 Okay welcome to your 12th session of the online tutorial session and today we're going to continue and do activities based on study unit 6 which is normal distribution. Do you have any comments, question, query, anything you want to share? Anything? Except for the flaves I saw on the assignment too. This thing is a lot of paper I have to catch up and try it again. I'm sure I'm going to send you more questions after that. Okay so remember you can always re-watch the recordings and go through the activities and it always will work best if you practice and do a lot of activities before you attempt to do your assignment questions as well because then you would have practiced and practiced and then when you go and look at your assignment question you might recall some of the things that you have learned while you are practicing. So like sessions like today, the Saturday sessions where we go through a lot of activities, I also make sure that I have additional activities that I know that we're not going to cover in class but they will give you an opportunity to practice. So you need to go through them. If we don't finish everything in one class you just need to continue do that. Let's have the discussions during the week while you are still practicing and not wait until you have to submit the assignment as well. So please remember this is state is like math. You have to always constantly practice and practice and they say also practice makes perfect. So the more you practice the more you're going to get good at it. Any other comments or questions? If they are done so let's quickly recap on what we did on Wednesdays because then everything we've learned on Wednesday is going to form a base for all the work that we're going to be doing today in class as well. So let's refresh our mind. We discussed the basic concepts when it comes to normal distribution. We looked at how we compute the probabilities and remember we looked at the three ways that we can compute probabilities for normal distribution. We said we can find the probability of the standardized normal distribution for a less than a value or we can find the probability for a greater than a value or we can find the probability where the standardized normal distribution lies between two values and we learned that if it's for less than the value on the table it's that probability that we're looking for because the table which also shows the positive and the negative contains all the less than probabilities. And if we find the probability of a greater than then we need to say one minus the value we find on the table and if it's the probability of the between we've learned that we take the probability of the second part minus the probability of the second of the first part of the value. So if for example that one was the standardized normal distribution lies between a and b so we take we first go find the value on the table for z less than b then we go find the value of z less than a. We've learned the table we know that the table contains the probability inside and the z values at the at on the outer side whether at the top or on the left hand side and we've learned how to use that probability table we're going to do that as well today. So we learned that a continuous probability distribution comes from a continuous variable and and we know that a continuous variable is a variable that it's been measured it's like your height your temperature time taken things like that. We also learned that with a normal distribution the curve of a normal distribution takes the shape of a b shape curve and the normal distribution is a symmetrical graph which has the mean the median and the mode which are equal and what we have learned also is that with normal distribution your mean will give you the the location of your your your data around the mean and the spread will tell you or the standard deviation will give you the spread of your data across the mean as well and we also learned that with a normal distribution the random variable has an infinite range which starts from positive infinity to negative infinity and we know that it will never touch the horizontal line so it will never the the normal distribution graph will will never ever touch the x axis and what we also learned is that the values underneath the graph that we see and we call that the area underneath the curve it is where we're going to be calculating our probability and that will be the probability we call it the area underneath the curve what we also learned was that when we shift the value of our mean the graph will move from left to right and if we change by increasing or decreasing the value of your standard deviation your graph will be flat or it will be taller or we say it will be narrower or taller or it will be flat or shorter things like that so by increasing the value of your standard deviation then it means your graph will be so when your when your your standard deviation increase your graph will be flat will be flatter because the standard deviation will be the values will be far away from the mean and if we'll say it's short flatter or shorter and if we if we decrease the value of your standard deviation then we say the graph will be narrower because then it will come closer to the the mean value or we can say it will be tall we say it's narrower or it's taller and what we also learned is because we're working with x units we want to standardize them in order for us to find the probability then we use what we call the z standardized normal distribution which is called the z score or the z distribution or the z value and we know that this z distribution is distributed or the normal distribution or the z distribution which is one and the same thing it's normally distributed with the mean of zero and the standard deviation of one and if we don't have the standard deviation they would have given you the variance and we know that the variance is the square root of the standard deviation so if the standard deviation is one therefore it means the variance is also equals to one and we use this formula which we call the z value or the z score to standardize our values or the values of x which we calculated by say your x value minus your population parameter mean divided by your population standard deviation standard deviation and that will give you your standardized normal distribution and then we use this standardized normal distribution to go and find the probability on the table and we looked at then an example on how we apply the formula to calculate the standardized normal distribution and we said if we're looking for the probability of a less than depending on where the value is so if it's the value beyond zero so like a so we're looking for the probability that z is less than a therefore it means it's everything underneath below the curve where it is red shaded and when we use the table that will be the value we find on the table we also learned that the table has the negative side and the positive side of the table and all these tables contains the values of z less than a so it means all these probabilities that we find on this table are the probabilities of z less than a value of a and the value of a will be the z score and we learned that always when we calculate the value of z we need to leave our value into two decimals because the first two decimals will be on your left and the last decimal which is six will be at the top as part of the last decimal or digit on your table on your left so we've learned how to use this table and we're going to be using it today also to answer a lot of questions we also learned how to find the probability of a z less than so if we have the question state text what we are given we given the mean we given the standard deviation and we are asked to find the probability of x less than 18.6 what we do we need to standardize that value of x by applying the formula and substituting the values that we are given and we found that the z value is 0 comma 1 2 and in terms of the probability on the table we're going to take our z value of 0 comma 1 2 we go to the left and look for 0 comma 1 and we go to the top and look for 0 comma 002 and where they both meet that is the probability that we are looking for and that's how we use the table we also learned about the probability of a greater than a value and we said for a probability that z is greater than a we're going to say one minus the value we find on the table because we know on the table we always have the values of a less than so we know since the sum of all probabilities are equals to one so then it means the probability of z greater than of z greater than a is a complement of the probability of z less than a and we know that the less than a is the value we can find on the table okay we looked at an example to say if we are given the mean of 18 and the standard deviation of 5 and we need to find the probability that x is greater than 18 we've done the previous exercise and we calculated the z value and we knew that our z value was 0 comma 1 therefore to find the complement of 0 comma or z greater than 0 comma 1 we say one minus the probability of z less than 0 comma 1 2 because we know that this 0 comma 1 2 we can find it on the table so we say one minus 0 comma 5478 give us the probability of 0 comma 4522 and that is the probability of between so sorry the probability of greater than so then we also looked at the probability of between and we said if z lies between two values then we're going to take the value that we find on the table for the z less than b and subtract so we'll say the probability of z less than b minus the probability of z less than a because we know that z less than a is the value that we will find on the table and z less than b is the value we find on the table and we're going to subtract one from the other and that will give us that probability of between looked at an example where we were given the mean the standard deviation and we were told that we need to find the probability that x lies between 18 and 18.6 we first calculated the z value for 18 and we calculated the z value for 18.6 and we found that if the answer for z was zero and the answer for z for 18.6 was 0.12 and we're going to say the probability of z less than 0.12 we go find the value on the table minus the probability of z less than 0 comma 00 on the table and that's what we do we go to the table we go look for the probability of 0 comma 0 0 comma 1 2 and we find that it is 0 comma 0 5 4 7 8 and go find the probability of 00 we go find 00 and we find that it is 0 comma 0 5 5 5 0 0 then we subtract one from the other and the probability of between which is only the shaded area is just 0 comma 0 4 7 8 and that is the probability of between not only that we learned how to find the probability of between the probability of greater than and the probability of less than we also learned to work backwards by working backwards we learned that sometimes they might ask you to find the value of x from you can use the formula of z x sorry z x minus the mean divide by the standard deviation if we are asked to find the value of x and therefore it means they would have given us the value of z or the value of standard deviation or the value of the mean if they didn't give us that then they would have given us the probability and they would have told us what is the sign whether it is a greater than or whether it's a less than or whether it is between so that we can find the value of x by working backwards and we looked at one of the example where we were given the probability that your x is less than when your x is less than 20 percent which is 0 comma 0 to 0 and that probability was oh sorry that probably the x where it's less than I'm going to say that is find the x that such that so we know what the we don't know what the value of x is so we we need to know what what will be that probability where x is less than the value it will give us 0 comma 0 to percent so then it means since we know that this 20 percent that we have found it by finding the z of less than a value which we need to go find on the table which is 0 comma 0 comma 0 to 0 0 so we need to go and find this probability on the table and since we know that we're working with the actual table because we're looking for the probability of a less than we go to the table we find the value of z and we replace it by that value that we find on the table so if the 0 comma 0 0 2 we go outside the table and we find the value of z there and the value of z there and both of them we combine and we find that both of them so we go out and go out so we find that it is mine it's located that minus 0 comma 8 4 and that will be our z value so therefore it means our probability of z would have been minus 0 comma 8 4 and because we know what the formula of z is we know that that formula is x minus the mean divided by the standard deviation we can just make this equals to 0 comma 8 4 because that is our z value so we know that it's 0 comma 8 4 we can just replace it and then solve for x and that's what we have done we solve for x and we found that it is 0 3 comma 8 0 and with all that that we have learned we can now do some more activities in order for us to understand all this so our first activity I'm not going to go into the summary our first exercise is just theory remember also you need to know your theory in order for you to be able to answer all the questions in your exam or in your assignment as well not only going to do calculations but you also need to know theory so which one of the following statement is incorrect I expect everyone now to be unmute because here we can discuss each statement one by one and you just respond because I don't have to allow you to go and and do some calculations so number one says we're looking for the incorrect statement the standard normal table or z table is used when the random variable has a normal distribution of the mean of 0 and the standard deviation of 1 is that true or false true that is true that is correct the area under a standard okay I'll come back to that one let's go to number three a normal distribution kef never touches the horizontal axis true true that is correct a normal distribution has two parameters namely the mean and the variance false why why is it false it is two parameters the mean and the standard it has the mean and the standard deviation so where do you find the standard deviation from the variance from the variance so it also can can still be one of those because the variance is the square root so your standard deviation is the square root of your variance and your variance is the square of your standard deviation so this statement is true if they would have said it contains the median that then that would have made it incorrect okay and number five the shape and the position of a normal distribution depends on the mean and the variance is that correct it's true that is true as long as the mean we know that it makes your graph move from left to right depending on where your sorry my my mean was not supposed to touch the the y the x axis that is the mean it will shift from left all right and your standard deviation which we will calculate from the variance will make your slight circle higher or more narrower yes so that your your standard deviation makes your your graphs to go up and down in terms of the in terms of the shape so the mean moves it in terms of the position but the standard deviation or the variance will move it in terms of the shape which now leaves us with only number two so number two says the area under the normal standard deviation for z less than a zero is equals to one so if you are not sure about this question you can also go to your table i need to go to the z table you can go to the z table and look at how the z table looks so remember let's go back here it says where z is less than zero net so therefore we expect z to be less than zero comma zero zero so you go to the table and you look for z so this is in the negative so we must go to the positive side so let's go too much we go to the positive side and we go look for zero comma zero zero zero comma zero zero the probability or the area underneath that kef should give us 50 percent or zero comma zero zero five so if we go back to our question it says is equals to one therefore that is there incorrect answer and that's how you can answer questions like that otherwise if you don't want to use the actual table you can also look at the graph if you look at it here is your zero so we know that the area underneath the kef the area underneath the kef should always be equals to one because the sum of all probabilities will always be equals to one but if it's half where zero is and it's less than so we're only talking about 50 percent of the whole area we're not talking about the entire area so therefore it means this will be zero comma five or in terms of probabilities in four decimal it will be just zero comma five zero zero because they're not standardized normal distribution or cumulative standardized normal distribution is the cumulative probabilities okay so that's how you will answer questions moving forward the distribution for which the mean is equals to the median is equals to the mode is a which one is the answer symmetrical symmetrical distribution symmetrical distribution that will be the answer because a normal distribution has the mean the median and the mode and it is a symmetric a symmetrical distribution let's please I assume also the distribution can be skewed since only symmetrical applies yes so a a multimodal distribution will look like like that it will have two modes a skewed distribution will look like that because then it's skewed to the left on this manner for this one a uniform distribution will be that will be a uniform distribution we let we will learn about uniform distribution when we do the sampling distribution because a sampling distribution we convert this uniform distribution to becoming a symmetrical distribution which is a normal I'm not drawing it correctly but you get the gist of it a normal distribution so we can convert this to that we can also convert that to that we can also convert it to that and that's what we do in statistics when we want to do some analysis and infer back the results we make sure that all our numerical values we confirm them and they become normally distributed it makes it easier for interpretation and also for application papers okay but that is for those who are interested in being in the same environment as me and being statisticians okay so now let's learn how to do the probabilities for a random variables that from the standard normal distribution with the mean of zero and the standard deviation of one that just describes the normal distribution which one of the following is incorrect so I'm going to give you five minutes to go through each one of this and then post in the chat your option which one is the correct option but you need to go and and and work it out so we can do one together now and then you can do the rest on your own so the first one which says the probability of z less than minus 1.52 so we need to go to the z table and go to the negative side of the z table remember it says it's negative 1.6 so we don't have to calculate the z values here we are given the z so we just use the table we look for 1.5 it is right here at the bottom and we look for two it's in this column sorry two is in this column and that probability is zero comma zero six four three and going back to our question so I'm hoping you all have your your your tables with you so that you can work through all the responses and we are looking for the incorrect one so already now with my help this is correct so check if any one of them are incorrect you have five minutes and I will set the timer please post in the chat also uh when you get to the option mesle yes by any chance would you know what page is the table on the study guide in your normal distribution chapter it should be in there somewhere 121 there's 120 thank you there you go times up must be I think it's false you find your credits correctly so you say you calculated it then you found that it is five let's see what others are saying I have one person saying it's four four is the incorrect one so let's see if four is the incorrect one so we're gonna go to the table I'm going to make my screen I'm sorry go out of the presentation mode and I'm gonna make it smaller so that we can have the table as well on the view okay I'm gonna make it bigger is it visible can you see it yes okay so let's we know that number one was correct let me activate my pen number one was correct so let's look at number two number two says z less than 1.4 so we need to go to the positive side of the table scroll too much so we're looking for one 1.48 so we go to 1.4 on the side and we need to go to the last yeah it is right which is 0.396 which is correct then the next one says we need to find the probability of greater than minus 0.43 so here we first need to do one minus the value we'll find on the table there so we go to the negative side since we're looking for the negative side and we're looking for 0.4 which is this and if I count this is zero one two three we should be there so that column 0.43 so the three is in the last in that column so 0.43 is zero so we need to say zero comma three three three six so we need to subtract one minus zero comma three three six and that should give us zero comma six six four six correct right um let's see so I just did it the other way around please tell me if it was right so instead of going to negative having a bigger than negative number I just converted to 0.43 as a positive number and then I used the positive size of the side of the table so instead of going minus zero three three three which is the value from the negative side I just left the Z as sort of I just inverted the rank so I made a negative 0.43 as positive yes so you you will find that it don't be correct yeah it is correct that's why or is it incorrect that's why I'm doing it no it's not incorrect so you um if you go to the positive side of this table remember here is negative so if you go to the positive side of this remember the sum of all probabilities equals one yeah so in the positive side 0.43 will be bigger than the negative side and the sum of both of those will give you uh will give you the they will give you one so but if you come here and you say subtract then it will also give you the same as in the positive side so you can use both ways you can either come to the negative and then do the minus or you can go to the positive and the value you find on the positive will be the value you are looking for so it's a step shorter basically it's a shorter one yes yeah okay cool thank you so as long as your answer will be 0.66 664 yeah 664 so number four it says the value is greater than 0.74 so now it means we need to go to the positive side because it's greater than we can also say one minus the value we'll find on the positive let's start with that one the normal way the value we find on the positive was it's 0.7 so that is 0.7 on the first and the last digit is four so we need to go to four which is 0.7704 so that will be one minus 0.7704 which then gives us one minus 0.7704 which gives us 0.22 0.2296 2296 2296 therefore this is the incorrect answer but let's go to the next one oh let's let's do your method so because this is positive so we need to go to the negative side of the table to find 0.74 so when we go to the negative side 0.7 and we go to four so we know four is on that column so I can also count from here 0 1 2 3 4 and that will give us the same answer as when we do one minus 0.774 because we came to the negative so the inverse of that yeah the alternative of that so when we look at this two the last one we need to go find the probability of z less than 2.34 on the table and we need to go find the probability of z less than minus 2.70 on the table so the first one we go to the to the positive side we look for 2.70 which is 0.9965 minus and we go to the negative side to look for oh sorry I went to the wrong one that's the wrong one we're looking for 2.36 2.3 0 1 2 3 4 5 6 which is yes I think it's 2.34 not 36 34 yes 34 0 1 2 3 4 it's 0 comma 9904 so we go 0 comma 9904 minus we need to go to the negative side look for minus 2.70 negative side let's remove all this previous answers we got it's 0.0034 2 2.7 yeah and 2.0034 0.0035 so when you do the calculation will be 0.9904 minus 0.0035 give 0.9869 which is correct so the only incorrect one is option 4 yes okay so let's move to the next question sorry it works best when you work on power point slight okay so say I'm gonna do number one so since remember we're working with cumulative probability so if we're looking for the probability of an exact you are going to get the probability of an exact will be equals to zero so for this one it is correct we're looking for the incorrect ones just to make sure that we have the right question you're looking for the incorrect statement it's much better remember to post on the chat when you have the option so you have let me just not give you five four minutes let me give you three minutes on this one are we done okay yes yeah okay what will be the probability of z greater than zero 0.5 that will be 0.5 anyway I don't even have to go to the table to show you because the probability of z equals to zero on the positive side is just 0.5 therefore it means the probability of z will also be of z greater than 0.5 it's 1 minus 0.5 which is the same as 0.5 the probability of z less than 0 0.5 0.5 also I don't have to explain that number four the probability of z less than minus one negative minus one so we need to go to the negative side and go look for minus so this will be minus 1.00 so we'll need to look for 1.0 on the left and zero at the top so that will be 0.1587 correct yes next one correct then it means the probability of a of an exit cannot be zero I thought probability of z is equal to zero is 0.5 correct me if I'm wrong no it's not equals to zero it's not equals to 0.5 remember the table that you are looking at it looks at the probability of exact wait the probability of an exact is equals to zero yes that is correct the table that we're looking at right now looks at the cumulative probabilities so this does not have the probability of an exact value and as you can see that the probability of a negative one is almost like equals to zero it's zero comma zero zero zero zero zero zero so the probability of an exact as well will be equals to zero because you're looking for the probability at that point let's look at number five number five says the probability of z greater than one so we need to go to the probability of z one in the positive is zero comma that is zero comma eight four one three but we need to subtract one from that because this is the probability of a less than and that will give us one minus point eight four one three gives us zero comma one five eight seven eight seven now we need to go find the probability of one minus the probability of z less than minus one it's also number for the schedule so z minus one we did find it it was zero comma one five eight seven minus one so this question says the probability this site should be equals to the probability that site and we know this probability this site will be equals to zero comma eight four one three so these two are not equal so this is the incorrect one okay i've seen my mistake i used to put one instead of negative one yeah if they didn't put if they didn't put this value one minus the probability of a less than equals to that then it would have been correct or if this was positive then it would have been correct but because this site it's a negative one and it's one minus the probability of a negative one and mix this one is the incorrect question so you always need to know that for a standardized normal distribution the probability of an equal will be equals to zero the probability of a greater than we use the cumulative probabilities okay next see this question says if the z i must take it back to bigger screen now so that we can see if the z score is given by z of less uh z equal z standardized normal distribution z value of minus 1.96 the distribution of x is normally distributed with the mean of 40 the standard deviation of five then what is the value of z all you need to do is use your z equals your x minus the mean formula to find the value of x so that's what we need calculate the value of x you are given the z value you are given the mean you are given the standard deviation and remember to post your answer on the chat and if you agree with the option are we winning okay we have two different answers three so i don't know how did you get to that answer but anyway let's substitute and calculate what is the value of z 1.96 it's minus 1.96 the value of x is what we're looking for the mean minus 40 is 40 the standard deviation is five we can multiply minus 1.96 multiply by five the site and we will be left with minus x minus 40 on the other side and your minus 1.96 multiply by five gives us minus 9.8 minus 9.8 and we take 40 the site it becomes plus 40 x will be equals to 9.8 plus 40 equals to 30.2 which makes option one the correct one happy yes yes good next the owner of an appliance store uses a normal distribution with the mean of 10 this variant which is sigma squared of nine to model the weekly net sales calculate the probability that x is less than 3.5 what is the standard deviation the standard deviation is the square root of the variant which is equals to three and we need to calculate the probability that x is less than 3.5 so you need to find the probability that z because we're doing standardized substitutive values i'm just writing the formula for you you just continue are we done or you want a minute talk to me done okay so let's do this talk to me z less than what is my x 3.5 3.5 minus 10 divide by 9 i mean divide by 3 divide by 3 and that will give us the probability that z is less than 3.5 minus 10 gives you minus 6.5 divide by 3 the probability that z is less than what is 6.5 divide by 3 gives you 2.1666 the round is minus negative minus minus 2 2.1666 round it to 2 decimals 2 and 7 remember to always round it to 2 decimals so that you can use it to go find the probability on the table so let's go find the probability on the table we're looking for 2.1 2.1 and 7 and that is 7 there so the probability is 0 comma 0 1 5 0 which is option 5 let's see we've got it right people and whoever like their answer happiness yep mark shows a normal distribution with the mean of 56 and the standard deviation of 4 what is the probability that a random chosen student fails the module what are we given here we given the mean we given the standard deviation right what else are we given there is something that we are given told that we need to find the probability of x or not yes what will be the probability that a chosen student will fail the module so what do they mean if they fail a module then is it a greater than or is it a less than or is it an equal it's a less than and when it's a less than since we're working with percentages of pass so what is the percentage of a pass 50 so the student should get less than less than 0 comma 5 0 comma 5 so that's what they want so you need to calculate the probability of a student getting less than 0 comma 5 and that will be the probability that z is less than x minus the mean divided by the standard deviation okay are we still busy slowly posted her option and there's one like others i'm confused what are you confused about i'll need to take you back again why do we standardize why do we find z so that we can find the probability now when i'm looking at our probability that we have the it's already 0.5 no that is not the probability that is a pass mark when we supposed to say x is less than 50 no yeah yes because 50 is your pass mark it's not your probability it doesn't tell you the probability of student failing it gives you the pass mark it's like so that is that is the amount let's say let's say we're not working with percentages here we're talking about uh the number the actual number so let's say it's 56 it's 4% and then it's 50 okay so use x is 50 yes okay i use 0.5 yes it's 0.5 because it's 50 50 percent a pass mark is 50 percent which is 0.5 but that is 0.5 for my x now it's the one that confuses me no you need to also to convert your mean because your mean is 56 percent what is 56 percent is 0.56 what is 4 percent 0.04 yes so you need to convert it to decimals okay are we done our z our x is 0.5 minus our mean of 0.56 standard deviation of 0.04 1.4 i mean minus 1.5 z minus 1.50 if we go to the table is it is it your sorry point my bet so you said it 0.05 your 0.5 minus 0.56 is 0.06 divide by 0.04 gives you minus 1.50 right so we go to the table yes and what do we get 0 0.0 and 0.0668 which is option number the probability that z is greater than minus 1.96 is denoted by the probability of z greater than minus 1. and it is equal so you need to go find the probability so what do we need to do here z greater than minus 1.96 what do we need to do one minus the probability that z is less than 1.9 negative 1.96 yes so go find the probability of z less than 1.96 which is equal to 0.0 to 0.5 and get the answer what is the answer 0.975 which is option number two the average high school teacher annual salary is 43 000 let the teacher salary be normally distributed with the standard deviation of 18 000 calculate the probability that x is greater than 80000 so you need to find the probability that x is greater than 80000 which is 1 minus the probability that z is less than x minus the mean divided by the standard deviation I just made it easier for you that's the alternative way of doing it because we know eventually we will find the probability of 1 minus that because of the greater than I'm gonna give you some few minutes two minutes okay are we done so 1 minus the probability that z is less than our x is 80000 minus our mean which is 43 000 and our standard deviation of 18 000 and that will be 1 minus the probability that z is less than 1 did you calculate it what is 80000 minus 43? 37 000 37 000 divided by 18 000 it's 2.0545 so you run it up z of less than 2.0545 055 which means it will be 06 yeah okay since we're looking for the probability and not the z value because you were gonna choose the z value you're not gonna get it right so we need to go to the table and go look for the z of less than 2 comma 0 so we go to the table and it's in the positive side so it's positive 2.06 so we go to the positive side 2.06 I'm still on the right lane 0 comma 9803 isn't it that's very easy to do there is no complex questions other than everything that we have done so let's just continue for the next 30 minutes to see how many questions we can cover there are still plenty more the stats final mark is normally distributed with the mean of 56 yes can you kindly go back to the previous question did you once you've got the z value did you subtract the minus 1 1 you are right and everybody else is quiet no we didn't we didn't we're trying to finish the calculation yes so we we would have also gotten it wrong there you are right yo we nearly made a very big mistake every big one so yes you need to subtract it from one I don't know why I forgot because I always carry the one 1 minus 0.9803 the answer should be 0 comma 0197 which is option 5 so we would have gotten it all wrong in the exam that cost 4 max so I was wrong yeah I actually got on the right but I think that's because I swapped the the the x and the mean I swapped around when I did the calculation so I got the answer of 0.0197 so I don't know yeah so in the exam who would have gotten minus 4 that's a huge that's huge that's huge okay thank you for that let's move on to the next question what is the final mark obtained by a student if the z score value is equal to minus 0 comma 5 so they want to know what is the x value so you need to use the formula like we said with the other one x minus the mean divide by the standard deviation find x in short you can write the formula multiplies you can say x is your z multiplied by standard deviation plus the mean so we just multiply z we take the mean over and then you just substitute into that formula the answer you you just need to multiply by a hundred when you're done just multiply by a hundred to get it to a percentage multiply by a hundred and that will give you the answer you are looking for are we winning slowly and two other people says it's option five others agree let's see if we agree with them our z is minus 0.5 times our standard deviation is 4 percent which is 0 comma 0 4 plus our mean of 56 percent which is 0 comma 5 6 everything we need to multiply by a hundred so x everything 0 minus 0 comma 5 negative 0.5 times 0.04 gives us minus 0.02 at that 2.56 we get 0 0.54 multiplied by a hundred gives us x of 54 percent 54 percent which is option two hey you have five minutes to answer this one let me set you at loggy so that we don't go over but actually let me give you four minutes no it's on this as well okay the time is on okay we have an answer I'm not gonna go through each one of them let's see which one unless if we have multiple answers so are you still busy or you yes we're still busy are you still busy okay so let's look at the at the answers and see if we can assess those who are still lost number one says we need to find the probability of z greater than 2.64 so it's in the positive side and we know that we need to say one minus the value we find on the table so 2.64 is that value so we'll have to say 0 comma 9 9 5 9 1 minus 0 comma 5 9 9 and that is 0 comma 0 0 4 1 that is correct and we're looking for the incorrect value if I'm not mistaken yes we're looking for the incorrect value so the next one says we need to find the probability of z less than negative 0 comma 8 7 so we need to go to the negative side table we're looking for 0 comma 8 which is just here at the bottom and it's 8 7 so we need to count 0 1 2 3 4 5 6 7 it should be here in this column if I go up I should be in 7 yes it's in this column so 0 comma 8 7 it's 0 comma 1 9 9 2 since it's the less than that is the correct and happiness the next one we need to go find the probability of 1 comma 4 6 so we need to go to the positive side and find 1 comma 4 0 which is 0 comma 9 1 9 2 and we need to subtract the value we find in 1 comma 4 minus 1 comma 4 minus 1 comma 4 is 0 comma 0 8 0 8 and when we subtract 0.9192 subtract 0.0808 gives us 0 comma 8384 which means that is correct I'm going to jump the first the fourth one because I see yeah they said they agree with the fourth one so let's go to number 5 number 5 says z of greater than 0 comma 7 4 this applies to the same thing that we did earlier so we need to look for 0 comma 7 4 you find that it is 0 comma so we need to say 1 minus 0 comma 7 7 0 4 because this is greater than we need to say 1 minus that so when we calculate 1 minus 0 comma 7 4 we need to find we will find it's minus 0.7704 is 0.2296 0.2296 and it should be the same as the z of less than 0 comma 7 4 so we go to the negative side and look for 0 comma 7 for 0 comma negative 0 comma 7 4 0 1 2 3 4 and we find that it is 0 comma 2296 so therefore this is correct which leaves us with only number 4 we need to go to 2.8 go to 2.8 2.80 which will be that and that is the incorrect one so that is the value that we are looking for any questions we have 10 minutes if there are no questions we are alone guys are you here can you hear me yes we are here for a fact there I thought I am here alone I'm talking to myself okay area to the right of a positive z1 is 0 comma 8996 then the value of z is if the area to the right so they say let's say this is the area they say if this shaded area is 0 comma 0869 what is this value what is that z z1 value there so they have given you the probability they have told you where to look for it and they tell you it's positive to the right is a positive so when they say to the right therefore they are looking for the greater than so if they are looking for the greater than because they say to the right which is the greater than side then it means this probability when we found it we found the probability of 1 minus 0 comma 0869 so we need to go find the actual probability and that actual probability is 1 minus 0.0869 which is 0.9 0.9131 so we need to go to the table and go find this probability on the table 0.9131 so we go to the table we look for inside this table we need to look for this 0.9 so it will be in the negative in the positive side because they also told us that it's in the positive so we go look for the positive side of the table and then we go look for 9 and the 1.336 1.36 and that's how you do it so you need to read the question carefully so the hint that I used was that they gave us they said the the area we know that the area they refer to the probabilities underneath the calf the area is the probability so that is what one thing probability the other thing they told us is that it is to the right so therefore it means is to that side which is to the right and when it is to the right then it means it is a greater than to the left it would have been it is done so to the right is a greater than so that's one thing that you need to remember that and since they gave us this probability which is the area which is the same as that they told us it's 0.8 they wanted us to find the value of z which is at this point which then gives us this cumulative probabilities going there but the other hint that they tell us is that this value you will find it on the negative side of the table if they would have said negative then we would have went to the negative because then the probability here would have been in the negative side of the table but they said it's in the positive so it's in the right greater than and it's in the positive area so we'll find it in the positive z value table and we know that if it's greater than the value we find that 0.0869 we would have found it by saying 1 minus 0.9131 so we would have taken this 0.9131 subtracted from 1 and it would have given us that value any question sorry must bring one question that 0.91 can you just show me again on a table please okay so you come inside the table you look for 0.9 on the positive side table you go through you scan through these values like 0.99 0.98 and you go and until you find the value 0.913 which is there because we're looking for the z value we need to read the z values out which is that value 1.3 and the last digit always at the top 1.36 and that will give us 1.36 and that's how we find that value so you need to actually go and check for 0.91 and anything day after yes okay thank you I am going to stop right here for today's session every other question that is coming up after this is your homework please go through that practice with it and yeah and let's have a discussion on WhatsApp yeah because I think it's always going to be a two-way let's have a two-way communication on WhatsApp because you post nobody and the response to those questions and I just give you feedback but at least it will help you to to show you where you're going wrong when you're working out the solutions as well so I'm going to stop right here so from question 13 you can answer it so I'll just give you hints on how to answer some of these questions so question 13 is asking you to find the shaded area so this is the probability of between so we're going to assume that this is the z normal distribution values already calculated so they're giving you 0 and 1.25 so you need to take those two values and find the probability that z lies between 0 and 1.25 and that will give you an answer from one of those that's what you need to do because they say find the area under the curve and they give you the two z values this one straightforward as well they give you the z value they're asking you to find the probability go to the table you know you will say one minus the probability of z less than minus 1.44 on the table this one is a little bit more complex but it's not as complex as you think it's actually straightforward they are saying to you that the probability of b and x greater than b is that here they also give you the probability of a less than so we know that if we need to find this probability so therefore the probability of x less than less than b would have been one minus zero comma zero one seven so we would have found this probability the only thing that we need now is this probability so we know that the probability of a less than x less than b is equals to the probability of x less than b minus the probability of x less than a so that is why i'm saying it is as straightforward as it is it is if you don't see it i don't know what else can i explain it or what other hint i can give you you just substitute the values and then solve for the probability of a because that's what they're looking for they want you to find the probability they want you to find the value of a so now want to have found the probability of a because you already found the probability of a in on the table oh not on the table using the probability that you will find here after you solve this then you will go to the table and go find the z value and then calculate x minus the mean divided by the standard deviation and find x because our x in this instance is our a that's all what you need to do first step oh actually this was supposed to be your first step your second step is supposed to find the probability of a the third step is to go to the table and find your z value your fourth is to find a using this formula that will give you the value of a just a small confusion on the first step i have an impression that the probability of x greater than e is supposed to be one minus that failure am i wrong but that is what i have said remember you are looking for the probability of z less than so remember the probability of okay so that's why you're saying your confusion is a so we know that the probability of z greater than b is one minus the probability of x less than b which is the same as the probability of x less than b is one minus the probability of x greater than b these are both complements that is a complement of this and this is the complement of that you can work it out let me blank screen wait let me blank screen the probability of x greater than b is equals to the probability of oh sorry it's equals to one minus one minus the probability of x less than b let's take this value minus minus p to the other side probability of x greater than b plus because it's minus it will be plus x less than b is equals to one now let's take the greater than to the other side we are left with the probability of x less than b is equals to one minus the probability of x greater than b can you see that they are complements one and the same thing okay i live to watch it over and over which makes sense i'll ask what's up if i'm not coming right okay here i took a shortcut because i made my assumptions that by now you know how to move from here between the two between the less than and the greater than they would mean one and the same since we are given the probability of a greater than then it's easy to find the probability of a less than because it's the probability of x less than b it will be one minus the probability of a greater than because it's a complement and that is why i said that will be your first step so once you have that probability you have this probability of between because they've given it to you there it is number two they have given you that you would have calculated that so that let's change color i can't change the color so this one where i am making a big dot it is this the answer that you will get there it will be that and then you just solve for this right once you have solved for the probability of less than a you would have found that that probability let's say is 0 comma 5 8 4 7 i'm just making an assumption you're going to take this value because it's a less than already so you're gonna go to the table you're just gonna come to the table and look for 0 comma 5 8 0 comma 5 am i not fine 5 8 because i just make it up 5 8 4 7 5 8 almost let's say is that one 5 8 3 3 2 let's assume that we pick that one as our probability then you're going to say that is 0 0.2 and what else and one so that is your z value that is what you do in step number three you find that z value then you take this formula which is z is equals to x minus a because we know that we're looking for a as our value we're not going to use z x we can use a in this instance so a minus the mean we have the z we calculate we found it by doing all this step number one step number two step number three we found z we're looking for a we found mean they've given you the mean they've given you the variance we just do the standard deviation which is the square root of your variance you substitute you find a that's one two three four steps you will find the answer okay that is my hint okay so then you can also answer this is just a normal theory question that you need to go through you can answer question 17 18 19 20 wait let's go back to this one as well so on this one instead of them putting 52 and 66 into this question they made them as a and b as that so you just need to work them out in here replace b with 66 replace a with 52 also you have it's the same thing that we have been doing all along remember the probability of equal but we're looking for the incorrect one and I've already gave you the answer okay let me just move so this one is easy as well you just need to calculate the z score which means you use the formula z is equals to x minus the mean divide by the standard deviation this is a normal theory question and this you just need to calculate your x which means you use z is equals to x minus the mean divide by by the standard deviation as well on this one they are looking for this is a point decimal sorry they are looking for x so you just use z is equals to x minus the mean divide by the standard deviation that's what they are looking for the x value the last one you just use oh it's not the last one you just also find the probabilities here like we've been doing what is the probability that the z score below 1.5 you can read through the whole sentence but it's not going to be helpful the question is just what is the probability of a z score below so know what below means is it a greater than or is it a less than you must know what they mean by below and then go to the table and find the probability that's this is the simplest one question and that concludes today's session and any question any comment any anything now you have a flow in the absence of any question any comment and release you go have fun enjoy your weekend but remember your assignment 2 and assignment 3 you still need to do them and those who haven't submitted their assignment 1 you still have very few days to submit your assignment 1 and assignment 2 which is the adieu on the 7th and otherwise happy learning happy weekend I will see you on Wednesday when we do sampling distribution on Wednesday as well we're going to only have one hour 30 minutes or we might spend the whole two hours I'm not sure we will see how the session goes but what we will do on Wednesday will be almost similar to what we have learned that is why the session might not be as long as the other ones because we're just going to repeat the same concept so have a lovely weekend I will see you on Wednesday bye thank you thank you bye bye everyone