefully they want the area to the right left the area to the left the area to the right think about how you're going to find that value from the area to the right because from the area to the left it's the table value from the area to the right 1 minus the table value so it means this value they found it by looking at 1 minus the value that they found on the table you will have to do the same as well so that you can get the correct set value so you will have to take this value and subtract it from 1 and then use the answer to go and find us that's all I'm going to say are there any other questions no questions so then that brings us to the end of the session likewise always we from pambili analytics where we're trying to bridge the gap in terms of numeracy and literacy and analytical skills we do this by offering range of services which you can benefit from or your company can benefit from and our flagship skills development we still have an offer which is expiring in two days um of a one-on-one session otherwise we're going back to the normal rate you can also join or watch or sign up for the self-led online trade sessions but especially on the data literacy and research in order for you to gain access to the recordings there are numerous recordings that are on there that can support your learning free of charge you can subscribe to the youtube channel and make sure that when you subscribe you like our videos share our videos and comment on our videos help us improve the content otherwise if you want the recordings of these free sessions that we are currently offering right now you will need to join the channel as a member only these three levels of membership grant you access to the recording and this is a huge saving that you will be doing other than that if you want to get hold of us we are available in all the media platforms you can send us an email or send us a whatsapp enjoy the rest of your evening see you saturday goodbye thank you lucy bye goodnight