 the previous or earlier than schedule time. So welcome to your session. I don't know, maybe it's our session, they do a 14 hour, I can't even remember now. So today we're going to do revision of study unit 6 and 7, because that's the only notes I have, and your feedback on the questions are also posted on my UNISA and your lecture site. So you just need to go through them so that you can see where you went wrong and with the explanation from the lecturer as well. So we're going to look at the assignment questions. Likewise, let's start with normal distribution. So what we need to remember with normal distribution is that a normal distribution is a belly shaped calf, with the mean which is distributed with the mean of zero, and the standard deviation of one. That is the property of a normal distribution, and we also say this is symmetrical distribution, and it's asymptotic because with the calf it will never, so your calf or the area underneath the calf will never touch the x-axis which is that line, never touches the x-axis. What we also need to remember is that the area underneath the calf for a normal distribution, the area under the calf is what we call probabilities. It's the same as your probability and it is equals to one. So everything underneath the calf is equals to one, and if we split this probability into half, therefore this side will be 50% of the area and this side will also be 50% of the area. That is normal distribution. But now, when we calculate the, or when we want to find the normal distribution probability, it means we have some units that we want to standardize. And standardizing those units, we use the formula z divided by, which is the z score or the z value formula, which is z minus x minus the mean divided by the standard deviation. And this will standardize your x units into a normal distribution with the mean of zero and the standard deviation of one. And once you have your z value, you should be able to calculate or find which probability underneath the calf you are looking for. Now, when you look for the probability underneath the calf, what you need to remember is we use the cumulative standardized normal distribution table, which is this table that has the positive end, the negative values. It's called the cumulative standardized normal distribution with the z scores and the probabilities inside the table. So the z scores are those values on the left and at the top and there's values inside the tables. We call them the probabilities. So now what you also need to remember with normal distribution is the table that I just showed you contains the probability of z less than a value. So if you calculate your z value and you're looking for the probability of a z less than a value that you just calculated, then the probability you will find on the table. If you're looking for the probability of z greater than a, therefore you will say one minus the value you find on the table. For all the probability of a greater than, we're going to use the one minus the value we see on the table. Whether you go to the positive or the negative side of the table. If you're looking for the probability of z lying between two values a and b, then we're going to find the probability on the table for the second value, which is z less than b minus the probability of the table value for z less than a. So you're going to find the table value for b minus the table value for b. And that is how you will find the probabilities. Sometimes you can be given the probabilities and you need to work back and calculate your x unit. So if you're given the probability, you also need to take into consideration how that probability was calculated. Was it with the greater than or with the less than so that you can know which values of your z you're going to be using from the table? Whether that probability they gave you, they found it by using one minus so that you can take one minus the probability and go find the value on the table. Or if it was the value that the probability that they found for the less than, then you go to the table, you look for the probability within the table and you go outside. So you just need to know that it is very important to know the sign that you are given. The other thing, you might be asked question to calculate the total number of something. So you must just always remember that the sum of all probabilities are equals to one. That is the standard thing because we said also the area underneath the kef. So it means all the probabilities underneath the kef should give us one. And one we say it's the same as hundred percent. So if they give you a number or a value and they say the way 2000 people and then 95% of them received 80% of the max. So you just need to make sure that you know that if you're only looking at those who receive more than 95%, therefore it means those 2000 are the proportion of the people who received the mark. But they are selected from the whole population which is the total, which made up the hundred percent. So you just need to make sure that you know how to use your proportions or your percentages to calculate an X value amount. So you can calculate your X value in different ways by being given the proportion and go and find the X value. So if you go to find the X value you just need to manipulate your equation because at the moment, if your Z formula, sorry, let me put it back. If I'm given my Z value of X minus the mean divided by the standard deviation and I need to find X. So I just need to make X the subject of the formula. Therefore it means it will be given by Z. Sigma times Z and plus the mean and we'll find our X value by that. By just simplifying the formula in that manner. So that is if I'm looking for the X unit but also you need to know what probability was that so that you can go find the Z value. And by that it means you take your probability from the table you go out to look for the Z value that corresponds with that probability and then substitute it into the formula. Okay, with that said, let's begin the questions relating to normal distribution. So the first question is asking which one of the following statement is incorrect with regards to the normal distribution probability. So I'm going to assume that this one question is asking about the properties of a normal distribution. So we need to know our properties of our normal distribution. So we're looking for the incorrect question. So we can read each statement and check whether that statement is correct or incorrect. The Z score of the mean of a normal distribution is one. Is that correct or incorrect? True, that's correct. The Z score of the mean of the normal distribution is equals to one. Is that correct or incorrect? That says the Z score is equals to one. So the Z score cannot be equals to one. So this is the incorrect one but we can read through the rest of the other questions because this says the Z score is equals to one but the Z score can be any other value because the Z score can take any of the form of the Z value that we have. Only the mean is equals to one. That's what I was thinking about. Yeah, but yeah, they're talking about the Z score. The smaller the value of a standard deviation the lower the steeper the curve or that's the other thing that I didn't discuss but that is correct because the smaller the standard deviation the narrower the you'll careful be because if your standard deviation, let's look at this. If your standard deviation, your standard deviation, yeah, let's say this is where your mean is. If this is a normal distribution, we know that our standard deviation is equals to one. So for a normal distribution. So if my curve now looks like this so you can see that that standard deviation there reduced to zero comma, I'm gonna make it zero comma eight. And if I draw another one, as you can see that that is closer to the mean and that I can say it's zero comma two. So the smaller the standard deviation the narrower your curve will be as well. And so that will be correct. The mean of a normal distribution can be any value that is negative zero or positive. So we know that it can take any value on this line for your mean of your normal distribution. That is true because the mean can be minus one. One day or two day. The area to the right of your standard normal distribution of zero comma five and the area to the right, oh sorry, to the right the area to the left will be zero comma five. And we just say it in our explanation to say if we split it into half this area will be 50% to the left and to the right it will be 50% which is what we just covered when we did the explanation. A 95% of the value of a normal distribution are two standard deviation away from the mean. Oh, that's the other thing that I needed to discuss as well. So you need to also know your standard deviation your prop the properties of your standard deviations in relation to the normal distribution because for a one standard deviation two standard deviation and three standard deviation what do they refer to in terms of the percentage? So one is 89%, 84%, oh sorry, 84%, 95% and so forth in terms of the standard deviation away from the means. And that is also correct because 95.4% is two standard deviation away from the mean the only incorrect answer here is option one. Any question, any comments because I didn't give you the time of day to answer the question as well. You will get time. I will give you a chance to answer the question. And this time I think I'm going to just let you answer the questions. Consider the standard normal distribution for Z which one of the following probabilities is incorrect? So now they say you need to go and check whether these probabilities are correct. So it means we need to go to the table and remember the following that the probability of Z less than A is the value you find on the table and the probability of Z greater than A is one minus the table value and the probability of Z lying between A and B is given by the probability of Z less than B minus the probability of Z less than A. So now if you know all these things then you should be able to answer the question. So let's see which one is incorrect. So you need to go and find the probability of Z greater than minus 2.8. So it means you go to the Z table. You go look for minus two. So remember also with probabilities you can leave your values to two decimals. So way there is no another decimal way it's one decimal. Let's put it this way. When it's one decimal then we can just leave it as zero. The last decimal will be zero. So this will be minus 2.80. So we need to check if this value is the same as finding the value of one minus the probability of Z greater than or equals to 2.80. So first let's go find what this probability is. So that is minus 2.8. And then we look for zero at the top and that is zero comma zero zero two six. So I'm just gonna show you only one. So this is zero comma zero zero two six. We just need to check if it's the same as this. So this one says this side should be one minus. We need to go find this probability. How do we find the probability of a greater than by finding one minus the value we find on the table? So it means yeah, we need to say one minus. So even though there is a minus there but we need to validate this statement. So it's going to be one minus the value we're going to find on the table because of this sign of a greater than. Therefore we need to say one minus the value we find on the table. And on the table, we need to go to the positive side of 2.8. So we go to the positive side and look for 2.8. 2.80 which will be the value at the top 2.80. So that is the first column. So that is zero comma nine nine seven four. So that will be zero comma nine nine seven four. And this is equals two. So we just need to check if this side are the same minus one minus zero comma nine. What do you get when you calculate that? One minus zero comma nine nine seven four equals zero. Zero comma zero zero two six. And one minus zero comma zero zero two six gives you. Zero comma nine nine seven four. Zero comma nine nine seven four. zero comma nine nine seven four so this site it will give us zero comma nine nine seven four so unless this was a mistake or an error on the question that they gave you um no on the other side isn't it supposed to be one minus that's zero comma zero zero two six one minus zero zero two six oh yes because this site is also greater than yes you are right my bed today so yeah also it should be one minus i forgot to look at the sign on this side as well because it's greater than so also this site is zero comma nine nine seven four thank you for picking that one up so because this site there is a sign of greater than or equal so this site and that site are equal so therefore this question is correct that is the correct one so let's go to b b says z of less than minus 2.1 so it means you must go to the z table and go look for because the sign says less than or equal so when the sign is less than or equal the value we find on the table is the value we looking for so what is minus 2.1 we need to go to the negative side minus 2.1 which is zero comma zero one seven nine zero comma zero one seven nine equal and this site says we need to go find the probability of z greater than 2.1 what must we do here it's going to be one minus the probability of 2.1 so we must go to the positive side and go look for 2.1 and the value is zero comma nine eight two one zero comma nine eight two one so what is the answer there zero comma zero one seven nine zero comma zero one seven nine so they are also equal so therefore b is also correct now we're looking for the probability of between and if we're looking for the probability of between therefore it means we need to go and find the probability that z is less than zero comma zero zero minus the probability that z is less than minus 2.28 so going to the positive side and look for zero comma zero zero is zero comma zero zero it's zero comma five thousand that is zero comma five zero zero zero minus the probability of z less than minus 2.8 we'll go find it on the negative side of the table two comma eight it's zero comma zero two six zero comma zero zero two six and what is the answer that we're getting it's zero comma four nine seven four zero comma four nine seven four which is the same as what we are looking for they it's the same so that is correct so d go find that probability let me give you a chance to also answer at least one are we done okay so what is the probability of z less than two point one is zero comma zero zero two six no for two point one because we first need to do the probability one is zero comma nine eight two one sorry minus the probability of z less than minus two point eight so here we find zero comma nine eight two one minus zero comma zero zero two six and that is zero comma nine seven nine five nine five which means that is also correct so moving to the last one the probability of between two point one and zero that is the probability of z less than two two point one minus the probability of z less than zero point zero what is the probability of less than two point one zero comma nine eight two one zero comma nine eight two one and what is the probability of zero zero zero comma five zero zero zero comma five zero zero and that should give us zero comma four eight two one zero comma four eight two one and this one says so number e is the incorrect one and that's how you will answer the question so you'll have to go and validate each one of them so that you find the correct one given the z is a standard normal distribution what is the value of z such that the area to the right the area to the right of z is zero comma two zero six one choose the correct answer so yeah they say given that the z value is normally distributed what is the value of our z to give us the probability of zero comma two to give us the probability of zero comma two zero six one and they say the value to the right so the value to the right we must always remember that the value to the right meaning on that side what is the z value that's what we're looking for to give us this zero comma two zero six one one now you must know that the value to the right it means it's greater than so therefore it means the probability that they are looking for the value to the right which is our a which is the value we're looking for should be two comma zero six one so how would they would have found the value to the right they would have said one minus the value they found on the table remember this value is given by one minus the value they found on the table to give them zero comma two zero six one because the sign says to the right which means greater than so what is that value that they would have found on the table to give them that so it means we need to find the value on the table by saying if we move table value that site one minus zero comma two zero six one is equals to the table value so that we can know what probability we are looking for on the table value so our table value will be equals to what is one minus two zero zero six one it's it's a zero comma zero comma seven nine three nine seven nine three nine three nine so we need to go to the probability table and go find zero comma seven nine thirty nine so zero comma seven nine thirty nine so we also need to because we're looking for the z value so that z value is zero comma eight and two so it's zero comma eight two so that will be the probability do you understand how to get there so yes you need to make sure that you read the question and understand the question that they are asking you because the question yeah they said if we're looking for the area to the right therefore it means we're looking for the greater than we're looking for the greater than value and they say that probability which is this area to the right is zero comma two zero six one so it means in order for them to find that probability they would have said one minus the value they found on the table they would have gotten the answer of zero comma two zero six one because the the probability to the right is one minus the value on the table and that's what we've learned the probability to the right is one minus the table value and we go and find the table value and the table value is zero comma seven thirty nine then you go and look for the z value that corresponds to that probability and you will find that it is zero comma eight two so b is the correct answer okay moving on to the next one consider a normal random variable with the mean of three hundred and the standard deviation of one thousand three hundred calculate the probability that a random variable is at most so you also still need to remember what the signs mean at most three thousand eight hundred and choose the correct answer from the option so we're looking for the correct answer from the option so because they say at most and we need to calculate the probability we need to be calculating the probability that x what is at most at most is less than remember at least it's greater than at most is less than three thousand eight hundred so it means you need to go and find the probability that z is less than x minus the mean divided by the standard deviation so what is our mean i mean our standard deviation sigma our x value substitute into the formula and calculate your z value and then let me know what the probability is so i can just sum up substitute the value on your behalf our x since i've already identified them is three thousand eight hundred minus our mean of three thousand divide by our standard deviation of one thousand eight hundred close bracket getting a yeah same six thousand seven hundred so what is z less than what do we get what is our z less than zero zero point four four four four four okay we keep two decimals we can leave it as zero comma four four then we need to go to the table and look for zero comma four four let's remove all this ink zero comma four is zero comma six seven zero zero which is option A upon conducting a research the education department found that learners travel time from home to school at one of the remote rural is normally distributed with the mean of one one four minutes and the standard deviation of 72 minutes what is the probability that the learner travel time from home is less than hundred minutes so also they want you to find the probability that x is less than hundred i'll give you some time to do the calculations remember you will need to go use the z formula to go find your your z very blue first so identify what your mean is your standard deviation is and your x is substituted into the formula and calculate t so let's see our x is hundred minus our mean one one four our standard deviation 72 72 and our z value is negative zero comma one nine four one nine one nine so we need to go to the negative side table and look for zero comma one zero comma one nine i think is the last column she is zero comma four two four seven four two four seven let's go back quickly let's see okay so the next one consider again the findings from the department that the learner travel time from home to school at one of the rural areas school is normally distributed with the mean of 14 and the standard deviation of 72 and education consultant has recommended no more than no no more than a certain minutes of learner travel time if the department like to ensure that 15 15 percent of the learners at years to the recommendation what is the recommended time travel so they're here they're asking you to cut to find x what is x that's what they are asking you to find so the first thing that you need to understand is what is no more than is it less than no more than is less than or equal so what they're asking you there is no more than is less than or equal so they say what is the probability what is the probability of x i'm making x a small x x not equal but less than or equals to x of what is 15.15 in decimal so take 15.15 divide by 100 15.15 divide by 100 is equals to zero comma 15.15 so what they are saying is if this is the probability what is the probability that x will be equals to this this probability what is the value of our x if our probability of traveling x kilometers is zero comma 15.15 what is that x kilometers that the learner should or be recommended for that's what we're looking for so we need to work in reverse so it means we need to go and find because at the end of the day we need to go find the probability because in a way if I put it this is the same as the probability that z less than a which is the a is the value that we are looking for which is the z value that we are looking for here is equals to zero comma 15.15 and if a is the value that we're looking for we can actually move from the value on the table so you need to go to the table because this is the less than we need to go to the table and look for this z for this z value so we need to go look for zero comma one five one five zero comma one five zero comma one four and have you found it it's minus one point yeah three one point zero three that and you go up is three one minus zero comma three so we know that this is z of minus one comma zero three is equals to zero comma one five one five so now since we have the z value we need to work it back so we can remove the probabilities because we no longer interested in the probability we can work back and go find our x value that we are looking for minus the mean divided by the standard deviation our z value it's minus one comma zero three our x that's what we're still looking for minus the mean is one one four divide by the standard deviation 72 and you can just simplify so that will be minus one comma zero three times 72 is equals to x minus one one four and if I move one one four to the other side so I can just since I'm going to run out of space I can also move one one four this side it will be plus one one four is equals to x and if you look at let me just run down run up again to what we did yeah so we said if we want to find x we can just multiply by z and move the mean onto the other side so it will be z times sigma plus the mean which is equals to x and that's what we are doing on that question on that question so solve the equation minus one comma zero three times 72 plus zero comma four what do you get 39.84 39.84 and because it's in kilometers and it's rounded off in two minutes we can just round it off to 38 is it 39 39.84 39.84 and you can round it off to 40 minutes and see will be so the recommended travel time is 40 minutes okay now the next one the emotional intelligent quotient or EQ score of high school learners is normally distributed with the mean of 80 and the standard deviation standard deviation of 20 if there were 2969 learners with the score of higher than so yeah it's another thing it says higher than 95 how many students took the test so the first thing that we need to do is we need to go find the probability of those who took the test and because it's greater than so we need to go find the probability that z is greater than our x minus the mean divided by the standard deviation it's greater than so it means the answer we will get we'll have to subtract it from one so z greater than our x is 95 minus the mean is 80 the standard deviation is 20 z greater than what is our z value 95 minus 80 is 15 divided by 20 what do you get 0.75 0.75 0.75 so what we need to do is we need to go and find one minus 0.75 on the table 0.75 we'll go to the positive side and look for 0.775 and that is the probability which is 0.7734 0.7734 which means what is the probability of those who took the who's got higher than 95 1 minus 7734 0.7734 0.002 0.002 that's what you're giving me okay so now we know the proportion of those but they say if there were 200 2969 of those who scored that how many took the test we know those who took the test there are many ways you can calculate this but let's let's first use the proportions that we are given so we know I'm going to show you one way of doing it we know that 2969 is equivalent to 0.2266 percent if 100 percent which is one I'm going to call it 100 percent if 100 percent of those who took a test because this is 22 percent remember that is 22 22.23 percent 23 percent of those who would who got an IQ of 90 higher than 95 100 percent of them is equivalent to one or I could say it's 100 percent of them they are equivalent to one you can say it that way if I want to know what is my x or in this instance I can use my n let's use n here let's use n and I'm not going to use 100 percent I'm going to use one because I'm working with decimals 100 percent is the same as one one is the same as 100 percent so if I have 100 percent what will be my n so my n is equivalent to 100 percent of those who did the score they took the EQ test so in order for me to find the n I need to cross multiply so I'll multiply so find n so if I cross multiply n will multiply there and I will have n and one will multiply with two nine six six nine multiply by one and I'm going to divide by because I'm cross multiplying I'm moving things around I'll divide by zero comma two two six six what do you get two nine six nine times one divide by zero comma two two six what do you get which it should be two nine six nine divide by zero comma two two six six it's 13102 it's 13102 that is one way of doing it the other way of doing it you could you could find the other part that is that did not score 95 percent so if you don't do it this way if you don't do it this way you can because I know that I have which will still be the same so you have your two nine six nine which are equivalent to zero comma two two six six so one minus zero comma two two six six two two six six will give me the others that did not take the test isn't it this will give me my x of those who did not take the test so if I want to get those who did not get get 95 higher than 95 so that will give me one minus zero comma that is we know that this is the same as zero comma seven seven 39 so that we know that because 34 34 that is that until we got so I'm going to cross multiply this way with two nine six nine multiply by zero comma seven seven three four divide by divide by zero comma two two six six and that will give me my x and that will give me x will be equals two so remember I'm not using n because my n is my grand total so I'm not using n so I'm using my x so that will give me two nine six nine multiply by point seven seven 34 equals divide by point two two six six equals and that gives me one oh hundred and 33 so in order to find n I must take x plus my two nine six nine so I know what my x is my x is ten one three three plus two six two nine six nine and that will give me plus two nine six nine one three one oh two so the first method is the shorter version so you can use this method for a shorter version or you can go the long route find use the the one that you feel comfortable with they both they both will give you the same answer and our answer is number c and I think that concludes our normal distribution so we're moving into the sampling distribution which is study unit seven so with sampling distribution what do you need to remember with sampling distribution it means you have different samples whereas with normal distribution we're only working with one population yeah we're working with multiple samples so the properties of a sampling distribution is that you creating different samples and use you you you some out of those sampling the samples you create you can calculate your mean and then you can also standardize the mean of those samples so that it takes a normal distribution calf because usually the data will take a form of a uniform distribution what you also need to know about the sampling distribution is that the population of a sampling distribution mean is the same as your population you also need to know that your sampling distribution standard deviation which is the standard let's call this the mean so that you don't get confused the mean and yeah we have this sampling distribution of the population so the standard deviation of the sampling distribution mean is given by your population standard deviation divided by the square root of n which is called the standard deviation of sampling distribution of means because there are different means that you select or we call this what we call the standard standard error mean one and the same thing the mean of the sampling distribution of the means is the same as the population mean but the standard deviation of the sampling distribution of the means is equivalent to the standard deviation or the population standard deviation divided by the square root of n which is also called the standard error in order for us to standardize the means and also to go find the probability we use the z score which is your sample mean minus the population mean divided by the standard error which is the sampling distribution standard deviation we can call it the sample mean minus the mean population standard deviation the know how to write the standard error divide by the square root of n and that's what you need to know about sampling distribution the probabilities we're still going to find it the same way the probability of a less than a the value we find on the table we're still going to follow the same the probability of z greater than a still going to be one minus the value we find on the table the probability of z lying between two values a and b we're still going to find the table value for b minus the table value for a we're still going to follow the same the same process okay so now let's get down to it a random sample of 120 is drawn from a normal distribution population with the mean of 160 and the standard deviation of 50 determine the standard error and remember the standard error is your population standard deviation divided by the square root of n that is the mean that is your standard deviation and you are also given the sample size are we winning so that will be 50 divided by the square root of n which is 120 4.56 4 comma 5 6 which is option b consider a normal distribution population with the mean of 190 and the standard deviation of 120 with the sample size of 50 a sample a sample size of 50 is drawn from this population what is the probability that the sample mean is between 150 and 190 so we need to find the probability that the sample mean lies between 150 and 190 so what you can do is calculate the probability that z lies between our sample mean minus the population divided by the standard error and you do the same on the other side so substituting the value our sample mean are always in the question so we start with the 151 our population mean they have given it to you it's 190 and your standard deviation is 120 and your n is 50 so we just substitute 190 divided by 120 divided by the square root of 50 less than z less than 190 minus 190 divide by 120 divided by the square root of 50 the first one do you have an answer for 150 minus 190 minus 2.36 2 decimals minus 2.36 36 it's less than z less than on the other side 190 minus 190 is 0 so therefore the site will be 0 comma 00 so we need to follow the same process you need to go and find the probability that z is less than 0 comma 00 minus the probability that z is less than minus 2.36 you need to go to the table I know this on the table is 0 comma 500 0 comma 500 minus we go to the negative side minus 2.36 go to the negative we look for minus 2.3 and 6 0 comma 009 91 0 comma 0091 where is the answer 0 comma 4909 0 comma 4909 hc okay the emotional or the eq score of grade 8 class is normally distributed with the mean of 80 and the standard deviation of 20 a random sample of the six and of the six grade eight lenas are selected let x be the eq score of the grade eight class it is further known that the probability that the mean of eq is more than 0.8849 determine the value of x such that the sample mean is equals to 0.884 the sample mean is greater than x of and that probability is 0.8849 choose the correct option so similar to what we did earlier so we know that how did they find this probability the probability of z greater than a value the probability of z greater than a value they would have found it minus by minus the value on the table which gave them 0.8849 so now we need to find this value on the table so therefore the value on the table will be the table value will be given by 1 minus 0.8849 because if I move table value this side and I move this one this side is the same it will be positive and that one will be negative it will be 1 minus 0.89 so what is our table value 1 minus 0.8849 so it means we need to go to the table to go find our a so we need to go find a 0 comma 0 comma 1 1 5 1 so inside the table we need to look for 0 comma 1 1 5 1 1 1 5 1 day so it is minus 1.20 so our a which is the probability that z is less than minus inside minus 1.2 1.20 was given by 0 comma 8849 so now since we have our z value we just need to substitute into the formula so that we can go find our mean divided by our standard error which is the population standard deviation divided by n what is our z value our z value is 110 minus 120 so it's 1 minus 1.20 is equals to the mean is what we looking for our population mean then gave us is 80 and our standard deviation is 20 so 280 divided by 20 divided by the square root of our n the inside is 36 divided by the square root of 36 so cross multiplying 20 divided by 6 I could just say I'm just gonna write an error that goes there so we can rewrite this as minus 1 comma 2 0 times 20 divided by the square root of 36 plus because I need to move 80 to the other side plus 80 and I'm left with my x body so what is the answer 76 so it means our answer we saying it is g most school blah blah blah blah in a sample of 90 school from sikukuni district 72 schools reported a decline in the number of absenteeism or absentee glenus calculate the standard error of the proportion of schools that reported a decline oh that's the other thing that I didn't talk about so when we do sampling distribution also we can do sampling distribution for the proportion and let's do the summary on the site and then we can answer on the so sampling distribution for proportions so when we do sampling distribution for proportion we are going to be given the sample proportion which is P if they are not giving us the sample proportion P therefore we will be given the observation that satisfy that proportion and we can calculate it so this is our sample proportion the population proportion will be given by population proportion it's represented by the pi side and if we go into standardized the the proportions then our formula we use is P minus the population proportion divided by the standard error which then our standard error is our population proportion one minus the population proportion divided by n so our standard error or standard deviation of the population proportion will just be all that underneath the square root under your underneath the fraction so everything underneath the square root so that population proportion one minus population proportion divided by n that is what we call this is what we call standard error or sampling distribution of the standard deviation of the sampling distribution of the proportion it's called the standard error and that is all what you need to remember with this so now they're asking us to calculate the standard error so we just need to calculate the population proportion one minus population proportion divided by n so now the challenge with this is they gave us the sample proportion because they didn't give us the population value so we can calculate it we can also use the sample proportion to calculate so what are we given we are told what the number of samples ah i'm gonna assume that that 72 is 72 percent of the schools because proportions are decimals or percentages so i'm going to assume that that is a typing error in terms of the proportion they forgot to put the percentage they should have put there a percent sign because the ASA 72 reported a decline maybe they should have said 72 percent of the school reported a decline and if that is the case then our proportion will be 0 comma 75 72 0 comma 72 so we can then just substitute onto the formula because there is no other information provided on this so this will be 0 comma 72 times 1 minus 0 comma 72 divided by our n is 90 what do you get when you calculate you get zero point zero four seven just double check double check oh it's my calculator as well i got the same value you got the same weight as well i i i i got zero comma zero four two what i did is i calculated the sample proportion i said um 76 divided by 90 then it gave me 0 comma eight then when i ah yes you're right then yes i just wanted to say if you do the the x divided by n you also get 0.8 yes so that is the sample proportion because i assumed that they didn't give us the population proportion so we we'll have to use the sample proportion standard error for this one because this is the x value so if that is the case thank you for that if that is the case then we need to calculate our sample proportion which is our x divided by n first which should be which that should be i'm just clearing out all these things that we put in which should be 72 divided by 90 because our 90 is n and what is our sample proportion 0.8 0.8 so then it's 0.8 times one minus 0.8 which gives you 0.042 which is 0.042 which is option e but lizzie i would have picked that one anyway with a previous answer so we're still being right it was the closest but yeah if that is if you had two options here one was 0.47 then you would have chosen the wrong one yeah true yeah thank you for picking that one up because i thought that is an error because i was looking at the formula and not applying my mind that is why when your mind is tired anything is possible you just find shortcuts so yeah okay so that is are we not finished oh gosh okay so the next last probably this is the last one consider population proportion yes there we go population proportion of 0.66 and the sample proportion of 72 are given with a sample of 99 calculate the value of the test statistic and choose the correct option so yeah they just want you to calculate z now i'm using the wrong formula which they say calculate p minus the population proportion divided by population proportion 1 minus population proportion divided by n that is the formula so don't get confused with the this one because on this one since they didn't give us the population values we use the sample proportion but the formula for z we always use your standard error is calculated using your population proportions so let's calculate the z test statistic substituting the values into the formula sample proportion is 0.75 minus population proportion is 0.66 divide by the square root of our population proportion of 0.66 times 1 minus 0.66 divide by n our n is 99 i got 1.89 a yeah 1.8 1.89 which is a a previous study has shown that 71 percent of schools in sikukini district municipality have reported the decline in learner absence since the start of learner transport and the school nutrition program by the education department suppose a sample of 99 sikukini district municipality is drawn at random what is the probability that at least which is greater than or equal now 80 percent of the schools have been reported so what they're asking you is find the probability that the sample proportion is greater than or equals to 80 percent which is 0 comma 8 0 let me not write it as 80 percent and write it as 0 comma 80 therefore you need to find the probability that z it's greater than the sample proportion times the population proportion divide by the standard error which is population proportion 1 minus population proportion divide by n so our population proportion which is our pi is 71 which is 0 comma 71 our n is 99 so that will be z of greater than 0 comma 8 0 minus 0 comma 71 divide by the square root of 0 comma 71 times 1 minus 0 comma 71 divide by 99 of greater than do we have the answer it's easy 0 comma 244 what do you get 0 comma 0 244 0 comma 0 244 oh no sorry uh 0 comma 9756 are you giving me the probability already or not the z value yeah i'm looking for the z value okay of 80 0 comma 80 minus 0 comma then divide by the standard error what do you get 1 comma 97 yes it's correct and then we need you now my equal my mind is upside down now my equal is 11 sorry i want to write the equals 1 minus the value we're going to find on the table one from 1 comma 97 go to the table on the positive side of the table we look for 1 comma 9 and go look for 7 at the top 1 comma 97 which is 0 comma 9 7 5 6 0 comma 0 comma 9 7 0 comma 9 7 5 6 and that probability is equals to 0.0204 this was the last one oh gosh finally i apologize for today but we are at the end of the road thank you any question we are done with revision of study unit six and seven do you have any questions thank you have any other uncertainty or anything no questions okay if there are no questions i will see you then on saturday when we do study unit eight and nine which is confidence intervals and hypothesis testing a revision we almost done with revisions if there are no questions then thank you for coming and have a lovely evening bye good night bye thank you bye thank you