 Fine. I'm not going to upload that one. We'll upload this one that deals with their assessment. So at the moment, I hope you see my screen which has the questions that we're going to be doing today. So we're doing study unit 7 till 11, but we're not gonna finish all of them. We'll carry on tomorrow again. So I'm gonna take this one as a revision as well. So you haven't done it. I will give you the formulas. You wake it through. We go it through it the same way as we did with assignment with the self-assessment number one where I show you and then we wake it out together so that there's no quietness. We don't have people being quiet for long. So we talk through it Okay, so everybody should be unmute. So and have your calculator ready because we're going to wake it out together. The whole exam paper. Okay, the first question. The mean annual cost of the automobile insurance is 95 rent. Assume that the population standard deviation is 14 rent. What is the probability that a simple random sample of 30 for the automobile insurance policy will have a sample mean less than 90. So remember study unit 7 is sampling distribution. With my pants. So this is a sampling distribution question. And since we are asked to calculate the probability, so remember we find the probability by using the z score, which is the sample mean minus the population population sample mean divided by the standard error, which is your population standard deviation divided by the square root of n. Remember that. So what are we given? We are told what the mean is which this is our population sample mean. And we are also told what the population standard deviation is. And we are given the sample size, which is our n. And we are told that the sample mean is less than 90, which the stand means is sample mean is less than 90. And we always need to remember the following. That for a probability of z less than a value, that probability we're going to find on the table. Remember that. Probability of z greater than a value, that probability we find one minus the value we find on the table. And the last one is the probability of between where z lies between a and b. We find that by the probability of z less than b minus the probability of z minus a. So now since this is the less than, we're going to find this probability of z less than a value. So it means when we go to the table and I have my tables here, so we'll use our table. When we go there, we go into find our probability. Substituting the values z less than our mean is always 90, yes minus our population mean. 95. 95 divided by the standard deviation of 14. Which is 14 divided by the square root of the sample size which is 13. Yes. So now calculate the value. I think minus 196. Minus 1.96. We need to keep it to two decimal because when we go to the z table, we only using two decimals. So now we go to the minus side. We are looking for z less than minus 1.96. 0.9. We go find it there. And at the top, it's not on there. And the probability is 0.0250. Agreed? Yes, my son. Yeah. Yes for me too. Yes. Okay. Moving on to number two, which is also the proportions. It's also sampling distribution. It says the proportion of eligible voters in the next election who will vote for the ANC is assumed to be 0.55 in housing. What is the probability that in a random sample of 500 less than 0.49 says that they will vote for the ANC? And also since this is also a z sampling distribution for the proportions. So we're going to use z is equals to our P minus our population proportion divided by the standard error, which is your population proportion times 1 minus the population proportion divided by N. Go to what we are given. We are given our population proportion is 0.05. We are given our N is 500 and we are told that it's less than. So therefore the sign here changes to a less than. And this is 0.049, which is our sample proportion. Remember, if they don't give you the sample proportion, they will give you your P, they will give you your X and you can calculate the sample proportion. But in this instance, they gave us the sample proportion. So we need to calculate the probability of this proportion. P is 0.49 minus our population proportion to 0.55 divided by the standard error, which is 0.55 times 1 minus 0.55 divided by our N, which is 500. Do the calculation and let me know what is our Z value. Do you have an answer? Minus 269, 2.69, but I just need to check. It has to be a negative answer. But I'm not sure. Mine is minus 2.69. So I'm not sure if it's right. I'll just do it now quickly again. Minus 2.69. Okay, we wait for the others to say what is their answer. I can also check. I have minus 2.7027, do you round it off to two decimal points? Yes. Okay, so minus 2.70. Yes, I think that is the answer. Let me quickly also calculate it and see. I think which gave me option A. Okay, divide by the square root of 55 times 1 minus 255 divided by 500. Minus 2.70. And that's what I found as well. Yeah, that's also what I have. Yes, then we go to the table. We look minus 2.7 and zero at the top. Therefore, that is our probability. 0.0035, which is option number one, which is A. Yes, to move to the next question, it asks the average cost per night of a hotel in what Elizabeth Township is 273, assume the estimate is based on a sample of 45 hotels and that the sample standard deviation is 65 for a 95 confidence interval and half. Then it means we are now in confidence interval, which is the unit eight. We're no longer in the sampling distribution. So with the confidence interval, there are certain things that we need to always remember. Remember that we can calculate confidence interval for the mean. When the population standard deviation is known, it means they would have given us the sigma or the population standard deviation. Or we can calculate it for the mean. When the population standard deviation is unknown, it means they would have given you their sample standard deviation and also for the proportion. So if we go through this question, we can see that they give us the average cost, which is the mean. They also give the sample size, which is our N. They give the sample standard deviation. Even if they didn't have said sample standard deviation there, you need to be very careful when you read the questions because sometimes they don't put it, sometimes they do put it. So for example, with this, when they didn't give you the sample standard deviation, the fact that this is based on a sample of 45 hotel that has a standard deviation of this much, then it means they would have, you can assume that this standard deviation comes from this sample because they are all in one line and it says the sample of this and the standard deviation of that. But I guess in order for them not to confuse you, it will always give you the one way that will tell you whether population standard deviation is known or unknown. And in this instance, they gave you the sample standard deviation. Therefore, the population standard deviation is unknown. And we use the point estimate plus or minus the critical value because this is where the population standard deviation is unknown. We use the T table and we do T alpha divided by 2 and the degrees of freedom. And remember the degrees of freedom is N minus 1 times the standard error, which is the sample standard deviation divided by the square root of N. But the question is asking for the critical value, sorry. So it means we're not going to do all this since the question is only asking for the critical value, not the confidence interval per se. So the question says, what is the critical value? And that is the only thing we need to be calculating. Our alpha, we're going to get it from a 95 percent confidence interval. Remember, this is the same as 1 minus alpha is equals to 0 comma 95, which alpha will be equals to 0 comma 0 5 if we make alpha the subject of the formula. By moving alpha onto the other side, we take 0.05 minus 1 minus 0.95 will give us alpha of 0 comma 0 5. Then we can find our critical value by saying T, 0 comma 0 5 divided by 2 and our N is 45 minus 1. Therefore, this will give the critical value of 0 comma 0 250 and 84. How do we find this? Go to the T table, critical values of T and we go find, remember, we don't use the cumulative probabilities at the top. We look at the values closer to the table, which is 0 comma 0 25. Remember, we do 0 comma 0 250 and 24, which is our degrees of freedom. So I'm going to go up because I'm going to lose that column at the top. That's why I create the line. We need to go to 44 and that is 44 and the answer will be that 2 comma 0 154. Let's look 2 comma 0 154, which is option number B. Yes, thank you. Yes, thank you. Okay, then the next question is asking you, oh, this is a repeat, the same thing, was asking the critical value as well. I think it's the exact same question. Yeah, I think it was a mistake there. Okay, so the next question, a survey of a random sample of grocery shoppers in Kimberley found that the mean of the grocery purchase was 78. Assume that the population standard deviation of the grocery purchase value is 21. The 95% confidence interval for the mean grocery is they want you to calculate or find the confidence interval. So remember this, I'm going to give you the formula based on the information that we read. So it says the population standard deviation. So this one, the population standard deviation is known and when it's known, we use the Z critical value. This will be your population standard deviation. I don't know how to write this, to write by the square root of L. Remember to find the critical value, which will be 0 comma 0 5 divided by 2, which is the Z of 0 comma 0 2 5 0. To find this critical value, you go to the Z table. We look inside the table on the negative side of the table, not the positive, but the negative side. Inside the table we look for 0 comma 0 2 5 0 inside the table. 0 comma, which is that and go get the critical value, which are your Z values and those are your critical values 1 comma 9 6. In a nutshell, what I'm trying to show you is also always remember they might in the exam, they like asking for 95% confidence interval. So you by now you should know that that is 1 comma 9 6. If your critical value is alpha divided by 2, then it's always going to be 1 comma 9 6. So please solve the formula. Remember when you're doing this, the minus first and then the addition second, because we're doing upper and lower critical values. Given n, we're given mean, we're given the standard deviation. Substitute into the formula. I'll give you five minutes to substitute. And then you can tell me how you substituted the values and then we move on into the calculation. I'm getting to option A. Okay, let's substitute the values. The mean is 78 plus minus your Z of 1.96. The standard deviation is 21 divided by the square root of 300. Then I went and said 78 plus minus 1.96 and I solved the value in brackets, which my answer was 1.212436. You read them too quickly. I hope I captured them right. There is, sorry, 1.212436. 1.212436. Yeah, not 34. And then I again did the 78 plus minus and I solved the 1.96 times 1.21. So, sorry, I'm just making sure I had it right. Yeah. And then I got to 2.376374. 2.376374. From there, I said 78 plus the 2.36374. So, we need to start first with the minus. Oh, okay, 78 minus the 2.376. And then we can go to the plus side, 78 plus 2.376. You get 78 minus. 78 minus 2.376 was 75.62. On the other side. And on the other side, it is 80.38, sorry, 80.38. Which makes option number B. You need to be very careful when you do this confidence intervals as well. If you have started with the plus and you put the plus first, the value of the plus, you would have chosen A, which is incorrect. Yes, I see that now. Okay. So, we always do the minus side because the minus side tells you the lower limit and the plus side tells you the upper limit. So, you cannot put the upper and then lower. Okay. So, going on to question 6. Question 6 is still also confidence interval. If I look at the options as well, they just tell me they showed confidence interval. Yeah. And they have given you at the 95% confidence interval for the developer confidence interval with the level of confidence of 95% where the population proportion, if a sample size of 200 had a 40 successes. So, you need to first calculate your P and substitute into the formula plus or minus. Because it's for the proportion, oh, that's the other thing I need to mention. Because of the proportion, we use z alpha divided by 2 times the standard error, which is your sample proportion, one minus sample proportion divided by n and substitute into the formula. So, we are given our n and we are given our x. Are we winning? Are we doing? I keep getting syntax error. Remember, if you look at the answer, they only want you to work out everything on the right hand side. And since you are given a 95% confidence interval, your z alpha divided by 2 is 1,96. I told you that they like using 96%, 95%. So, you must now know that at 95% confidence interval, confidence level, your critical value is 1.96. So, let's start with the P. It's 40 over 200. What did you get? What is your P? 0.2. So, 0.2 plus or minus 1.96, which is our critical value, times the square root of 0.2 times the square root of 0.2 times 1 minus 0.2 divided by 200. So, do what is inside the square root, multiplied by 1.96. What do you get? 0.028. 1.96 multiplied by the square root of the values that you get underneath the square root. I'm just calculating quickly. 1 minus 0.2 divided by 200 is 0.028. 55. 55. Sorry, I was at I first. Yes, and then I'm multiplied by the 1.96. Oh, sorry. Yeah, sorry. So, it was a 0.028 and then times the 1.96 is a 0.05. Okay. So, this is 0.028 multiplied by 1.96. Okay. Yeah. All right. So, let's do it that way then. 1.96 is 0.028, which then should give, leave it to 3 decimals, okay, to plus or minus. And when you do that, 0.055. 0.055 to 3 decimals. And that is option number one. I think this will be our last one, and then the rest will discuss tomorrow. And this is confidence in, sorry, hypothesis testing. So, already I can see you on the question it asks about hypothesis. So, this is hypothesis. And, oh, before we move to the hypothesis, if I can go back one up to show you. So, for the sampling distribution, we had two questions, except for the other one, which was an error. And for confidence intervals, we had three questions. So, we have one, two, three, yes, we had three questions for confidence interval. The first one was asking, so this is also part of the confidence intervals. It was asking about the critical value. And then the other one was asking about the sample standard, the, sorry, the probability, oh, not the probability, the confidence intervals themselves. And then we had one now that asks about the proportion. So, confidence interval, roughly you will get three questions. And you might also get three questions on the hypothesis testing, because it also has three, where the population standard deviation is known, when it's unknown, and for the proportion. So, now, this is for the mean, because the hypothesis that they gave, they stage the mean in there. Remember, the sign in your alternative tells you whether you're doing a one-tail test or a two-tail test. It also tells you where your region of rejection will be. It also tells you how you will find your region, your critical value, which also determine your region of rejection. Okay, so, since this is a hypothesis testing, I'm not going to say the six steps, because this question does not ask you to follow the six steps. So, they give you the hypothesis, the null hypothesis and the alternative. And we can see that it's a two-tail test, because it's not equal in your alternative hypothesis. And they also give you the population standard deviation. So, yeah, the population standard deviation is known. They also give you your mean x bar, which is your sample mean. And they give you your sample size and the alpha. And then they say, suppose that the test statistic was calculated, and they found that that test statistic is 1.96. So, it means they have calculated z, and they found that z is 1.96. The question they're asking you is for the p-value. Now, two things you need to remember. When finding the p-value is the value inside the table. Remember, the table contains the less than the values. If we're doing a two-tail test, it means we're having two areas of rejection. Therefore, it means our p-value. To find the p-value, we will have to multiply the value we find on the table by two. That's one. Number two, if the value of z, if z is positive, if z is positive, we're going to say one minus the value we find on the table. And once we have the value we find on the table, that value will replace the table value that we have inside there. Only for when z is positive, if z is negative, it's fine. The value you see on the table is the table value, and then we start that, what do you call that thing? Then we multiply by two because of two-tail data. So now, how do we find the p-value? We go find the value on the table, which is 1.96. We know that 1.96. Oh, sorry. We go to the positive side of the table, of the z table. We go find 1.9, and then we go to the top to look for the six, and the value we find on the table is 0.9750. Come here, we say one minus 0.9750, and it gives us the answer of 1 minus 0.9750 gives us 0.0250. So we're going back to number one and say two times 0.0250, and this will give us the p-value of 0.05, which is option number B. Do you understand what I just did? Yes, I do. If I can repeat this, let me repeat it, but repeat it in a vice versa manner. Let's do this, can delete all these values. So they gave us our z is 1.96. Since z is positive, number one, when z is positive, when z is positive, we go into, say, one minus the value we find on the table, especially when we're doing a two-tail test. And then step number two, to find the p-value, we go into, say, two times our one minus the table value that we have. So we go this way, one minus the table value that we have. So we know that z is 1.96, we go to the z-table, we go find 1.96 there, and we find our p-value there, which is 0.9750. So we'll say 1 minus 0.9750, which then this gives us 0.0250. We're going to say two times, because that is one minus the value we find on the table, which is 0.0250. That is the value we found on the table. And this gives us 0.05, which is option number B. And this is only for a two-tail test. If it's a one-tail test, the value you find on the table, it will be the value you see on the table, regardless of whether it's in the positive or the negative side of the table. Only this is applicable when we have a two-tail, because when we have a two-tail test, we are interested in only those small areas there. And both combined areas of 1.96, they give us 0.250, 0.250, 0.520. But when we combine both of them, they give you the p-value. This will be a 0.0250, and this side also will be 0.0250. But we need to combine them, because the p-value is the one-value. So we combine both of them, and that is why we multiply by two. Okay, we have eight minutes. Let me see how many questions are here. We have one. Since they are numbered, I know that one was eight, 10, 11, 12. So we have eight, nine, two, four questions that are remaining. So we'll do them tomorrow. The four questions. And since this is hypothesis, we'll start with the hypothesis again. The next question is on hypothesis. And we can end here today, and then tomorrow we'll start with another question of the hypothesis, and then we move into the chi-square test, and then we do the regression. Any question? Any comment? Noa, thank you for your time. Thank you very much. I will post today and tomorrow's videos tomorrow. If you want to watch them, or maybe I can do it today. No, I'll do them tomorrow, both of them. Thank you for coming through. I enjoyed the rest of your evening and see you in the afternoon tomorrow. Thank you so much. Have a good night. Thank you for all the help. Good night. Goodbye.