 Hey so welcome to your last session. We're also going to go through 13 questions and that deals with study units 6 up until study unit 11. Always remember at least there will be two questions per study unit with exception of some where you might get three questions per that study unit because you're going to get dating questions and you are writing out of six questions per study unit. Or is it six or five? Six, seven, eight, nine, ten, eleven. Yes, six study units. Yeah, so let's, yeah. Okay, sorry. I didn't open the PowerPoint slide. Let me go grab the PowerPoint. I opened a PDF version. Okay, I've got it. Always remember to have your tables close by because we're going to use the tables to answer the questions. So this example raised to study unit 6 up until study unit 11. There are only dating questions in here. So this will help you also give, give you more activities to prepare you for the upcoming exam. So you will have to go and find more questions to practice, especially some people like yesterday they said they are struggling with hypothesis testing. Hypothesis testing is the easiest. Also, I'm, I'm going to assume of most of the study units because you just need to know the six steps of hypothesis testing. And once you master those six steps, then you can answer any question you want. But let's look at more activities relating to those study unit 6 up until study unit 11. So you need tables to answer this kind of a question. Which one of the following probabilities are incorrect with regard to a standard normal distribution? Sometimes you will require tables. Sometimes you just require the logic and you remember the logic that we have the probability of z less than a value that is the value you find on the table. If it's the probability of z greater than a value and it can be less than or equal or greater than or equal, that will be one minus the value you find on the table. And if it's the probability that z lies between two values a and b, then it will be the probability of the table value for the second one minus the probability of the table value of the second one. If you know this logic, nothing will go wrong when you answer questions, especially on normal distribution and sampling distribution. So let's try and answer this question and find which one is incorrect. A, it says the probability of z lying between negative 0.4 and 0.4 is equals to the probability of z less than 0.4 minus the probability of z greater than 0.4. B, the probability of z is equals to 0.4 is 0.00. C, the probability of z less than 0.4 is equals to the probability of z greater than minus 0.4. The probability of z greater than 0.4 is equals to 1 minus the probability of z less than 0.4. And the last one it says the probability of z greater than 0.4 is 0.346. Which one will you choose? Let's start from the bottom. We'll start from E. Starting from E. What must we do here? What is the probability of z greater than 0.4? We go to the table. Okay, we go to the table. 0.4. On a positive side. On the positive side, we're going to look for 0.4 and 0 at the top. And that is the answer. The number is 0.554. 0.6554. 0.6554. Is that all? What do we need to do? No, we have to 1 minus that amount. We have to say 1 minus that because the sign here it says it is greater than. The probability that z is greater than you're going to say 1 minus the value you find on the table. And the answer will be correct. Because the 1 minus 0.6554 is 0.3446. Right? Let's go to D. We're looking for the incorrect one. D. Is that statement correct? The probability that z is greater than 0.4 maybe because I'm using table value. Yeah, that's correct. That will be correct. The table value that I'm referring to. Remember that the table value is all probabilities of z less than a value. Right? Your table contains all probability of z less than a value. So this one is correct because you can see that it looks like the same as we did with E. Number C. How do we validate that it's correct or incorrect? So we can go and check on this table to see if that is correct. Because number C says the probability of z less than 0.04 and that is 0. This probability on this side is 0.6554. And what about this one? The probability that z is greater than minus 0.14 or 0.4 is going to say 1 minus the value we find on the table. Right? On the negative side. Yeah. Let's go to the negative side. 0.4, negative 0.4. The value there because at the top is 0. So it's the first column. 0.3446. 0.346 which will be equals to 0.654. So the left hand side is the same as the right hand side. Which means B is correct. Oh, sorry. C is correct. B, the probability that z is equals to 0.04. Because we're dealing with standard normal distribution, we're dealing with continuous variable. So if we have the probability of exact value, it will always be equals to 0. So this will be correct. Because it says z is equals to 0.40. This is only applicable only on standard normal distribution. Number A, we'll have to check if the statement for A is incorrect. Let's look at the rule that we know. In terms of between, how should we write the statement? The probability that z lies between minus 0.4 and 0.4. Today, the probability that z is less than 0.4. Because we take the probability of the second one, which is our B and this is our A. We take the probability of B minus the sign must change and be greater than, less from greater than to less than. Because if you look at that, it has to say. So then it means number A is the incorrect one. Was it that difficult or confusing? Sorry, Lizzie, can I ask you to explain the last two again? Those two confused me. B and A. B, we're dealing with cumulative continuous probabilities because these are normal distribution. For equal, it will always be equals to 0. A constant equal for exactly, it will always, the probability will be equals to 0. So for standardize normal distribution, it either has to be less than or equal or greater than or equal or greater than or less than. Because it's cumulative probabilities. And when you go to this table, this table contains all the cumulative probabilities. All this shaded area means cumulative, it's not at the point. If it was at the point it would have been at this line, then it will just be 0. So cumulative, if it's exactly, it will be equals to 0. As you can also see here, if you don't know that, you must always remember that for cumulative standardize normal distribution, we use, if it's equal or exactly, it will be equals to 0. Number A, you just go back to the rule. What does the rule? Do I call it the rule? Or do I just say the property of a normal distribution? It says, if it's between two values A and B, you will go and find the table value for A, sorry, for B minus the table value for A. And look at the sign. The sign for A, it didn't say it's greater than, right? It used a less than value. So if you look at this one, it's just asking you to take this equation, convert it into that format and see if it looks exactly the same as this. Now, our B is 0.04, so it will be the probability of Z less than 0.4. Our A is minus 0.4 minus the probability of Z less than minus 0.4. So you can see that the sign is incorrect in terms of this. The sign should have been less than or equal. Thanks, Lizzie. Thank you. Moving on, unless if there is another question, are we good? Moving on to question two. A random variable X is normally distributed with the mean of 85 and the standard deviation of 20. Calculate the probability that X is at most 125. Choose the answers below. Now, what I did here on this question, I highlighted the key things, the facts. In the exam, they won't be highlighted. You need to be able to identify those facts. You need to read the question and be able to identify what are the facts given in the question. And after you have identified the fact, ask yourself because this is study unit six. Go to study unit six, look at your equation or your formula and write it down. Because we calculated the probability, so we're going to find the probability that X is at most, what is at most? It's less than or equals to 125. Therefore, it means we need to find the probability that Z is less than or equals to. You write the formula X minus the mean divided by the standard deviation. Now, if this is your X, you already defined that because we wrote it. That is your mean, that is your standard deviation, and that is your X. Substitute into the formula and calculate 125 minus 85 divided by 20. What is the answer? The answer is 2.5. 2.25. 0.25. Let's see if it's 2.25. Not that I don't trust you. No, it's two. Not that I don't trust you. I'm going to check and use the calculator. 1.25 minus 85 divided by 22 is equals to 2. The chances are you might have said 125 minus 85 divided by 20. What your equation would have done is to show you what you would have done. 125 minus 85 divided by 20. What this equation does, it will apply both ways. Multiplication and division have priority over addition and subtraction. It would have looked at this subtraction and looked at the division and evaluated it and said, hold on. Division first before subtraction and it wouldn't have given you 2.25. It would have given you 125 as well. Be very careful when you apply or use your calculator. Use the equal sign as well. Equal sign of brackets. The top part in the bracket divides by the bottom part. The answer here will be 2, which is 2.00 because that's what I need. Go to the table. When we go to the table, we go to the positive side of the table and you're looking for 2.00 at the top. And the answer is 0.972, which is option D. Always remember when they ask you about the probability, you need to calculate the z value and then go to the table to go find the value. Always leave your answers to 2 decimal. Make sure that you know how to round off correctly as well. Question number three. An analytic company sampled 90 online meetings and found the meetings to have an average duration of 90 minutes with a standard deviation of 90 minutes. It is further known that the probability that the average meeting duration is longer than x is 0,8531. That is the probability of greater than or equals to x will be 0,8531. Calculate the value of x and choose the correct answer from the list of options below. Now remember two questions per study unit. How many questions we have already done? Two. We are in question number three. Already think about it. If it's two questions per study unit and I've already answered two and I was in study unit six, what study unit am I in right now? It will be study unit seven. So it means you will go and page through to study unit seven and find the formula for study unit seven. Now, with this question, I've also highlighted the things that affect that you needed to know and highlight before you start answering your question. The other thing that will make you understand that you have moved from study unit six, you are now in study unit seven is that thing at the beginning. In study unit six, we do not include the sample size. In study unit seven, we include N, our sample size. So the minute you see somewhere where they tell you about the sample size or the sample, you might know that now you have moved from study unit six, you are in study unit seven. So they say the average, which is the mean. It's also from the sample, because the question says an analytic company sample 90 minutes online meetings and found that the average, which is the average from the sample is 90 minutes and the standard deviation is 90. It is, we can also not use, we can also decide whether we, because this is sampling distribution. Let me also go back and retract my statement about that is the population. So I'm going to take this as a population average, and I'm going to take this as a population sample as well. Regardless, it doesn't matter. So we can use that because I'm going to look at the formula for that. So the sampling distribution is your sample mean minus the population mean divided by the sample standard population standard deviation divided by the square root of N. Now, the X that we are looking for is the sample mean. On this question, because we're dealing with sampling distribution, they have made a mistake here, they should have said X bar, sampled mean for that X, because that is the average meeting, right? If you look at that, the probability that the average meeting, which is your sample mean is longer than, which will then be your probability test. But that does not stop us from solving the problem that we have now. At the moment, they have given you the probability and they telling you that that probability is 0,8531. What they're saying is they went onto the table or they went somewhere and they found that the probability of A, which is our X, I'm going to use X. The probability of our X is 0,8531. How would they have found that probability? Because the sun says greater than. How did they find that probability? How did they get to 0,8531? Do a reverse engineer, because what we know that the probability of Z greater than a value is 1 minus the probability of Z less than a value of the same from the table. So then it means this 0,85, they have found it. We need to go and find what the Z value is, right? The only way to find the Z value is to subtract this value from 1, so that we can get the Z values. What we're going to say our Z is 1 minus 0,8531, because for them to get to 0,831, they would have subtracted the value they found on the table. They would have subtracted the value they find on the table from 1 to get to 0,835. So we're doing a reverse engineer. We are looking for the Z, for that Z value that they use. So let's go find that Z value. So this, the Z is, what is 1 minus 0,8531. 0.1469. So we're going to take 0,1469 and go look for it inside the table and go out and look for those Z values. So let's go. We are looking for 0, let's go back. The value is small. It's 0,14. It will be on the negative side, because the positive side has the bigger probabilities. So we go to the negative side and we're going to look for 0,1446. I wrote the draw. 0,1469. So let's go inside the table and look for that number. 0,14, 0,15, 0,1469. I found it. That is the probability we need to go out. Okay, because it's going to hide all the things. Let's take the first digits, which is minus 1,0 and we go up to the top to go find the last digit, which is 5. So we have found our Z value is minus 1,05. So let's go. So our Z, we found it. This Z we're going to replace it. I don't even have to put the sign like that. I can use the equal sign. Our Z value, we found that it was minus 1,05. The X is the mean that we are looking for. Remember, I said this is a mistake. They should have had an X bar there, which is the value that we are looking for, because that's what we are looking for. Minus the mean is 90 divided by the standard deviation is 90 divided by the square root of 90. And we can apply that to this. So it will be minus 1,05. Multiply. And so we take everything that is at the bottom 90 divided by the square root of 90. And we take 90 to the other side. Plus 90 is equals to X bar. So our X bar will be equals to minus 1,05. And then open bracket fraction 90 divided by the square root of 90. And use my arrow, arrow again goes bracket plus 90. What do you get? What is your answer? What do you get? I can also move my calculator. Slightly. Option D. And when I press equal, I should get 80.0388. 80.0388, which we estimate to 18.0. However, hold your peace. Questions. Be good. Be happy. Happy. You must ask if you're still not sure. In the absence of comment or question, then it means I am going to assume that you are all good. Moving on to question 4. No analytics from online meetings to show that the proportion of invitees that accept a meeting invite and actually attend is 0.888. In a sample of 30, what is the probability that the sample proportion of invitees that attend the meeting invite? And actually attend is between 0.83 and 0.93. Also, we're still in the sampling distribution, so you go and look for the formula that deals with proportions. Remember, if they didn't give you the sample proportion, you will have to calculate the sample proportion. In this instance, they did give you your sample proportions. It's between two values, 0.83 and 0.9. Since it is between, so you're going to say the probability that 0.83 is less than the sample proportion 0.93. So we're going to find that between P minus the population proportion divided by the square root of your population proportion 1 minus population proportion divided by N. P minus population proportion divided by the square root of population proportion 1 minus population proportion divided by N. P, our P is always in the question. The other sum proportion is your population proportion, which is the pi is 0.88. So yeah, we start first with the 0.83. 0.83 minus 0.88 divided by the square root of 0.88 times 1 minus 0.88 divided by our N is the sample, which is 8. We do the same, 0.93 minus 0.88 divided by the square root of 0.88 times 1 minus 0.88 divided by 8. Due to calculation, give me the one on the left first, the 0.83 minus 0.88 divided by the standard error. It's negative 0.8427. One, two decibels, so let's go. Zero comma. Negative 0.8427. I have only one, two decibels, so it will be negative 0.84. Negative 0.84. And 0.93, the distance is almost the same, then this side will be positive, but you can just double check. Because this is 80, this is mine, so same distance. 0.8427 also. 0.84. Now we need to go to the table. Remember, we're going to say the probability of Z less than 0.84 minus the probability of Z less than negative 0.84. So go to the table and find what is the probability of Z less than 0.84. So we go into the positive side and we look for 0.8. The answer is 0.7995. Then let's go to the negative side to go find 0.84. The negative side, I'm going to go first at the top and just highlight this column. 0.80. The negative 0.8 is the column, that is the value, which is 0.2005. 0.2005. The magic, what is the answer? 0.599, letter E. It will be option E. Are we good? Are we happy? I know, I'm not responsible, I don't know, are you still here? Maybe people left the session, let's see. Are people in the session? Are you tired? We are all tired. Today has been a very long day for me as well. Moving on to question five. A simple random sample of data items results in a sample mean of 200. The population standard deviation is known to be 100. Construct a 95% confidence interval. Oh, I forgot to ask how many questions we did on that. There were two, see? Two questions per study unit, with exception of one, that might have more than two. So it will be two, two, two, two, two per study unit. Okay, construct a 95% confidence interval estimate for the population mean and choose the correct answer. I don't even have to tell you which study unit you are in, because clearly they have given everything to you. They told you that it's confidence interval. Now with confidence interval, remember to go to the summary, note there is that table with confidence levels, right? You can use that especially for the Z when the population standard deviation is known. You just need to know that. Otherwise, by the time you go write the exam, do not forget this. When you construct a 95% confidence interval, or when you do a two take, especially for Z, your critical value, because you will need the critical value, Z alpha divided by two, when it's 95%, it's always going to be 96. 1,96. If you can learn that, you can know that, you can memorize that. Nothing will go wrong, but it's only applicable for Z when alpha is divided by two. Okay, so what is it that they have given you, a random sample of 30, a sample mean of 200, the population standard deviation, which then is known, then it means we are doing the Z. Construct a 95% confidence interval, then you go immediately to the formulas, the formula for confidence interval for the mean. The population standard deviation is known. It's the mean plus or minus the critical value times the standard error, which is your population standard deviation divided by this way root of N. Your sample mean is 200 plus or minus your critical value. I already told you that it's 1,96. Times your standard deviation, which was known to be 100, divided by the square root of the sample, which is 30. Remember, you do the minus first, and then you do the plus. So it will be 200 minus 1,96 times 100 over the square root of 30. And 200 plus 1,96 times 100 over the square root of 30. Our negative side is 1,64 comma 2,1 comma 2,3. 1 comma 2,2, yeah. 1,64 comma 2,2, yeah. And on the plus side, 235.78. Which is option? Easy, isn't it? This one is more easier than the other. In a sample of 150 online meeting participants that accepted the meeting invite only 185 attend. Construct a 95% confidence interval estimate for the true population proportion of participants that attend the meeting and choose the correct option. What is this one? Is proportion because you are told. So do the thing. If your sample proportion is not given, remember they would have given you observation satisfying your X. Identify which one is X, calculate that. Look for the formula for the confidence interval, sample proportion plus or minus. Because it's proportion, also it will be z alpha divided by 2. And we're still doing a 95% confidence interval. So you should know what is that. P times 1 minus P divided by A. So have you calculated P with this P? What is the value of X and what is the value of N? The high performance formula has a powerful combination of five oils and the three, such as almond, aloe, macadamia, and aloe. Somebody is having their TV on or some something, something. Okay, so let's go. What is our X and what is our N? They are both highlighted. So in the exam, you won't have made any new or highlighting things for you. You need to know which one is X. I think X is 135. X is 135 and N will be 150. Always remember N is the biggest one. If you have two values, one of it is small and the other one is big. The biggest one is N. That's how you will remember this. And what is the proportion? The sample proportion will be what is 135 divided by 150? 0.9. 0.9, so we can just substitute 0.9 plus or minus 1,96 times the square root of 0,9 times 1 minus 0,9 divided by 150. And then do the Y's. Do the minus first and then the plus side first. So let's do it together. 4.9 minus 1.96 times the square root of fraction. 0.9 times 1 minus 0.9 close bracket down 150. A rule, A rule, A rule again, A rule again. A rule, close bracket and equal. 0.852. 0.852. 0.852. On the plus side, on the plus side will be ... And the plus and equal 0.948. 0,948. Which is option D. Easy, isn't it? I even did it for you. If you don't know the critical value, which is z alpha, please make sure that you have the summary notes next to you when you go right so that then you can use the table that is on there to help you identify the critical values. The only time you will need or require a statistical table is if you're going to find the critical values of T. Then I will say use the table because on the summary notes the table with critical values it's not there. It's not there for T but it's there for Z. Moving on to question 7. I hope we will make it already halfway through. Do you now connect? A startup internet provider would like to submit a proposal to Google to consider expansion of its next Wi-Fi services to remote areas in South Africa. As part of their market research, the NEO Connect needs to meet your assistance in testing the hypothesis that the average fiber internet cost in South Africa is less than 600. In a sample of 70 households with fiber internet, the main fiber internet cost 550 and the standard deviation is 220. What is the value of your test statistic? Where are we at? We dealt with two questions from Confidence Interval. So this one will be from the next unit which is hypothesis testing. So if they're asking you, you're going to find out what your test statistic is. Why am I writing a T instead of a Z? Why am I not saying Z? Anyone? Because sigma is unknown. They have given you the standard deviation from the sample. In a sample of 70, the mean is 550 and the standard deviation is 220. Both the mean and the standard deviation comes from this sample. So it means those are your sample statistics. So here I'm using 80. Sample mean minus population mean. So you're going to get the formula. I know the formula by heart. You don't probably, but you need to know that it's the formula that you need to write. So you need your formulas. Let's substitute the values onto the table into the formula. Our sample mean, sample mean is 550 minus our population mean was less than 600. I don't care about the less than and the greater than because I'm just looking at the test statistic. If I was asked to calculate the P values and all that, then I would have to worry about that. Divide by the sample standard deviation is 220 and the square root of sample size of 70. What is the T test statistic? What is the answer? Negative one comma nine zero. Negative one comma nine zero. She is option A. Moving on to the next question. A quality control manager would like to test at 6% level of significance. If the prototype sensor of self-driving cars have an average lifespan of 300 mega meters, nine sensors were manufactured tested and found that they have a mean of 341 mega meters. I assume that the lifespan of the sensors is normally distributed and that the population standard deviation is 70 mega meters. What is the critical value for the test? Are we going to use a T test or a Z test? Z test. We are going to use that Z test and the reason being is the population standard deviation is known. Are we going to divide? Is it going to be divided? The critical value that we are looking for is it going to be Z alpha or is it going to be Z alpha divided by two? Has any way when we read the sentence, have they ever mentioned greater than or less than or they just said they are equals to this much or they are this much. When you read the sentence, you need to also think about what is it that they are giving you. Like remember in this one they said it's less than 600. On this one it says the driving car have an average of 300. Nothing is mentioned about less than or it's exceeding or it's greater than or at least or at most. So it means we're going to be doing a two tail and therefore it means our Z we're going to divide by two. So our alpha which is our level of significance that is our alpha Z of zero comma zero six divided by two which is the Z of zero comma zero three do not get alarmed by you always using zero comma 95 percent or the level of significance of five and all that. The table also has the probability remember that your alpha over two is a probability on the table. So we're going to look for this probability inside the table and go out to look for the Z value because the critical value is your Z values. So go to the Z table and inside it will not be on the positive side because positive side has bigger values. It's going to be on the negative side. We're going to look for zero inside a value close to zero comma zero three. It can be zero comma zero two nine or something like that. So as long as it's closer to zero comma zero three inside inside inside inside zero comma zero one zero two zero three that's big zero three two nine it's big zero three one four it's big that can be but there it is zero comma zero three zero one but also this can be zero comma zero two nine very tricky because we can choose between those two values and let's go check the Z values. We can use this as a guide whatever the value is close the closest one from their option we're going to choose that one. So negative one comma eight I'm just going to go there just to see if we have more than negative one comma eight we only have one value with with with one comma eight right one comma eight it's only one. So we can choose that and go and choose the value there but if you go up with those two values it's eight eight and nine so it would be one comma eight eight. I would have said nine I don't know why they have that or maybe they took the average between the two values and find eight five I don't know I don't know how they derive and go to one comma eight eight five because that's the only closest value in terms of the critical value so I would have chosen option a that's the question is asking for a critical value. If it was z alpha so we would have when they looked for zero comma zero six on here zero comma zero zero six you could have been on the positive side zero comma zero six also this is zero comma you could have been somewhere around those two values which is zero comma two then zero comma two you can see there okay so that's how you will find the critical value based on the the table values. Question nine in a random sample of 200 a sample proportion of 45 percent you are required to test at five percent level of significance if the true population proportion is equals to 40 percent calculate the p value now reading the question ask yourself are you doing a one tail test or a two tail test one tail it means they you should hear words like greater than or less than a two tail you should have words like equal right because when you find your p value for the one for the less than the p value will be the table value for a greater than it will be one minus the table value all right if you are doing a two tail it will be two times the p value only if the answer was negative if it was positive it will be two times one minus the p value remember that we dealt with this we did this when we were refreshing so but before you can get to the p value you need to calculate that well let's go calculate that we need to calculate the z step and use a digital say detail it one tail separate which means sample proportion minus population proportion divided by the square root of or the standard error which is the population proportion one minus population proportion divide by n and substitute the values in the sample proportion is 0 and is on strike 0.45 minus the population proportion 0.4 divide by the square root of 0.4 and 1 minus 0.4 divide by 200 you have the answer what is our z test what is the z test one one comma four one comma four four one comma four four that is not the end now let's go back to our question are we doing a one tail test or are we doing a two tail test are we doing a one tail test or are we doing a two tail test ski wakey i just gave you the answer you guys you are exhausted me we're doing a two tail test now the answer there is a positive or negative this needs to guide you does that test there is positive so if it's positive we're going to go to the table even though we're going to the positive side of the table we're going to look for one comma four and four where they both meet that is the value we are looking for but that's not the answer is zero comma nine two five one and we say two times one minus zero comma nine two five one and that will give you your p value so what is your p value zero comma one four nine eight zero comma one four nine eight I should just say two times two times one minus point nine nine two five one zero comma one four nine eight and that's how you will answer the question all this you just need to think what does the question ask you to do what are the things that they have given you there are a couple of things that you need to get right am I doing a two tail test is my z test value is it negative or positive if it's two tails all those things if they are not in that summary table you need to write them in your summary table with your notes by now you should be just adding things that you need to make yourself aware of be aware of this right question 10 which then it means we are left with three more questions now we're getting to a contingency table consider a four by four contingency table you are required to test the independence of rows and columns variables with a one percent level of significance which one of the following statement is incorrect we're dealing with a four by four which means we have four rows and four columns let's go by one by one and eliminate we're looking for the incorrect statement number eight how do we find the critical value remember your criterion waiting for chi squared is alpha and the degrees of freedom and your degrees of freedom is your number of rows minus one times the number of columns minus one that's what a is is a correct or incorrect so it means you do you need to do some calculations number of rows you are told how many rows and columns you have number of rows minus one times number of columns minus one that is your degrees of freedom what is our degrees of freedom four minus one nine times four minus one it will be equals to nine and our alpha we are told what level of significance is that is your alpha so is a correct or incorrect correct a is correct b the null hypothesis is that the two variables are independent how do we state the null hypothesis and alternative hypothesis for contingency table chi square for independence your null hypothesis always has you forgot about this we did it last week Sunday and equal sign sorry here we're dealing with contingency table it's either independent or dependent so null hypothesis will always state that the two categorical variables are independent independent so null hypothesis independent so therefore it means statement b is correct the critical value is 21.66 so you said this is correct in terms of your critical value so go find out on the table if this is correct so we go into look for 0 comma 0 1 at the top and nine degrees of freedom on the left critical values of chi this is critical values of t critical values of chi we're looking for 0 comma 0 1 at the top and 9 where the boys meet the correct answer that is the same moving on to the next one there are 16 degrees of freedom is that true false that's not correct so the incorrect statement is option g e if they observe then the expected frequencies are the same for each cell then the test statistic will be equals to zero we dealt with this the last time i did show you the example of it that is true consider the following contingency table of observed calculating the chi square test statistic required to test for independence of rows and columns variables choose the correct answer from the list below so we have a contingency table with rows and columns there are two rows and three columns this is when you can use your template we're going to use our template we know that we have a two by three i'm going to use my two by three template and i'm going to take it to the side and this is my two by three i can change the headings at the top if i want to change a b c it's finally p q r p and i can just change the values only the white area that i need to remove of the observed left and this is 50 and this is 60 and this is 88 and this is 34 and this is 75 and this is 41 okay and we're looking for the test statistic if i go back i can double check that i have captured all the values correctly and if i scroll down down down down down that and i will get my test statistic here and it is 15.51 which makes option number three correct easy by now you should know how to use the template if you download that if you get access to the template make sure that you download them don't work on them on the google drive download them onto your your machine and work from there sorry regardless of where you find the the template sorry Lizzie uh-huh my sister can i please ask you to go back to the template once again please and this is the template oh sorry you're taking a picture of the template no is the template on your website Lizzie yes it's they are all the okay thank you Lizzie on the site right if you go onto the site you will find the template they uh there are two of them okay so we dealt with two questions from contingency table so i'm not going to assume that the next one will be from there we are left with only two questions question 12 and question 18 so they will be from regression okay so this the following simple regression equation estimate the relationship between the number of the nail connects unlimited data subscriber waking remotely which is our independent variable and the average data usage in megabytes which is our dependent variable and the regression line is given by y is equals to 15 times x where 15 is your slope and your slope in this instance is positive huh i'm already giving you answers why am i doing that which one of the following statement is incorrect a we can do a process of elimination to get to the incorrect statement a there is a positive relationship between the number of unlimited data subscribers waking remotely and the average number or the average data usage is that correct or incorrect i'm already gave you the answer it should have been a quick answer that will give me back it's correct it is correct because the slope is positive so there is a positive relationship that is correct when 18 000 or 1800 denial connects unlimited data subscribers are waking remotely the average data usage is estimated to be 27 gigabytes is that correct how do i know that your 1800 is your x you just substitute it back into that equation so you will say 15 times 1800 do you get 27 gigabytes which might be 27 000 on your calculator do you get that or is it another variable is it 27 000 yes then it means this is correct go on and do the second the third one when is 22 000 or 2200 33 000 which is 33 gigabytes which that means this is correct let's check if they are 1500 he is incorrect because the answer there is 22 500 and he is incorrect he says when there is no unlimited data subscriber when no unlimited data subscriber are waking remotely the average data usage will be 0 megabytes so you just replace x with 0 and that will give you 0 because 15 times 0 will be equals to 0 there we are almost done the last question the following sum square total of squares and sum squares of squares due to error were calculated from the previous regression equation where the sum square total which is sst is 33 000 800 and sum square error which is sse is 6760 which one of the following statement is incorrect this one a it says what is ssr now to understand what ssr is you can use the formula of e because that should be the correct formula that you need to to use because your total variation is equals to your sum square measures of regression plus your sum square measures of errors which is ssr plus sse is equals to st so if you have e and i told you that e is correct because e is just the formula to calculate sst so if you go to the regression section of the summary sheet or the summary sheet or table or document you will find the formulas as the summation the sst and the ssr use that so i've i've just told you that e is correct based on the formula use e to calculate a what is your ssr ssr will be sst minus sse is a correct or incorrect but here you just say 33 000 800 minus 6760 it's correct it's correct which is 27 000 for 0 27 000 and 40 that's that's that i'm gonna skip b and c i'll come to it later d the sign of the slope of regression equation determines the sign of your correlation coefficient the sign of your slope and your coefficient of correlation should always be the same so it means if for example i'm using the ssts and the ssr right to calculate the coefficient of correlation and the coefficient of determination the answer i will get on there will always be positive i will never know whether the regression is negative or it's positive whether the the slope is negative or positive or not the slope i will never know whether the relationship is negative or positive based on the ssr rather than the sst formula to calculate the relationship but if i calculate the regression line i will be able to know whether the relationship is negative or it's positive based on the slope question d is just an a confirmation to say do you know that the sign on your slope should be the same as the sign on your correlation of coefficient the sign of the slope of regression of equation determines the sign of the correlation coefficient is it correct or incorrect based on what i just told you so it means if you calculated the slope and the slope is negative therefore when you calculate your coefficient of correlation using the ssts and the ssr you can put you can determine that the sign of the coefficient of correlation on your formula will be positive or it will be negative based on the sign that you have on the slope that is d which is correct because here they just giving me the statement and they say you must say whether the statement is correct or incorrect if you agree with the statement or you not number b and number d are based on because we are not given the some summations we are given the sst and the ssr you cannot calculate the coefficient of of the coefficient of correlation you have to calculate the coefficient of determination and the coefficient of determination which is r squared is given by your ssr divided by your sst using that formula is b correct and what is the value of your coefficient of determination because that will determine what the answer would be is as well so you're going to say 27 040 divided by 338 800 what is the answer it's 0 it's 0.8 that it is not correct that is our incorrect statement how do i calculate my how do i calculate my coefficient of correlation which is r so it means r is the square root of 0.8 and because it's the square root i'm going to put the plus or minus because i don't know whether it's negative or positive because i don't have the slope on this so i cannot say for sure it's a negative call it's a negative relationship or a positive relationship so when you take the square root of 0.8 what is the answer it's 0.89 right 0.89 this is correct you need to know because the questions in the exam might be different you need to know if they didn't give you the ssr's and the sst's if they give you your x and your y remember to use the template right remember to use your template to calculate your regression and if you need to calculate your sst's and your ssr you must also do the same and there is the formula i just wrote as well in terms of ssr so you can use the template to find your formulas because you can see there is your ssr divided by sst and you also have the formulas on on the site on the template to help you navigate and when you use the template remember if you are adding you start from b you highlight the rows you want you start from b and you highlight the rows that you want to to insert or delete and you only up to that point you don't delete the entire row and you go up or down right that is that and that concludes today's session because that was the end of the session and we are five minutes into okay someone asked me just give me a second i didn't uh i didn't write the question but we're gonna improvise someone asked they are calculating the the coefficient of variation now remember that coefficient of variation your coefficient of variation is your sample standard deviation divided by the sample mean multiplied by a hundred all right so if you have data let's say it's two three four eight nine and seven i'm not gonna be a lot so that then i have time so if this is my data the first thing is to use your calculator to put it onto state mode so that it becomes easy and quick calculate so we're gonna put to state mode and because we only have one variable we're using one right and you're going in the data so you're going to say two equal three equal four equal eight equal nine eight seven equal so you have all your data in you press the ac button and you press the shift and you press that which is number one and you go to var which is four you press four and you follow what this formula says is your standard deviation is s so we're going to press four again and you press divide and you go shift that form and you press two for me you just follow what the the calculator is telling you to do so there it is and because multiplication and division have the same priority i can also just say multiply by a hundred on the on-site it will apply both must move from left to right because they have the same priority and say equal and that will give you your coefficient of variation always remember to know to recognize the formulas that are required if you are going to use the templates know how to use the template if you are going to use your your calculator know how to use your calculator quality as well for a sharp calculator for a case your calculator or for the online version calculator or for whatever the calculator you are using hp whatever make sure that you know the steps of how to use your scientific calculator especially if you are going to put it to state mode whether you're going to calculate for one variable or you're going to calculate for two variables remember for two variables it will be a plus b x and remembering that b is your slope and a is your intersect based on the formula that you have okay and that concludes my engagement with you i wish you all the best of luck with your exam there's not much i can say i've been saying this since from the start in january when we started with the sessions and there's nothing more i can tell you that would change create miracles all you need to do is go and practice yes i my sister my apologies i'm the one who who asked about this so it's a problem now i'm trying to uh to cancel some of the things i my story is full on my video i can't take video for this and i'm no i know i need this information what okay uh it's not part of the content are there any is there any other question relating to what we just went through anything you want me to explain even more on so that then it becomes part and puzzle of the video otherwise i'm going to stop the recording i want to ask for the link to the templates that you are referring to okay no problem if yours is not part of the content so the going once going twice and forever do peace let me first stop the recording