 Welcome to your ninth session. Please also remember to complete the register. I've posted the link in the chat. We can start with today's session. Today we're doing sampling distribution. And the following two weeks, I am thinking and contemplating because we are way ahead of almost where you need to be in terms of all your modules, like 1501, 1610 and 1501. I'm thinking maybe we need to take a break. If it's possible to cancel those two last session and we will resume again in July. Why I'm saying that it's no good use if you are still busy with your assignments and your assignments are not even yet at this point where we are discussing because then by the time you start doing the work related to this two session confidence interval and hypothesis testing. It will be sometime some way in August. So, if I'm, but you can post in the chat or we can discuss when we get to the question and answer session just now. Let me know what your feeling is so that we don't start on those two sessions. We wave at them until July and then we can start to properly in July. With confidence interval and hypothesis testing. Okay. So today we're going to deal with sampling distribution. Are there any questions or query or comment before we start with the session for today? Based on what I just said, do you guys feel you want to be here the next two weeks or can we wave at them and start in July the sessions? Are you good? Are you happy with the suggestion? Thank you. I will inform you, Nisa, as such. Okay, so I'm going to share my entire screen for these people because we're going to be using the table. I hope you have your tables with you. So for today's session, you meet your statistical tables. We're still going to rely on the formulas to calculate and we need to calculate. So by the end of the session today, you will learn how to compute probabilities from a sampling distribution and going to look at how to find the probability of Z less than Z greater than a value or Z lies between two values. We're going to also calculate the mean and the standard deviation of a sampling distribution, which is called also the standard error. But also we're going to rely mostly on the Z table that we used the last time. Remember when we were doing the normal distribution, the Z table that we used for the normal distribution, we're going to use it today because sampling distribution as well. We are trying to normalize a uniform distribution and making it a standardized normal distribution type data. So what are the things that you need to know since I told you? With sampling distribution, we're working with multiple samples. With normal distribution, we're working with only one population. Yeah, we're working with one population, but multiple samples taken from that population. And once we have those multiple samples, we calculate the average of each of those samples and then we calculate the average of all the samples. The sample means and that will give us the sampling distribution mean, which it is denoted by mean of X bar. Mean subscript X bar will just be the same as your mean of your population. They are the most likely to be equal. The standard deviation, however, on the other side, the standard deviation of a sampling distribution will be calculated by using the population standard deviation divide by the square root of your sample size. And this is also referred to not only as the standard deviation of a sampling distribution of mean, but it is also referred to as the standard error. So if they ask you calculate the standard error, they're asking you to find the standard deviation of the sampling distribution. Later on, we will introduce for the proportion. So yeah, we're talking about the standard deviation of the sampling distribution of the means. Let's get an example of how we will calculate them. Suppose a population has the mean of eight and the standard deviation of population standard deviation of three. Suppose that a random sample size n of 36 is selected. Calculate the mean of the sampling distribution. So what they're asking you to find is the mean of the sampling distribution and we know that the mean of the sampling distribution is the same as the population mean. So therefore, our mean of a sampling distribution will be equals to eight. Question number two asks, calculate the standard deviation of a sampling distribution. We know that the standard deviation of a sampling distribution is given by the population standard deviation divided by the square root of your n, which is also referred to as you remember. This is so referred to as the standard, the standard error. Our population standard deviation is three, our n square root of 36, which will be three divided by what is the square root of 36 is six, which is one over two, or we can say 0.5. And that's how you will find the mean and the standard deviation of a sampling distribution of means. And how will I identify that this is what I need to be calculating in this, the standard distribution of means, it's because they will give you the mean and the standard deviation. And you will see when we talk about the proportion, you will get the proportion or you will get the values that associate with the sample. Okay, you are also expected to know how to find the probability of a sampling distribution. So to find the probability of a sampling distribution of means, we use the z score. Remember, in normal distribution, our z score was your x minus the mean divided by the standard deviation. This is to find the probability we use z score to find the probability of a normal distribution, probability distribution. To find the probability of a sampling distribution, we use the z score, but the formula then changes to because now we have the means of the standard error and the population means. So the formula is the sample means minus the population of sample means divided by the standard deviations of sample means. Which is the standard deviation of the sampling distribution of the mean and the mean of a sampling distribution of the means. Which is also given by the sample mean minus the mean population mean divided by the standard error, which is the standard deviation divided by the square root of n. So you just need to remember this formula. Suppose a population has the mean of 8 and the standard deviation of 3. Suppose a random sample size of n is selected. What is the probability that the sample mean is between 7.8 and 8.5? You can calculate the two z values or you can say is they are asking us to find the probability that the sample mean lies between. I'm just going to put the between 7.8 and 8.2. Therefore they're asking us to find the probability. I'm going to write the formulas of the sample means minus the population sample means. But now because we know that it's the same as the population means. The standard error, which is the standard deviation divided by the square root of n. And this is our z lies between the two values. And I'm going to write the formula again on this side for the same thing. And I'm going to substitute now going back already our values are highlighted. So it makes it easier. So to find the probability that z lies between the first one we're going to calculate for the 78 7.8. So 7.8 minus our population mean is 8. Divide by the standard deviation. She's 3 over the square root of 36. And therefore it means I must expand my bracket. And the same 8.2 minus the mean. She's 8 divide by standard deviation of 3 divide by the square root of 36. Close the bracket. And that's then you just calculate the values. I've got a solution already. Then we know that from the left hand side 7.8 minus 8 divide by 3 divide by the square root of 36 will give us 0 minus 0 comma. 4 8.2 minus 8 divide by 3 divide by the square root of 36 gives us 0.0.4. This is between we know with the between the probability of z lies between the value of A and B. We know that we're going to find it by using the probability of z less than B minus the probability of z less than A. Because those are the values we can find on the probability table. Then we go and find this value on the table and subtract this value from the table. So let's go to the table and go and see if we can find minus 0.4 and 0.4. It's the same the same thing. Z cumulative standardized normal distribution table which has the positive Z values and the negative Z values. Now we're looking for minus 0.4. So it means we need to go down to the minus 0.4 and minus 0.4 Z. And we need to go to the top because we need to just add another zero there. Remember we need to keep two decimals. So our next column is just the first column and the answer we get is 0.03446. And we go to the positive side and we go find 0.4 and 0 will give us 0.6554. Moving back, we know that from the positive side we got 0.6554 from the negative side we got 0.3446. And we subtract one from the other and we get the probability of between 7.8 and 8.2 to be 0.3108. So we took our population distribution, we did a sampling distribution of it to convert or standardize it into a normally distributed population. And that is how you will answer the question as well when you get them from your assignment or your exam. So you can be asked as well to find the probability of a less than. You know that when we find the probability of Z less than a value, that will be the value we find on the table. If we find the probability of Z greater than a value, remember it's one minus. The probability of Z less than a value that will be on the table. Okay, so let's do an example. Yes. How are you? I'm good, thanks Justice. The table I see next to me here is the standard normal distance. Can I look on it or they are the names for because they are different tables. No, the table is called standardized normal distribution. And the normal distribution. Yeah, so you must look for a table named cumulative standardized normal distribution table. On on in your module, you have two tables. So you just need to find the one that looks like this that will also it's a rating. It represents cumulative standardized normal distribution values. So it should have those kind of information. But your table needs to have negative. Distributions and positive distribution. If it's a rating standard standard, if it's written like this without, without the other weights, it's not the one that you need to be using. You need to look for the exact table. Are you using your standard, your, your study guide. Using your study guide, you need to go to sampling distributions. And in there, probably the tables are somewhere in between. The section where it's, it deals with sampling distribution. If you're giving 15 or 15 or one, please help justice where type in the justice can't see the chart. But let us know if you're using a study guide. What page number. And if you're not using a study guide, where can you find the table. Okay, so let's do an exercise. Suppose Africa check conducted further research using a sample size of 100. The number of times an AI algorithm is successful at detecting fake news is normally distributed with a different mean of 900 and the same standard deviation of 100. Ex be the number of times the algorithm is successful at detecting fake news. What is it that we are given and what is it that they're asking us to do. So the question is, find the probability of X less than 920. They have given us the sampling, the sample size, which is n of 100. They have given us the mean, which is 900. They have given us the standard deviation, which is 100 and they gave us the sample mean, which is 900. So let's go and find the probability. The probability that z lies between X bar minus mu divided by sigma divided by the square root of n. Substitute and calculate. I'm going to give you some few minutes. Those with no tables do the calculations. When you are done, ask me to get to the table and tell me whether you want to be on the positive side or the negative side so that I can. I can navigate the screen for you. Let's see if I can also get a calculator. Are we done? Let's see before I go to the table. P z of less than our sample mean is 920 minus our population mean is 900 divided by standard deviation is 100 divided by. The square root of 100. What do you get? What is the answer that you get? I'm going to get the calculator I'm going to use. We have 920. I don't know why is my calculator acting up minus 900. I go to the bottom and again another fraction. It's 100 square root of 100. And the answer we get is 2. So it's 2,000. It's positive. 2,000. Good. Justice. They say on your study guide you must go to page screen acting up. Maybe I need to restart mine. They say you must go to page 770 to 771. That's where you will find the table. 77. 770 and 771. So we're going to look at the table. It's positive. So we go to the positive side. We're looking for 2.0 and 0 at the top. 2.0 and 0 at the top. And the answer is 0.9772. And that's how you will find the probability. Any questions? If there are no questions, then now I will give you time to do some exercises later. Now let's look at this. Can you please just elaborate how you come up on 2.00? Oh, we calculated that. I put it down. It's my calculator not visible. I use this. So it's 920 minus 900 divided by 100 divided by the square root of 100, which gives me the answer of 2. And because I know on the table we need to have 2 decimals. So 2 is the same as 2.00. And that's how I got to those 2.00. Okay, for some of us we are using financial calculator. I don't think I can. Okay, so if you're using a financial calculator, you can do the same because you just need to do it manually. So you will say 920 minus 900, right? And you will find the answer, which is 20. And you just go into write your answer and say your PZ less than 20 divided by, and then you go to the bottom and calculate what is at the bottom. And then you go and say 100 divided by the square root. So on your financial calculator, you will press the square root. Where is your square root on the financial calculator? It is on button number three. So you will press shift, second function. You will do second function and press button number three. So on my site, I have already the square root and you will press 100. Once you've done that and then you press equal and it will give you 10. So it will be 20 divided by 10. And we know what 20 divided by 10 is. It is the same as this. 20 divided by 10 will be Z of less than. You just say 20 divided by 10 equals 2. And you say, because I'm going to the table, I will need to find 00. Okay. Okay, thanks. Now let's look at the population proportion. So with population proportion, it can be denoted by the pie. Don't get confused with what we already did. Somewhere where we were talking about the pie being the probability of success and probability of failure. So here we're talking about something. So the population proportion and from now on going forward, you always need to remember that your population proportion, those who are doing 15 or 1, it will be different. So this is for those who are doing 15, 10, 16, 10. So the population proportion is denoted by pie. 15 or 1, your population proportion is denoted by a capital letter P. Going forward, when I talk about pie, you need to remember that it's a capital letter P. So we will use the population proportion to calculate the mean of the population, which is the population proportion in a way. The sample proportion, which sometimes they will not give it to you as the sample proportion, but they will give you observation satisfying the sample proportion. You need to be able to calculate it using the observation satisfying sample proportion divided by N. Those who are doing 15 or 1, yours will be P guppy, P with a cap on top. 15 or 1 and 16, 10, sorry, 16, 10 and 15, 10, yours will be P. 15 or 1, it's P with a cap, it's P guppy. So if you're not given the sample proportion, you must know that you will need to calculate it by using X divided by N, which is something that we have been doing from basic probability, from discrete probabilities, from bilomial probability and from here. And always remember that your sample proportion should always be between 0 and 1. It can never be a value bigger than 0 and 1 or it can never be less than 0% or more than 100%. Similar to what we did when we were looking at the sampling distribution of means with sampling distributions of proportions, our mean of the sampling distribution of proportion, it is your population proportion. So if they ask you to calculate the mean, you need to know that that is the same as your population proportion. The standard error or the standard deviation of a sampling distribution is calculated by the square root of your population proportion times 1 minus the population proportion divided by N. The standard error or the standard deviation of the sampling distribution of proportions is calculated by the square root of your population proportion times 1 minus the population proportion divided by N. Let's get an example. Obviously, I don't have an example. I go straight to the exercise. From the past knowledge, Africa checks knows that this true proportion of ghost profile on Facebook is 0.2. Suppose we take a sample of 200 Facebook profile and found only 34 to be ghost profiles. What is the value of the population proportion and the sample proportion? Yeah, we go back to our question. What is it that they have given us? They say from the past, they know that the true proportion of ghost profiles on Facebook is 0.2. So this should be our population proportion. So that is our population proportion time. Suppose that we take a sample of 200. That should be our N and found that on Facebook and find out only 34 to be ghost profile and 34 is our X. Then I can come here and answer the question. What is my population proportion? I've identified it from the question or from the statement. It is 0.2. So therefore, this one is correct. That one is correct. It means these three statements are incorrect. What is my sample proportion? They haven't given it to me, so I need to calculate it. So your P gap is given by X divided by N. Your P or P gap is given by that. So what is our X? X is the 4. What is our N? N is 200 because I've already identified them in the question. It makes it easy. So what is the 4 divided by 200? It is 17 divided by 2 by 100. And I can change it to a decimal because my answers here are decimals. So that is 0, 17. 0, 17. And the 4, the correct answer that is not correct. So the correct answer is only number 4. And that's how you will answer the questions. Any question? None. Okay, so now let's look at how we calculate the probability. So with probability, we're also going to use the Z score. But yeah, the Z value for the proportion is given by your sample proportion minus your population proportion divided by the square root of your population proportion multiplied by 1 minus the population proportion divided by your N, all we can say is divided by the standard error. So you must also remember that everything underneath here, this is also what we call this standard error. The square root of pi times 1 minus pi divided by N is your standard error. So let's get an example. If our population proportion is 0.4, our N, or going back to the formula, let's go back there one shot. This is for STA, 1610, STA, 1510. For STA, 1501, you must know your formula. And Z is equals to P gap B minus capital letter P divided by the square root of capital letter P times 1 minus capital letter P divided by N. It's the same. Okay, so you will use your formula because in my notes I'm using the one with the pi and a P letter. So just be aware of that. Don't get confused. The principles are the same. If population proportion is 0.4 and our N is 200, what is the probability that P sample proportion lies between 0.4 and 0.45? The first step I will do is to calculate the standard error because it's very long calculation. So remember our formula Z is equals to your P minus the population proportion divided by the population proportion 1 minus population proportion divided by N. So the first step I do if I'm calculating everything manually is to find my standard error, which I find, which is 0 comma 03464. So it's to find the standard error. And once I have my standard error, then I can go and substitute into the formula to find my Z. So I know that my P. Yes. This pi into 1 minus T, you say it's a standard error. No, everything, this is the standard error. All this formula. Yes, is to calculate your standard error or what we call the standard deviation of a sampling distribution of your proportion. Okay. And this Z score is symbolizing the table. Yeah, the Z score will be the value that we will use on the table after we have calculated all these values. Okay. So you substitute into this formula. So remember justice, your formula will be P Gabi. So yours will be Z is equals to P Gabi minus capital letter P divided by the square root of your P 1 minus your P divided by N, where this is your capital letter P and this is your P Gabi on your side. Right. So then we just do the substitution. So 0.4 minus 0.4 because our sample is 0.4. Our population is 0.4 divided by the standard error we just calculated, which is everything that is underneath the square root. This whole thing underneath the square root and taking the square root underneath the division sign. We get 0.3464. And on the right hand side, our sample is 0.45. Our population is 0.4 divided by the standard error. And the answer we get from solving all these fractions is on the left, we get 0. On the right, we get 1.44. Now the next step, which is missing here, that you don't see is that we went and we said P on Z of, we went and said Z of less than 1.44 minus Z of less than 0.00. Then we went to the table. We can start with 1.44. So we went to the table 1.4 and 4 at the top where they meet. We found 0.9251, which is the value that we found there. Then we went to the 0.00, which is also on the positive side 0.00 at the top. And the answer we get is 0.05. And that is 0.500. And we subtract one from the other and we get the probability that it lies between 0.4 and 4.5 to be 0.4251. So we have moved from a sampling distribution to a standardized normal distribution. Let's look at an example because I always like giving not only the example, but an exercise. And I want you to do this exercise and we'll start. From the past knowledge, Africa check knows that the true proportion of ghost profiles on Facebook is 0.2. Suppose we take a sample of 200 Facebook profiles and found only 34 to be ghost profile. What is the probability that the proportion will be greater than 0.17? Some of these things we used before. Remember in the other question that we were asked to find what is the population proportion and what is the sample proportion. Then we said our population proportion which is 0.2 and our sample proportion was 0.17. Do you still remember that? I can go back two more steps. That's what we did because it's the same question. If you look at it, I'm lazy to calculate again the sample proportion again. We've already calculated that. So we are asked to find the probability that your P is greater than 0.17. From here, you can already start by saying minus the probability of Z less than and convert everything to a less than. You can do that or you can write it out and say this is the probability that Z is greater than and I can write the formula which is your P minus your population proportion divided by the standard error which is the square root of 1 minus the population proportion divided by N. From here, we can substitute into the formula. We have all the values Z. It's greater than. Our value is, what is our value? Our sample is 0.17 minus our population is 0.2 divided by but also one thing I forgot to mention. Also, the sample proportion will always be given to you in the question as well. So you don't have to go and calculate it based on this information that is given. It will be given to you in the question. And here will be the square root of 0.2 times 1 minus 0.2 divided by our sample is 200 and you're going to go and calculate the answer and give me the answer when you have it. Otherwise, I can do it from here as well. You also get the same answer. Otherwise, I'll also calculate it manually and see what we get. 0.17 minus 0.2 0 will be minus 0.03 and some 0's and the bottom part will be the square root of and I'm going to do the bottom part. 0.2 times 1 minus into bracket 1 minus 0.2 close bracket equals and I get 0.16 divided by 200 and I get 0,000 I'm not going to take the square root here. I'm going to put the actual value and I get 0,0008 and I take the square root which will be second function, the square root of the answer which gives me 0,028284 I need to write all the digits so that I don't run off quickly. Now I can take my minus 0.03 divided by 0.02824 which is equals to the probability that z is greater than minus I need to only keep 2 decimals so it will be minus 0 minus 1.06 and if I look at I got the same when I was using the case you now this is not the answer I need to do 1 minus the probability I find on the table of z less than minus 1.06 so we need to go to the table on the negative side to the negative side we need 1 minus 1.06 so we need to find minus 1 and we need to go to the top and we'll find the last digit 6 that is happy 0.1446 then 0.1426 and we're good and that concludes me talking and you doing the work now just to recap on what we just did today we looked at the sampling distribution for the mean always remember that your population mean is the same as your mean of the sampling distribution means your standard error which is also called the standard deviation of the sampling distributions of means it's given by your population standard deviation divided by the square root of n in order for us to find the probability we need to use the z score and the z score for the sampling distribution of means is given by your z is equals to your sample mean minus the population's sampled means divided by the standard error which is your standard deviation divided by the square root of n and remember always when you find the probability of a z less than a value that will be the probability you will find on the table if you find the probability of z greater than a value you will need to subtract one minus the value you will find on the table based on that probability of a less than the probability of between if z is between a and b you first find the probability of z less than b and you subtract the probability of z less than a from it we also looked at the sampling distributions of proportions and we know that the mean of the sampling distribution of proportions is given by the population proportion which is the pi or the capital letter p you are also required to be able to calculate the standard deviation of the sampling distribution of proportions which is also called the standard error which is given by the square root of your population proportion minus the population proportion divided by n if we need to find the probability of a sampling distribution of the proportions we use the formula z is equals to your sample proportion which is always given to you in the question your sample proportion minus your population proportion whether it's a capital letter p or it's a pi depending on which module you are using the standard error which is the square root of your population proportion times one minus the population proportion divided by n let's look at exercises and I'm going to give you one minute to answer that question if you know the answer you can shout it out you don't have to calculate the calculator this which you are using is it the smart one or is it the normal case is the normal case that you can buy from I have it with me here but I'm struggling to get this answer which you have done for the previous exercise but did you put the values correctly yes I have done that but where I'm struggling is that p is greater than or p into z that one I don't know how to put it in the shop calculator you only put the value you only do yes I was putting only these values but I don't get this answer okay I don't know how to help you but maybe probably we can connect after class so that I can show you how to work on your calculator okay it's a list what is the answer what is the mean of a sampling distribution of the mean you are asked to find the meaning of the sampling distribution of the mean and we know that it's the same as the population you don't have to calculate anything yet it should be easy and quick to answer no one that is 900 option one it's 900 because it's given to you there in the statement it's option one we know that the standard error is calculated by that the answer is 10 that is option 3 okay how did you calculate that I took the standard deviation which is 100 which is divided by the square root of the sample size which is also 100 and then I got a 10 which is option number 3 some of these things are easy and quick to calculate and answer you just need to know the basics things you just need to make sure that you are able to identify what is given in the question and substitute into the formula that the reading time of adults with ASD is normally distributed with the population mean and standard deviation of 90 and 18 respectively so therefore this is our mean and this is our standard deviation if she selects a sample size of 30 which is our n what is the probability that the average reading speed of an adult with ASD will be at least so what is at least will be at least 95 per minute what is at least at least is greater than or equal right so what they are asking you is to find the probability that your mean sample mean because this says the probability of those means of the average average is your sample mean is greater than or equal 95 therefore we can find 1 minus the probability of Z less than and we substitute our Z value remember our Z is given by the next mean minus your population divide by your standard error and we can use that formula so our 95 minus our population is 90 divide by standard deviation 18 divide by sample size 1 minus the probability you can see that I'm saving time in terms of or I'm making my life miserable by putting the 1 minus but I will always remember that I need to subtract from 1 answer I only need to keep 2 decimals 1 comma 5 to 1 minus I need to go to the table to go find this probability on the table and the probability on the table we go to the positive side looking for 1 comma 5 to 1 comma 5 and 2 0 comma 9357 we see your answers on the chat if you did answer on the chat nobody has answered 1 minus 9357 is equals to 0 comma 4 3 and that's how you will answer the question are there any questions that you have I'm going to give you time now to answer some of this question on your own we have more time if there are no questions is your first question I'm going to give you 5 minutes if you have an answer you can type it on the chat and then we will do some feedback together no answer on the chat this should be the easiest and the quickest questions to answer what is and what is the standard error remember if they didn't give you the sample proportion which is the bigger p or your p you can calculate it by using x over n and your standard error will be the square root of your population proportion times 1 minus the population proportion divide by n or you can use p 1 minus your capital letter p divide by n are we done? okay based on the information given a random sample of n of 100 people 25 are classified as emitting characteristic interest that is success suppose the proportion of success is known to be 0.3 which means this is our pi and this is our x so to find p x divide by n will be 25 divide by 100 which is equals to 0.25 right so coming in that is not correct that is not correct it leaves us with those two options number the other one says we need to find the standard error which is the standard error of the proportion will be given by population proportion 1 minus the population proportion divide by n our population proportion is 0. the square root of 0.3 times 1 minus 0.3 divide which is equals to the square root of 0.3 times 1 minus 0.3 close bracket by 100 which is equals to 0.458 if I keep 0.0458 0.0458 and some number this is 0.33 this is the correct one I'm going to give you some few minutes to answer the next one autism South Africa knows two population proportion of children with ASD in special need school is 0.74 that is your population proportion assume that a sample size of 100 which is our n of ASD is selected what is the probability that the sample proportion of children with ASD will be at least which is greater than or equal so they are asking you to calculate the probability the sample proportion will be greater than or equal to 0. therefore you need to go and find the probability of Z greater than or equal to your sample proportion minus population proportion remember the sample proportion is always given in the question divide by the square root of your standard which is the square root of your population proportion one minus the population proportion divide I'm going to give you some few minutes are we winning still calculating are we done are we winning we have an answer that will be Z greater than or equal to our P which is given in the question minus our population proportion which was 0.74 divide by the square root of our standard error which is 0.74 times one minus 0.74 divide by 100 A4 our P Z will be greater than or equal to 0.7 minus 0.74 divide by the square root of 0.74 times one minus 0.74 close bracket divide by 100 minus 0.9 is it 0.91 0.91 we go to the table go look for minus 0.7 0.91 that's what we looking for that's what we looking for so we need to go to 0.9 and we looking for one at the top which is column number two 0.1848 but because this is the probability of a greater than so going to say one minus the value we find on the table 0.1848 which will be equals to one minus 0.1848 equals 0.8152 0.8152 0.8152 which is not on the list this is one of those errors on this table yes it's actually option three why is it option three what did I do your last row there should be one minus 0.1814 did I type it wrong oh one four yes I typed it wrong so it's 0.1814 sorry we can just fix that one minus 0.1814 0.8186 0.8186 0.8186 0.8186 0.8186 which is option three thank you a random selection of 64 household was selected for a survey the question was asked do you or any member of your household own a cell phone that you can use or access the internet of the 64 respondents 32 said yes and 32 said no the population proportion was known as 0.75 which one of the following statement is incorrect which one of the following statement is incorrect the sample proportion is 0.5 so what sample proportion we can take either the yes or the no so we know that this is our population proportion and we know that there were 64 respondents therefore it means there are our n is 64 either the yes or the no so at this point because they've got the same answers so they are both our x is 32 for both of them so the sample proportion will be x divided by n therefore 32 divided by 64 would be 0.5 so that is correct the standard error of the proportion is 0.062 let's see if that is true population proportion 1 minus population proportion divided by n we root off 0.75 times 1 minus 0.75 divide by n of 64 and that is the square root of 0.75 times 1 minus 0.75 divide by 64 equals 0.0541 0.054 which is not that the standard error of the proportion is that the sample proportion is half which is the same as 0.15 because we can say 32 divided by 64 we say 32 goes time into 32 and it goes to time into 64 the probability of proportion p is less than 0.0 so there we need to go find the probability of less than so we need to go and calculate the probability that z is less than 0.65 minus 0.75 divide by the standard error we did find it was 0.0541 and therefore our probability that z will be less than we can go and calculate that just go to the front of this table and say into bracket since I don't have my fraction thingy on I'm going to do 0.65 minus 0.75 and then go out and say divide and then press equal and the answer is minus 1.85 minus 1.85 and then I go to the table look for 1.8 and the top look for 5 when they both meet is 0.322 0.0322 which is correct and that's how you will answer the questions but in the exam you would have already found your answer there you stop I'm going to give you 5 minutes to answer that question it looks like it's if like all my questions are like proportion or there is the mean one autism South Africa knows that the population proportion of children with ESDS schools is 0.74 therefore that is our population proportion a zoom a sample size of 100 that is our end you need to go find the probability that let's write it right here at the top find the probability that the sample proportion lies between 0.7 and 0.84 therefore you can go and find your Z lies between since we know our formula is Z of proportion minus P minus proportion divide by the square root of population proportion minus should divide by N so we can just calculate it or substitute 0.7 minus our population proportion is 0.74 divide by the square root of 0.74 times 1 minus 0.74 let me rewrite that again should never write in the Z value first 0.7 minus 0.74 divide by the square root of 0.74 minus minus times 1 minus 0.74 divide by our N which is 100 0.84 minus 0.74 divide by square root 0.74 times 1 minus 0.74 0.74 minus 0.74 minus 0.74 divide by the square root of 0.74 times 1 minus 0.74 close bracket goes up equal minus 0.9 0.90 go back the only thing that changes is the first value so I just scroll scroll scroll to the top delete delete delete and I need 0.84 and I press equal the answer is 0.2 2.7 2.2 0.28 2.28 therefore you need to go and find the probability that Z lies 2.28 minus the probability that Z lies 0.91 here we go to the table 0.91 we know that it was 0.8114 right that one we did this is different is it no it's not different 0.91 yes it was 0.814 I hope I'm going to write it correctly as well 0.1814 and we need to go find on the positive side 2.282.2 and we're looking for 8 at the top let's do that see if we can find that 0.9887 and that is 0.9887 do the calculations there which will be 0.9887 minus 0.1814 which is equal to unless if I've typed something wrong 0.8073 did I type something wrong did I type something wrong wrong this one I did everything to the latter yes for the probability of the Z is less than 2.28 I'm that's alright I see where you rounded it off I think I was the one I did the wrong calculation sorry that is Z of 2.28 is 0.9987 I didn't round off anything unless I'm standing on the wrong color you mean in terms of the answer of 2.28 oh no yeah I rounded it off because the answer was 2.799 some number 0.08 what I see here it seems like in that question they would have said one of those but that is not the right so there is no answer on here I will fix the slide as well the answer is 0.8 0.8073 0.8073 I will fix the slide one of this must go probably I will change that to 0.1814 and I will change that to 0.8073 I will change that to 0.8073 so to make the just give me a second that I can remember what to do I will take a screenshot of this I will fix the slides thank you and upload the latest slide as well okay let me check what time is it we still have 13 minutes also a dimeter brand of a ping ball is approximately normally distributed with a mean of 1.3 and a standard deviation of 0.8 if a random sample of 4 ping-pong balls were selected the mean and standard deviation of the distribution of the mean in respect to that so remember the question is asking you to find the mean and the standard deviation should be easy to calculate and find since we are told what the mean is and what the standard deviation is do we have the answers what is the mean the mean is 1.31 1.31 and what is the standard deviation we substitute 0.08 did they give us a sample of 4 yes they did I forgot to do that and the answer here is 0.04 because the square root of 4 is 2 0.08 divided by 2 will give you 0.04 and the answer is option 4 using the same information you are asked to find the probability so let's go find the 1.3 and 0.4 so we know that our mean is 1.3 our standard error we did calculate it you don't have to calculate it again is 0.04 to find the probability that your sample mean is less than 1.28 therefore you just need to calculate z of less than 1.28 minus 1.3 divided by your standard error of 0.04 remember z is your sample mean minus the population divided by the standard error sample mean minus the population mean divided by standard deviation over the square root of n is 1 in the same formula so what I am doing is I am using this shortcut one so therefore your p will be as done and what is the answer it's minus 0.75 so we need to go to the table minus 0.75 we go to the positive side of the table also the negative side of the table we look for minus 0.75 so minus 0.7 and we look for 5 at the top they both need 0.2266 see if it's the 0.2266 the last question is on the standard error of its sampling distribution so you given standard deviation you given the mean and the sample size so this 1000 is going to throw you off which is you but because they say a population is normally distributed with the population of a 1000 observation this is our capital letter n which is the sample size not the population size that has nothing to do with any of the calculations so you can ignore has the mean of 100 say they give us the mean and the standard deviation of 10 and the sample size which is what we need to be using is randomly drawn from the population what is the standard error and we know that the standard error is your population standard deviation divided by the square root of n do we have an answer that's option 2 you can divide by the square root of 200 which will be given which will be equals to are you saying it 0.707 yes please we have 5 minutes let's see I'm not going to ask you to answer this in 5 minutes because it's the probability of between it will take you time but you can take a screenshot of it and you will be able to answer the question oh then in the next 5 minutes you can answer this question I'll give you time since that is the last question are we winning okay so I've already wrote the formula I've substituted into the formula we know what the formula for the z score looks like and for the first one the answer is 0.5 and I'll just put a 0 at the end because I need to keep 2 decimals so remember your z is your sample mean minus the population mean divided by the standard error which is the square root the population standard deviation divided by the square root of n calculating the other side we just want to use the same the only thing I need to remove is 70 and I just do delete and I put the 1 71.5 and I press equal and the answer is 1.5 1.5 we know we need to go find the probability that 1.50 minus the probability that z is less than 0.50 so you go to the table let's first find the probability of 1 1.5 so 1.50 it's on the positive side so we go to the positive and we look for 1.5 and we know the answer will be the 0 so the answer is 0.9332 0.9332 minus next one 0.5 it's also in the positive and we need to go to 0.5 0 it's at the top it's 0.6915 0.6915 and the answer we are looking for is 0.9332 minus 0.69 0.15 and equals option number 4 which is 0.241 0.17 0.18 0.16 0.17 and that concludes today's session are there any questions any comments any query any absence of such yes if you can do the same with regard to attendance register like last week I just logged in as a guest I tried to log in with my units on the WhatsApp group it will still be there you can just leave the same link that was posted there thank you thank you please remember as well to please complete the register before you leave today other than that I will keep you posted in terms of when our next session will be enjoy the rest of your Saturday and I will see you in July enjoy your Saturday as well thank you very much thank you now this because on the that page which you talked about 770 I don't have it in my study guide I need to add to something okay so I will need to check again all right no problem thank you bye maybe if any one of my colleagues can assist