 Okay, so let's continue with hypothesis testing for today. We're going to do a quick recap and then we going to look at what we're supposed to be covering today, which is hypothesis testing for the mean when the population standard deviation is unknown. It's going to be a short short session, I hope. Okay, so just to recap so that we can remember some of the concepts that we've learned on Wednesday. We looked at what hypothesis testing is, which is the claim that the researcher wants to make. We also learned that with hypothesis testing you get two sides of the story. We've got the null hypothesis, which is the actual statement that the researcher wants to claim or to prove. And we always know that, or we need to always know that in your null hypothesis, we always use the population parameter. And when we make decisions, we refer back to the null hypothesis. And the null hypothesis can be rejected or cannot be rejected. The alternative, which is our most important side, not that the null hypothesis is not important, but yeah, what we do with the alternative hypothesis is very important because alternative hypothesis, which is the opposite of the null hypothesis, should not have an equal sign. We use this to make decisions because the sign that is underneath all the sign that links the null hypothesis and the alternative. The sign in the alternative helps us in a way. One, to know how we're going to make the decision. Two, how do we find the critical value? So that sign is very important. For not equal, it refers to two take. So it means two decisions. Then it means when we find the critical value, it's alpha divided by two. When it's less than or greater than, then it is one take. Then it is one decision, side, rejection area, whether on the left or on the right, but there is only one decision area. Then it means when we find the critical value, we just use alpha, not alpha divided by two, but just alpha. Now, we also let that, we can also make what we call, when we make decision, we can make a or commit a type one error or a type two error. A type one error is when we reject the null hypothesis, when we reject the null hypothesis, the true null hypothesis, we reject the statement that the researcher wants to prove. The type two error is when we fail to reject the false hypothesis, which means the researcher wanted to prove something, but because we could not put it in that null hypothesis, we put it in the alternative. Therefore, when we make a decision and do not reject the null hypothesis, therefore we are rejecting the false hypothesis. We also learn that with hypothesis testing for the mean, when the population standard deviation is known, which is what we did on Wednesday, we use the Z test. Today, we're going to look at when the population standard deviation is unknown and we're going to use the T test. That is the T test and which is the same as your sampling distribution Z value. When we make a decision in terms of two-tail test, we have two rejection areas and anything in between, we do not reject anything that falls in the rejection area, we reject the null hypothesis. One-tail area is one side of the rejection area and we need to also remember that with a null hypothesis for a one-tailed area, it doesn't matter whether we have a sign which is a less than or equal or whether we have an equal sign because the null hypothesis should always have an equal sign, so it doesn't matter what sign we put there as long as it's a equal sign. But you cannot just because I say it doesn't matter there and put there a less than or equal and then also in the alternative put less than or equal, then it means you do not understand the whole concept. Okay, then we also learned that there are six steps of hypothesis testing. The first step, you need to state the null hypothesis and alternative hypothesis step number two, you need to state what you are given in terms of your n, your alpha, your standard deviation, whatever the things that they are given to you in in the statement. Step number three, you need, oh sorry, in step number two, you also need to state what you are given in relation to your sigma because in step number three, it will tell you whether the population standard deviation is given or not given, you need to state the appropriate test statistic that you're going to apply. Then step number four, once you have, once you know what test statistic you're going to apply, then you need to find the critical values so that you can have the two region of rejections and then do the calculation to calculate your test statistic that you have identified in step number three and the last step is our favorite step when we make the decision by taking your critical value and your test statistic, comparing the two and see where your test statistics falls on the rejection area or non-rejection area and make a decision in this instance. And we looked at the examples, I'm not going to go into details about the example, I'm going to look at the hypothesis testing approach when we use the p-value. So when we use the p-value, when we make a decision, if the p-value is less than the alpha, we reject the null hypothesis. There are also similar steps, but in these steps we do not use the critical value so step number four now becomes where you do the calculations and we did this. So we're going to continue on that note with the hypothesis testing for the mean when the population standard deviation is unknown. That is what we're going to be looking at today. And on Wednesday we're going to look at hypothesis testing for the proportion. When we do hypothesis testing for the mean, like we did with confidence intervals, you remember when the population standard deviation is unknown, then we go and find the critical value on the t table by using alpha divided by two and the degrees of freedom. And our degrees of freedom is n minus one. So similar to this, we're going to apply the same concept. Our test statistic comes from the sampling distribution z-score value or t-value. When the population standard deviation is unknown, we use a t and since we use a t, therefore it means they have given us the sample standard deviation that is the only difference between the z and the t distribution. Also with the t values or when the population standard deviation is unknown, we are not going to do p-value. You cannot find the p-value, it's very complex. You can only find it if you use a statistical program. So we're not going to use p-value to make decisions when it comes to the hypothesis testing for the mean when the population standard deviation is unknown or when we're using a t. Okay, so the steps, we know them. First step, state null hypothesis and alternative hypothesis. Step two, state what you are given. Step three, determine the appropriate test statistic. Step four, find the critical value. Step five, calculate the test statistic. Step six, make a decision. So you need to know all those steps because in the exam, all these steps can be an option in one of the answers that you need or all the questions that you get in the exam might be one question after the other based on how you answer the hypothesis testing. You'll never know, so you need to know all six steps. So let's look at an example and here we're going to use a two tail test. So with this example, the average cost of a hotel in New York is said to be 168 per night. To determine if this is a true, a random sample of 25 hotels is taken and the resultant in the mean of 172 rent, 50 rent, and the standard deviation of 15 rent. Determine the appropriate hypothesis testing at alpha 0.05. Step number one, state your null hypothesis and your alternative hypothesis. So since the researcher here wants to, the researcher statement claims that the average cost of a hotel room is 168 per night. Here they didn't say it's less than or it's greater than or it's more than or it's more like or it's nothing like that. So yeah, we're talking about the equal. So it's equal. So the room is equal. It's exactly. So since it is equal, therefore our null hypothesis will be the mean is equals to 168, the alternative will say the opposite. It is not equals to 168. So to answer this question, step by step, six steps. Step number one, we have already done that. State the null hypothesis and alternative hypothesis. Step number two, state what you are given in relation to the question, everything that we are given. So we remember what we are given here. We wrote it down somewhere so that we can remember that. So the alpha of 0.05 we're given and is equals to 25 we're given the sample statistic. And they said, let me go there. Our S is 1540. So they give us our S of 1540. So already here, I can also calculate my degrees of freedom because we're going to use it later on. So my degrees of freedom is n minus one, which is 24. And step number three, what type of a test statistic am I given? So since I am not given standard population standard deviation, I'm given S, which is my sample standard deviation. Therefore I'm going to be using a tree test or a test statistic. Find the critical value. Step number four, critical value. We find the critical value by using t alpha divided by two and the degrees of freedom. So we were told alpha is 0.05. So alpha divided by two. So some of these things I'm not going to go to the table as always, but I'm going to show you this one now. Unless if you still remember how to find the critical value. So alpha divided by two will give us 0.05 divided by two, which is 0.025. So that is alpha divided by two. So if you go to the table, the t table, so we go to the t table and we look at the top for 0.025 at the top. And on the side we look for the degrees of freedom, which we will find as 24 where they both meet. There we will find that it is 2.0639. We will find that the critical value is 2.0639, which is that critical value that we have there. So once we have that, that is step number four. Step number five states that we need to calculate the test statistic, calculate the test statistic that we have identified. And our test statistic is the sample mean minus the population mean divided by the standard error, which is the sample standard deviation divided by the square root of n. So our sample standard deviation is 172.50 was given to us minus 168, which is our population mean, which is always stated in the hypothesis testing, divided by the standard error of the population standard deviation of 15.40 divided by the square root of the sample size, which is n, which is 25. Calculating the whole thing, you find 1.46. So now I have my critical value, I can go and define my region of rejection because it is a two-sided test. It's not a one-sided, it's a two-sided, so therefore I have got two region of rejection plus or minus 2.0639. And I'm going to use that to locate my 1.4. So I create my region of rejections, it falls in the white area, it falls in the do not reject area. And then I can make my decision and conclude that we do not reject the null hypothesis at alpha 0.05 because there is insufficient evidence that the true mean cost of different from 168. And that's how you do a T-test. If we look at one tail test, and yeah, we're going to look at the upper tail. So yeah, the researcher says, a phone industry manager thinks that the customer monthly cell phone bills have increased, and now it averages over, over 52 run per month. The company wishes to test this claim. So since keywords are very important when you read the statement, because it says over, so it means the researcher wants to prove that it is more than. Since this statement is more than, we cannot put more than in our null hypothesis. Therefore, it means if we're going to fail to reject this statement, then we are committing it to type two error. So we're going to put what the researcher wants to prove in their alternative hypothesis. And that is why it appears under alternative hypothesis, not on the null hypothesis. And in the null hypothesis, we can do less than or equal, or we can just put the equal. It will still mean running the same thing. Okay, so since we have our null hypothesis and our alternative hypothesis, step number two has to go find state what we are given. And also step number three was to identify what kind of a test is this. We know that we're going to do T-test. Let's identify what we are given, and then go find the region of rejection. So what we're given is alpha of zero comma zero one. And now, very important, this is a one-tailed test. Therefore, it means the critical value that we're going to find is only going to be alpha, T alpha, and the degrees of freedom. No division by two, because this is only one site, because we're only going to find the rejection area on that one upper site. So our alpha is zero comma zero one. So we're going to go to the table, look for zero comma zero, zero comma one zero on the T table. Remember, you go to the T table, zero comma one zero T table on the T table. You find that and you're going to go and find the degrees of freedom. And our degrees of freedom in this instance is our n minus one, which is 25 minus one, which is equals to 24. You go find 24 on this way they both meet at that point. You will find that it's one comma three one eight. And that is your rejection area. You identify that. Anything that falls above that area, we're going to reject. That's what that statement says. So anything that falls here, we reject. Now let's go calculate our test statistic. We know that we're using a T test, so calculating, substitute the values. We were given what the mean is. Our population mean was given in the question. It was 52, remember. And our standard deviation is 10. Our square root of n is 25. Substitute into the formula calculate, we get zero comma five five. Now let's go make the decision. Remember we got the test statistic of zero comma five five. Remember that zero comma five five. We need to take this and we need to take this critical value region and go and locate where zero comma five five is zero comma five five falls in the do not reject area. So we do not reject another hypothesis. Since our test statistic of zero comma five five, it is less than our critical value of one comma three one eight. There is no sufficient evidence that the bill mean is over 52 rents. Any questions? Can you guys hear me? No questions for now. Oh, okay. I thought there for a minute. You guys don't respond to me. If there are no questions, just say no questions so that I can feel that there are people here with me. If we were in a class, it would have been much much better because by now I would be looking at your faces and seeing all your confusions and or whether everything is overwhelming. I will be looking at you and be feeling like, oh, you want me to repeat that? But now I can. So communicate with me too, please. Okay, so let's look at more activities or exercises. Like I said, today's session is going to be very quick because most of the things you already know is just to emphasize on the on this area that you need to know how to identify the questions and know what you need to do in that. So let's look at this question. It says the security department, I removed all the other options that were there actually because I wanted to do this as clean as possible. And we're going to do this together because this is for everybody. I will need you to also open your table so that you can go to the tables. You will do all the calculations. I'm not doing any calculations. So I need your help here. So we're going to use all the six steps. So step number one, state your null hypothesis and your alternative hypothesis. So reading the whole question, which I am lazy now to read. I'm going to read it. The security department of a warehouse wants to determine whether the average time required by the night men or night watch men to walk around is 12 minutes, which is the average time to do the rounds. In a random sample of 24, the night watch men average is 12.57 minutes with a standard deviation of 1.23 minutes. The null hypothesis, which H naught equals, the mean equals 12 minutes is tested against the alternative, which states that the mean is more than 12 minutes. Assume the distribution is normal, test the hypothesis at 95 percent confidence. So this statement makes it easier because for step number one, they have stated the null hypothesis and the alternative hypothesis. I don't have to state it again because they have stated them. Or I can say here it is a one-tail test because of the sign in the alternative hypothesis. So since I know that it's a one-tail test, makes things easier. Step number two, state what you are given. What am I given? What are we given? So reading the question, you need to be very, very careful because in the exam you will not have me telling you all this because you will have one question after the other after the other. You will need to be able to read the question and identify that they did not give you the population standard deviation, but they gave you the sample standard deviation. If you read this sentence, it starts with in a random sample, therefore it means everything that follows belongs to that sample. In a random sample of 24, the night watch man averages, so therefore they have given me my sample size. Let's start with the sample size because that's the first thing that they gave me, my sample size of 24. The average which because it comes from the sample, it means my mean sample mean is 12, sorry about that, it is 12.57, and they say with standard deviation. So therefore the standard deviation also comes from that sample. So my standard deviation S is 1.23. If they would have said with the population standard deviation of this, then it means they would have given you the population standard deviation. So therefore here we say the population standard deviation is unknown. So when the population standard deviation is unknown, we know that step number three, we know that we're going to use a T test or a T starch steak. Step number four means we need to go find the critical value. So you're going to help me find the critical value because I need T alpha divided by two and the degrees of freedom which is n minus one. Our 95 percent confidence interval, our alpha is zero comma zero five, correct? Is it not T alpha only because it's a one-sided test, sorry. Thank you very much. You are right because we're not going to find T alpha divided by two, we're only going to use T alpha. Thank you. It's a one-tail test. So therefore our T will be, our alpha is zero comma zero five, and our degrees of freedom which is 24 minus one, then our T of zero comma zero five, and you need to go find that critical value. Which you must use zero comma zero five. Go find zero comma zero five at the top and then find 24 or not 24 but 23 on the side. 23 and tell us where they both meet. And remember you go to the T table. I'm waiting. 1.717. In flow motion. 1.701. 1.714. Is it the amazing and old question paper that I found? Is it the table for T distribution? Yes, T distribution or T critical values of T. So it can be critical values of T or T distribution. It's on mine, it's 2,807. It's not the same. I think it's 28. Hi, wait. Let me also go check on my site. So let's see where is my tutorial letter. Sorry. The one that I have here doesn't have. Oh no, sorry. It's my apologies. I was looking at zero point zero zero five, not zero five. All right. So you have found where you went wrong. Okay. So the critical value is 1 comma? 714. 1 comma 714. There you go. So that is our critical value. Now move to step number five. Since we have the critical value, we can draw. I'm going to use this site to make my decision. I don't have to write the whole sentence, but I can just use this. So where is the region of rejection? It says it's greater than. Is it on the left or on the right? Left or right? Greater than is to the right. It will be on the right. And this will be your region of rejection. So our critical value at this point is 1 comma 714. That's our region of rejection. So anything that falls this site going to reject the null hypothesis. That's for later on. We're not making any decisions because we need to first calculate our T statistic. So our mean is 1, 2.57 minus our population mean is always the one that is stated in there or in the null hypothesis, which is 12 divided by our standard deviation, 1.23 divided by the square root of our N, which is 24. Calculate and give me the answer. I'll get 2.2702. Let's write it with another color. Just give me a sec. You get 2.2702. So we can leave it at two decimals or one decimal in diameter at this point. So where does it fall? It falls somewhere in the rejection area somewhere there. So it means since it says it is greater than that, so it will be somewhere in that rejection area. And therefore we reject the null hypothesis. So our answer here, our decision will be we reject the null hypothesis at alpha 0 comma 05 or at 95% confidence level or at 5% level of significance. Is there any questions? Anyone who is the lost? Speak now. Okay? If there are no questions, so we're going to move to exercises. You have, I'm not going to give you a lot of time on this one. You have three minutes to read the question and remember you can also use the chat function to post your answer. Let's see. Let's use the chat function. Are we doing? I will talk for myself. Confused. Are we meant to be calculating anything? No, it's not meant to be calculating anything. We just used the information given and stayed there. Option two, if we're not calculating anything. Others? I also agree with option two, even I'm not sure. Okay, others? Quiet. Okay, so you've read the statement, the only thing I'm going to read is just the end part, which says that the pH of the swimming pool is greater than 7.5. The null and the alternative hypothesis will be. Number one, what you need to always remember is you can use a process of elimination when you look at questions in the exam or assignment as well. Remember that your hypothesis testing, your null and the alternative hypothesis, we always stated with regards to or with respect to the population parameter. So looking at this, everywhere where I see the sample me, I will know that that is incorrect, that is incorrect. Process of elimination. The other thing you need to also remember is in your null hypothesis, there should always be an equality sign. There should be an equality sign, so question number two, there is an equal sign that should stay. Question number three, there's no equality sign, therefore question number three is out. We already eliminated question number four. Question number five, there is an equal sign or there is at least an equality sign in there. Now the only thing that is left is for us to look at the two statements with regards or in relation to the main question that was or the statement given. So the statement set is greater than, so greater than means the statement should look like that. So if the statement should be greater than, this is less than, therefore this is incorrect. And then they will have your answer. There you will have your answer. So option number two is the correct one. So you can either apply the process of elimination or you can actually go to the question where it says it's greater than and look at only the statement that has a greater than because this would have not been correct. This would have not been correct. This would have not been correct because it says greater than and this would have not been correct. And that would have saved you a whole lot of trouble already by just looking at the alternative statement which says greater than because you don't care about the other one. So if you get multiple, then you do the process of elimination there as well. So you could have done it that way. As you can see that all statement would have been incorrect, incorrect, incorrect, incorrect because of just the alternative sign which would have had the greater than. Anyway, we move on to the next question. You have five minutes. A study was done on a daily cash balances of a bank to investigate the hypothesis that the average cash balance did not exceed 25,000. The sample of 100 days yielded a mean of 24,920 and a standard deviation of 300. A hypothesis test with a level of significance of 0,05 was done. The critical value in this case is so you need to find the critical value. Remember the critical value we find it by using t. I'm not sure whether you're doing a two-tail or a one-tail. So I'm going to write both of them because I don't want to give you the answer. Or are you doing a one-tail? It is very important to know the difference. Any questions? Are you done? Do we have an answer? Mama said I give you a little bit time. I got option four. Are we all happy? Yes. Am I supposed to be getting 1660? You're using the table with three decibels, right? Oh, yeah. Yeah, then it's fine. Okay. So are we doing a one-tail or are we doing a two-tail? One-tail. Two-tail. Okay. I hear two answers. Let's defend our arguments. Our answers. Those two said they are doing a one-tail. Why are you saying you're doing a one-tail? I'm saying we're doing a single tail because the claim is that the average cash balance did not exceed 25,000, therefore meaning that it is less than or equal to 25,000. So it means our null hypothesis states that since this is about the average mean, so it will state that the mean is less than or equals to 25,000. Those who said they are doing a two-tail, sorry, you said you're doing, you answered the question for the one-tail, isn't it? The two-tails, why are you saying it's a two-tail? I was assuming if you use less than or equal to, since less than or equal to 25,000, equal sign I must use a two-tail, but I'm wrong. Okay. So when you read the questions, you need to make sure that you understand what exactly are they asking as well and also apply the site. Like we did when we were doing the discrete probabilities, remember, I said sometimes they will give you the weights. You need to convert the weights to a mathematical function or a mathematical symbol. So this is one of those cases. So it did not exceed, it means it's not more than, it's not bigger than, so it means it's less than or equal. So since it did not exceed is a less than, therefore it means the alternative, the alternative will be, it is more than. So it will be more than that. It would have, the alternative will say it exceeds. So when it exceeds, it will be greater than 25,000. So since this is a one-tail test, therefore we're not going to use the critical values. We're not going to find the critical value there. So we're going to find the critical value by using T alpha divided by two. So what is our alpha? Our alpha is T0 comma 05. What is our degrees of freedom? N minus 1. So it will be 100 minus 1, which therefore will give us T0 comma 05 and 99. And the answer would be option number 4 when you go to the table. Any questions? Okay. No questions. Copy and paste. So question 3, 4, and 5, they are the same, exactly the same. So based on this question, question 3, which is the same as this, answer question 5. So if I take the same statement, discard, because it's a picture that is the reason why I couldn't copy and paste. I just want to bring it to the same question so that you have everything on one slide. Sorry about that. That will be the last question. Otherwise, let's bring this here, unless if I find another question. There is your exercise. Are you able to see the presentation? Is it shared? Yes, we can see it. Okay. Okay. Remember the six steps of hypothesis testing when you answer the question. Sorry, Miss Liz, can I have the question again? Did it disappear? Yes. Miss Liz, I think it's on slide 32. The answer is slide 32. No, the question is on slide 32. You are on slide 34. Can we see the question again, please? So aren't you seeing the same slide as me? Okay. Exercise 5. Yes, now it is. Thank you. So are you are you seeing exercise 4 or exercise 5? Now we're seeing exercise 4. Okay. Are you winning? Miss Liz? Yes. The question says which statement is incorrect. We don't even have to calculate because the second statement would be that it's greater than and not less than. I think it got less of what we did. We actually interpreted that one incorrectly. Did you do all the calculations? Did you calculate the test statistic as well? I did. Did you go find the? I got, for the test stats, I got negative 2.67. Yeah, so did I. Did you find the critical value? I used the one from the previous question. The one we are now confused is option 4 and 5 of the drawing of the graph now. Okay. So we need to make a decision and therefore it means if we didn't state, sorry, if we didn't state the hypothesis correctly, we will draw the graph incorrectly as well. What if we didn't do this correctly? So what is, what if did not exceed means less than? What if exceeding means greater than or equal? So because if exceeding is less, is less than without the equal, if this represents exceeding, it will be in the alternative. So did not exceed would be less than, which will be in the alternative. So this will be our alternative and our null hypothesis will be greater than or equal. So what did we say? Oh, because I removed the, the marking in our statement then we said it is less than or equal, whereas we should have left it as less than. So this it should say exceed did not exceed is less than. Then it will be in the alternative irregardless because if even if we want to put it in the alternative, it will not be in the alternate because in the, sorry, in the null hypothesis because the null hypothesis has then equality sign to it. So then, then alternative, if it's less than, then it means our area of rejection will be on the site, which the critical value you went and found was one comma six. What did you find? Six, six, zero. That will be our, that will be our origin of rejection. Taking the test statistic, you need to come and make decisions. Your test statistics will fall in the rejection area. So therefore it means this will be, we're going to reject the null hypothesis and the last statement says we fail to reject the null hypothesis. That is, that is this scenario. Yes. Is it possible if maybe we can do the six steps step by step? Yeah, but I'm still trying to figure out where we missed it. Is it in the alternative hypothesis where we missed it in the previous one as well, but it doesn't change the answer we got in number one. What changes is the state, the stating of the alternative and the null hypothesis. So what does did not exceed mean? In our case we said did not exceed is less than or equal, therefore this will be greater than. Then which, if it's that, then when we come to option number five and option number one, because then number one will be incorrect in a sense, because we would have said our alternative is greater than. And when we come and make decisions here at the bottom, we will be using this site. Remember we're using the greater than. So we would have said this is one comma six six zero. And where does this fall? It will fall in there. Do not reject area and therefore number five will be correct because number five says we fail to reject the null hypothesis. So because also number four in this instance number four will also be incorrect. But if we use number one, number four and number five, sorry, number four and number one will be correct. And number five will be the only statement that is incorrect. But if we use number two, number four and number five will be incorrect because they cannot be the same. So what does exit mean? On our, let me just say, I think exit does not have an equality to it because it doesn't say more than or equal. We just need to know what exit mean. This is English converted to meds. I don't know what exit will mean. So let's Google. Let's ask our friend Google. It means it will not be more than. Exceed mathematical symbol. Exceed in meds. Not to exceed. So yeah, we need to know the weights and convert it because I don't think also when we did the, when we did what we call that, when we did the discrete distribution, we didn't deal with exceed. So now the question is asking exceed. So ah, there we go. I found it. I found it. It's great. Don't worry. It's greater than. Exceed means greater than. So exceed does not have an equal sign to it. So we need to be very careful when we, when we do this. Sorry. I want to discard. And then we will do the, I just want to bring it in here. New slide. It's not going to be nice, but at least one thing that you can use to refer back as well in terms of the questions. Since I'm going to email you back the slides. So I just want to also include this in there. So it means exceed. If it says did not exceed, therefore it means it is less than. So did not exceed. It will mean less than because exceed does not include an equal sign to it. Okay. So it means in the previous, the one that we did, it doesn't change the answer, but you just need to correct the hypothesis testing. Okay. So, okay. So let's do step by step as you have requested. Step number one, state null hypothesis and alternative hypothesis. Our null hypothesis will state that they mean because this says it did not exceed. So it cannot come into the null hypothesis. So in the null hypothesis, we can just say it is equals to 25,000. The only important information will go in the alternative, which will state that it does not exceed, which will be less than 25,000. That's step number one, step number two, state what you are given. We're given alpha of zero comma zero five. We're given n of hundred, which we used to go find the degrees of freedom, which is n minus one, which is hundred minus one, which is 99. Then we also used the critical, oh, we need to go find. So step number three, we state that we're using a t test. Step number four, we need to find the critical value t. Since it's exceeding, which is one tail test, like I said, it doesn't change what we did there. So it's alpha and the degrees of freedom still, which it was t alpha of zero comma zero five and 99, which we went and found what was the critical value, one comma six, six, zero four. Step number five, we need to calculate the test statistic. Let's write it some way here. Step number five, so that we can have enough space, which is x bar minus the mean divided by s divided by the square root of n. And I'm going to assume you did all these calculations. Our x bar is with the sample yielding 24. So that will be 24, 920 minus 25,000 divided by standard deviation, sample standard deviation of 300 divided by the sample size of 100. And the answer we get here is how much? Did you get the same as minus 2.7? So I'm going to assume that is the answer. Then we go to step number six, where we make decisions. We draw our thing, our diagram. And because it says it's less than, so therefore our rejection area will be in the less than side. And this is one comma six, six, zero, because it's on the site, it will be negative one comma six, six, zero. Then go to your test statistic. Let's go find our test statistic. It's minus 2.67. It's somewhere in the do not reject. It falls in the rejection area. So therefore we can make it a claim here and say reject the null hypothesis at alpha equals to zero comma zero five. So that is our rejection area. And we can say there is no enough evidence to support or to state that it did not exceed 25,000. Okay, so now let's answer the question. Statement number one, state in the null hypothesis and alternative, that is correct because we've learned that exceeding means greater than, therefore it means did not exceed, it means it will be less than. Number two, we calculated the test statistic. Number three, we used the t table in this case because the population standard deviation is unloaded. Is that correct? Yes, it is correct because it's step number three. We made it conclusion and when we make conclusion, we make it in relation to the first statement, there is no evidence. Yes, that is correct to reach that decision because we reject in the null hypothesis. Step number, step number five, it says we fail to reject the null hypothesis but we are rejecting the null hypothesis so that is incorrect. Sorry, Mr. Leslie. Just ask quickly, the rejection area of one comma six six zero four. Why did we move it to the negative side? We didn't move it to the negative side but it is in the negative side because the sign says less than. Okay, so for hypothesis testing, four hypothesis testing is not like with the other probability like when we were doing the sampling probability where we are able to find, oh, what am I drawing now, where we are able to find the area, this area because we do not have the less than or equal site so we always use the smaller site area so it's always going to be this small area because we use the table and the table contains the small site, the small portion so we either going to use that portion or if it is less than or greater than we will only use that. If it's a two-tail we use both of them so we'll never use the area in between for a hypothesis testing. Okay, so if it's less than therefore it falls in the negative side of the table. If it's positive, if it's greater than it will fall in the positive side of the table. Okay, thank you. Thank you and remember also because at this point your mean is zero so anything that goes this side of the mean, anything this side will be negative that is the other reason. So how you find the hypothesis, the critical value we actually changes, you pointed the right question. So we need to go back and fix our critical value is this. So we need to go and change your answer, your critical value will be in the negative side because your alternative hypothesis that is the other thing we need to remember. So therefore our critical value will be in the negative side of an area so it will be negative one comma six, six, zero, four. So option two is your correct answer. It does change everything. It's very important that you get the signs right because you see one mistake you're going to get everything incorrect so we would have missed the mark here because we would have gotten zero and lost four marks and we would have lost so much on the other. I need to find a better picture. This is very blurred. I'll find another picture which is visible. So we need to be very careful with the alternative side. We need to put the right one, the correct one. This is our last question and then we are done for the day. Then I will see you on Wednesday. We will still have another, I think one more session before you submit your assignment just to recap on everything and do lots of other activities. I'll look for past exam papers. It's very hard to find past exam paper with the T critical value because it takes longer to find the T critical values in order to do the calculation but I'll find more exercises for you on that last session so we'll do multiple activities. You see statements like this are very tricky. No more than. Let's go back to our graph and see if there is a no more than. There is a no more than and you see why I'm saying statements like this are very tricky. Yeah, it says greater than greater than less than equal not equal. Let's go back to our statement. The statement says no more than is less than or equal but a more than is less than. I will find more definitions of this thing because then in my assumption will be no more than we just tend this to a less than as to no more than. No more than. Yeah, I don't know. So I guess the person who also did this statement also applied the same logic as more than is greater than. More than is less than. No more than will mean what am I saying? More than is greater than. It's Saturday. More than is greater than. Then no more than. So which statement is correct? Number five. It will be number five because number three you can see they talk about the proportion. Proportion we're talking about the average time. So number three and four by process of elimination would have been incorrect. Number two. It says not equal. We know that it says no more than so that won't be correct. Then then we are left with two. We know number one says it's more than so that will be incorrect which leaves us with a no more than which is if this was a no more than because a no more than no. No, it's not number five. Gosh, no, it's not number five. Not number five. Not number five. More than is that no more than has an equal sign to it. We need to be very careful. Okay, I think I'm also very my brain is tired and it's killing it. It makes me rush things. So we need to be very careful because this is a hypothesis testing. We need to state the hypothesis testing no more than our null hypothesis has the equality sign to it. No more than will be less than or equals to less than or equals to 40 days. The alternative of what the researcher wants to prove is the mean will be more than because that is the alternative. So therefore, number one will be the correct one because we always need to look at the alternative since so you need to be very careful. Don't rush things state the null hypothesis and the alternative the way they're supposed to be because I think we only concentrated on the alternative than to concentrate on the hypothesis statement, which is that both the null hypothesis and the alternative hypothesis. Yes. So basically what you're saying is the age zero is the true statement and each one is should be the opposite of what that is. Yes. So they can both be less than in this case. Yes. So that we did previously. Because yes. So now the other thing is in terms of the statement given in relation to the statement given we always always have to relate it to the null hypothesis and if that statement does not fit into the null hypothesis, for example, if the statement that is given to us is like a less than or a greater than or anything that does not have an equal sign to it, then it will go into the null hypothesis. Sorry, it will go into the alternative hypothesis and not onto the null hypothesis. But in this instance, our null hypothesis is given to us with an equality sign to it because it says it's no more than which is less than or equal. Therefore, it means this statement goes on to our null hypothesis, not into our alternative. In the previous one, we put it in the alternative because it was exceeding, not exceeding, it was in the less. Yeah, not exceeding, it's in the very confusing. Yeah, this session hypothesis testing signs. Sorry, Ms. Lizzie. Yes. In this question today, computers is more than 40 days. Yes, but remember this statement, it's your null hypothesis. So this no more than, no more than it is less than or equal. Where is the less than or equal thing? I lost my pen now. So this is less than or equal, can fit into the statement, can fit into the null hypothesis. The opposite of this statement, that's what the alternative is. And our opposite is greater than, which will go into your alternative. And remember, your null hypothesis, we can put the equal sign or we can, this can be written the same way as this. Null hypothesis will be the mean equal 40 and the alternative will be the mean no more than, which is greater than 40, which will be the opposite of that. So the null hypothesis, you can write it with an equal sign or with the sign, the original sign it comes with really doesn't make any difference. The only thing that is very important is the one that is in your alternative. So this statement doesn't go into your alternative, it goes into your null hypothesis. And if in your null hypothesis, you cannot have an equal sign, then, sorry, if your null hypothesis, if what the researcher is claiming does not have an equal sign, it cannot be in your null hypothesis. If the researcher wants to claim any of this automatically, it goes into your null hypothesis, your alternative, all this part of your alternative. So in this instance, the researcher wants to claim that no more than, which is greater than, oh, sorry, no more than, which is less than or equal. Therefore, the opposite of it will be greater than. The opposite of no more than is more than. So we need to, yeah. With that, any question will look, I will check other questions when it comes to the proportion for Wednesday so that we can have more questions that relates to this. To the more than, less than different type of questions so that it can give you an idea that you do not have to rush into the answer in the hypothesis or looking only for the alternative, but look at both statements in relation to that. If there are no questions, so we can go home early today. We can go and enjoy our weekend early. Just to recap, what we've done today, even though we are very tired, we looked at hypothesis testing for the mean where the population standard deviation is unknown. And when it's unknown, the critical value, we find it on a t table. And since this is only for t, we do not calculate the p value. When we're making a decision, we only base the decision on the critical value and the z state. And with that, thank you for being present today. I will see you on Wednesday. Enjoy the rest of the weekend unless there is a question.