 Good morning everyone, um. Sure. Jack, I hope you haven't left since you still have little bit of minutes there. There are some comments in the chat before we start with the session. Yes, I've read Lizzy's comments. Maybe you can also repeat because I see that there are some people who just join. Maybe if they can also unmute and talk to you for the next 5 to 10 minutes before we start with the session, I'm going to allow that so that they they can have a discussion with you. Right, so I think some of us, um, you know they be saying that psychologists or future psychologists are actually, um, they they, um, they are linguistically able to voice and not just only to listen, um, so if you want to share some of your thoughts, you can unmute yourself, but when you look at research analytics, right, and the session that Elizabeth Boyce going to present now, why is it important, um, for support to be given at this level, at third-year level? Is there anyone that wants to unmute and just give us two or three sentences? Jacques, it's Lizzy speaking. Hi Lizzy, a very nice response I've read and I will consider that. Thank you. So Jacques, on a third-year level, so I understand the importance of the research, um, statistics and analysis, because if I'm going to do a survey or question, say if I'm going to work in a school environment or I'm going to work in a company, if I'm going to work and I need to do a survey on maybe stress, how stress is related in the workplace or how stress is related. So I would need to do, um, the quantitative, uh, I would need to get the quantitative information, so I can't just, it can't just be by having conversations with people, I need to work out how many people, and so I understand the idea about you have to have the quantitative, um, information as well, not just the qualitative. So if I'm going to do a survey, I need to need to be able to work out mathematically, how many people suffer from stress due to work-related incidents or stuff like that. So that is my view on why we need this, um, research analysis and statistics theory, especially in the third year, because you're coming to an end, so to say. Yeah, yeah. Thank you so much, um, Lizzy, for that. I will really, really consider that and integrate that into my report. You see anyone more, um, Andrea, Simpson, Andrea. Hello, it's Kim speaking, and I just joined it. I didn't hear what you said about the recordings and where we find the links and everything, because I had to go via Ms. Boy, I didn't even know where to get onto the link, so I don't know if I've missed something, but if you could just say that again, please. Okay, thank you so much, Kim, for that. Kim, I was just, um, I'm telling people that, um, you know, at this point we've got around about, um, 58 academic literacy sessions, um, running, and that is apart from the tutorials. Now, um, at this point I think a lot of people confuse the two with one another. Um, tutorials look much more on, in terms of content, um, so, so, um, they would take you through the study guide on certain areas that's difficult, but academic literacies look at the skills needed to jump from the different levels, um, and to have those skills consolidated from one level to another. So, um, when you, when you end, um, this, um, your third year, for example, um, we will have some research workshops on honest level that's not allowed in the tutorial, um, domain. Um, so, so, so, in terms of, um, academic literacies, a lot of people think that they are academic literate, which is right, but we want just to equip people a little bit more with the skills. So, that's why, um, we agreed, as from this year onwards, to, um, to integrate research analytics and we can't call it research psychology or anything like that because then you will borderline what the academics want. So, so, so we just say that we look at the skills in terms of research analytics, um, for psychology. Now, um, now each, um, every, um, second week, Elizabeth is going to present, um, um, some sessions, um, on how to use, um, scientific or whatever calculators, how to work out calculators, how to work out, um, information, qualitative information, and how to read for statistics, especially, um, for the human sciences. Um, but you can also ask some, some examples were to explain, um, um, for you to get a better understanding. At this point with the rain and everything the last two weeks, um, connection has been so bad and some of the recordings failed that we're trying to recover, but we are running a little bit behind on the downloading of, and the uploading of, of the, of the previous sessions. So, if you can just be patient for about two more weeks, then we will start, um, populating the research analytics videos that Elizabeth created, um, online and, um, yeah, so, so then we'll, then we will be able to, um, then you can have a repository where you can forward and, and rewind certain sessions and, and, and just pace yourself in terms of, um, of, of, of what you understand or what you don't understand. And remember we're also going to have, um, um, a group consultations, um, that is going to take place. Um, the group consultations will not take place in next two, two weeks, but very close to the examination. Um, where Elizabeth will have some, some individualized or, group, um, sessions on areas that you still, still find difficulties prior to entering the examination. And, but I will communicate that with all of you closer to the time. Um, Kim, I, I know it was a very lengthy answer, but did you, um, what, did it make sense to some extent? Okay. Um, I'm even more confused, but never mind. I just want to, Is this the stats lecture for 3704? That's what I need to know. Yes. Yes. I'm very, very, um, dinosaur-ish with all this. So, okay, just as long as I know I'm here and that there will be recordings coming up, that's cool. Thank you. And I don't, I don't see any, any links sent to me or any, um, emails about this. I just sort of came about it via Elizabeth. So if you could, I don't know where we must find you. Um, you know, um, Elizabeth, have you shown them the, um, the schedule on Mayonnaise? I will do that. Yeah, I will do that. Okay. Okay. Otherwise, I can also do that. Um, I, I can, uh, okay. I think it was, oh, yeah. Um, let me just say, let, let me just do that part now while you're still here as well. Okay. Okay. Okay. I will, I don't know what's happening with the lobbying. I think it's, oh, it's case. I think it's, it is your settings. You must go back to the setting and check if you said. I will do that. I will do that. But I'm now just admitting people. Um, but, um, but this is a class for PYC 3704. Okay. It is an, um, a little bit of extra, extra ordinary class or, um, an adult class because the link this week, there was, it was faulty. Um, so, so we're just trying to make up, but I will make sure that everyone that's in attendance today will receive the updated schedule. Thank you. There are other people on the WhatsApp group that are asking the same questions as me and I don't know how to get back from this onto the WhatsApp group to tell them. So, if somebody could do that, I am very dinosaur-ish of all of this. So I'm going to stay here now. Thank you. Okay. Okay. So to access the, the same, the schedule or the session plans every two weeks, you will have to go via my UNISA. Um, and I think, um, we will share with you the link as well. So when you go onto my UNISA, uh, you will, through that link, if you go via the link, you don't have to go through this whole process. Um, hmm, my machine is very slow. Um, I don't know why it looks like it's stuck. Um, okay, I still say it's loaded, but you will go through my UNISA and there is, oh, there we go. Um, you will get access to the Western Cape region because we will have given you that link. And if you are already registered student at Western Cape, you will have this link loaded I think on your machine or on your my UNISA site. When you look at the homepage, we are under the numeracy center. So you will click on the numeracy center. I think my browser is small, sorry. Um, on the numeracy center and under the numeracy center, there is the schedule. So every week you can come here and check the schedule for Tuesdays because I think the sessions will be every second week on Tuesdays. You will look under the Tuesday schedule will appear there and there will be a joint session, which is the link to MS team. And there is also the link to the notes and the recording. You don't have to go through that because this is only for the session plan or the schedule. This schedule only reflects per week. So for example, the previous notes and recordings, you will not find them if you only look at this area. You need to scroll down to the bottom. When you scroll down, you will have this section as well where it contains the links to the notes and the recording. So we're looking for the research analytics literacies, which is this link. When you click on it, it should take you to this session. Section of the site where at the top is the resources. If I click on the resources, it will take me to the notes and your recordings will appear at the bottom. And since we only had one session, and this is our second session, so today's session will be loaded here after 48 hours or more, depending on how quickly the UNISA people work in the back end. And if you want to check the notes, since the notes will be uploaded here, so the session plans. So under the session or the schedule, you will find all the schedules, so you can check when is the next session and what topic are we discussing on that week so you can find it there. If I go one, two up, and there are your resources. The summary decision three is the one that we used in the previous session. Remember that session that we had and today's session, which we're going to look at the literacies around hypothesis testing. How do we answer questions relating to the hypothesis testing and how do we formulate the hypothesis testing and make a decision out of it? The notes for today are here. So during the course of the week, I will upload all the notes for all the session because I'm also trying to catch up with other sessions that I do notes for or I do two facilitations for. So I will update this. So we should have all the notes on yeah, regardless of which session we are attending. I will have all the notes so that you don't have to ask for notes. And we can have all the notes well in advance so that you can also prepare for class so that you can come with those questions and make the sessions interactive as well. So I will post all of them. Yeah. So going back remember you go to the Western Cape region site and you will click under the numeracy and that's where you will find all the information. The session plan, the link to the session for that day will be under the join. I have joined the session and other than that, if you need the recording, then you go to the bottom of the page. Any questions because then we are done. Yeah, Elizabeth, I just want to remind everyone to complete the register so that I can just email them anything that they need to know. And because I'm going to use the student number and PCC them in the future. Thank you so much, Elizabeth. Do you be okay to handle Mary-Leave? Okay, then all the best to all the students. If you got any questions or support, just reach out to us. The seat into UT remember that is for learner support. So all of your questions will be placed in my folder and then I will just be able to respond to you. Thank you so much, Elizabeth. No problem, Jack. Enjoy your weekend. I will do so. Thank you. Keep home and keep safe, everyone. Okay, so let's start with the session. So welcome to your research analytics online session. Like I said, the session for July, we will have another session on the 20th, which is Tuesday between 6 and 8. And that session, we will look at hypothesis testing for two samples. So today we're only going to concentrate on one sample when we have one sample size. I'm not going to ask that question right now, so we're just going to dive in. Maybe also to add to what Jack said. Remember this is not about the research module, the site 3704 is about the literacies within the site 3704. So there might be some of the topics or some of the content that I don't dive deep into. And I think we did explain this in the first session that we had to remember the Q&A session that we had to say I'm only going to concentrate on more of the quantitative skills, more of the calculations and things that relates to statistical methodologies within your site 3704. Things that relates to the other part of your site 314, like your psychology content. If you want to understand and unpack those content, I think then we will have to have a separate consultation. Or you can consult your lecture as well with regards to those ones. Because also for me to also structure the the literacies, I need to find the challenging literacies or skills that are challenging within site 3704 and not dwell too much in terms of your content of 3704. So I hope you will understand that we're not doing tutorials. We're not going to go into concept explaining what constraints means, explaining what mediators are and we're not going to do that. So I'm here with an assumption that you know all those discussions, you know all those definitions. We're just here to do some of the technicalities or skills that are relating to answering some of those questions. Okay, so with hypothesis testing, by the end of today's session, you should be able to learn the basic principles of applying hypothesis question hypothesis testing in order for you to be able to answer some of the questions. You should be able to also use the hypothesis testing for the mean, which is what we're doing today to make a decision. And when we talk about the hypothesis testing for the mean, there are two ways that you can calculate or find the hypothesis testing. Sometimes the population standard deviation is given, sometimes the population standard deviation is not given. So you need to know when those times are applicable so that you can use the right formulas, the right calculations as well and make the right decisions as well. So what is hypothesis testing? So with everything, we need to also make sure that everybody is on the same page. Hypothesis testing is one of the statistical branches or technique that we use for one of the branches of statistics, which is inferential statistic. And we know with inferential statistics the parameters that we use to calculate, that we collect from the sample, we use that information to infer back to the population. So with inferential statistics, we make a conclusion about the population using the sample measures. So hypothesis testing is one of those processes that we use. And with hypothesis testing, the researcher most of the time wants to claim something or wants to prove the claim that they have. So they might want to prove that in South Africa, if we take the COVID-19, superspreader events are more likely to have more people or contract the COVID-19. That is the claim that the researcher is making to say when we have superspreader events, there are more likely people to contract COVID. We need to prove that because we cannot take it to phase value. We need to prove that and that is the role of a hypothesis testing. So there should also be another side of that story. It might be that what we need to prove is that the superspreader events are not. So we're going to prove the opposite of what the researcher is claiming that should be the norm. And that is hypothesis testing. So there are several steps that you need to know and learn when you do hypothesis testing as well. So when we do hypothesis testing, like I said, the researcher has a claim, but there should be an alternative to that claim. So there should be a two side to that. And this is also the same as you are innocent until proven guilty. So you have two sides of every story and we know that same. So once you stated your hypothesis testing, which will contain the claim and the alternative of that claim, then you need to also know how you're going to make a decision. What kind of a method you're going to apply to make those decisions. And this is based and this will be based on the things that are given to you to make that claim. For example, like I said, because we're doing a hypothesis testing for the mean, you need to know that your population is normally distributed and you need to know that the population standard deviation is given or not given. And if your population is not normally distributed, you need to know that you are given a huge sample size. So those are the things that you need to know when you're going to make a decision about something because we're going to infer the results that we get back to the population. Once you know all those decision methods that you're going to apply, then you need to calculate, you need to compute, you need to make sure that you create measures that can help you to make or reach those decisions because sometimes a numerical vain can assist. One measure can tell a huge story. And once you have those measures, you can compare it with other measures for example, we're going to calculate what we call the test statistic. Your test statistic is those measures that you calculate. Then you're going to find the p-value which is your probability value. Once we have the p-value from the test statistic, then we can make a decision because we can take the p-value, compare it to the level of significance which is our alpha and make a decision. Or we can take our test statistic, compare it to the critical values and make a decision. And when we make a decision, those decisions will be based on, depending also on the hypothesis testing that you are testing. And we're going to unpack all this in a short bit. So depending on your alternative hypothesis, which is the alternative claim of what the researcher wants to prove, depending on the sign that is located there, what I prefer to, in terms of the sign, we'll go into that later on, will assist with knowing what decisions you need to make, whether you're making a one-directional decision or whether it's a two-directional or two-tailed decisions that you need to make. So the sign that you put in your alternative will tell a lot of, or will assist with making a decision. And we're going to look at that shortly. As well as the p-value and the alpha value or the level of significance will also assist us in making the decision. So since we know the steps now, how do we make that decision? So making the decision, like I said, we can either reject or accept. We do not actually also say accept, but we can only say we reject or not reject. Reason being is we cannot accept something that we're not 100% sure of, but we can reject it or not reject it, but not automatically say we accept it. So the only two words you will use when you make a decision will either be do not reject the null hypothesis or reject the null hypothesis. When we do or when we make a conclusion, we do not rely on the alternative hypothesis, but we make use of the alternative hypothesis to assist us to make a decision. But when we state the decision, we state it in relation to the null hypothesis, because that is the claim that the researcher is making. Okay, well, sometimes let's put it this way. Sometimes the researcher might say they want to claim that it is more than. Okay, so when the researchers in their statement, they say something like more than or less than. So let's say we want to put it as a hypothesis testing statement. So when we do the null hypothesis, which is what the researcher is claiming, we always, always use the population parameters to state the claim. So the researcher wants to prove more than. We cannot put the greater than in the statement of the null hypothesis. The null hypothesis always contains the less than information. So in that instance, therefore, the alternative will contain the researcher statement, and I'm going to explain this just now. So the researcher, let's say they want to claim that more than 30 people or 30 on average. So this now becomes the alternative, but we know that this is the claim that the researcher wants to prove. So therefore, in your null hypothesis, we will say the researcher wants to prove the alternative of that. Can I also ask that in the absence of Jacques, when he's not here and you can see the pop-up comes through of people wanting to be admitted? Can you guys press the admit button if you have or if you can? So then we're going to state that the researcher wants to prove something else. So this we know that this is the false statement and I'm coming to that just now. That is the false statement. But if, so this is number one, remember that. So if the researcher wants to prove that less than take on average, less than 30 people contract COVID-19. So then statement number two, example number two. So this one states that the researcher wants to prove that on average, less than 30 people contract COVID. Then the alternative of that statement will be the mean of greater than 30 people contract COVID. And this is true of what the researcher wants to prove. So when we make a decision in these two statements, we are going to be committing some type of an error. When we make a decision based on those two statements, so let's say for example, statement number one. Statement number one, if we make a decision and we do not reject the null hypothesis, state number one says do not reject the null hypothesis. So if we do not reject the null hypothesis, therefore we are failing to reject the false null hypothesis because this is not what the researcher wanted to prove. So in this instance, we will be committing what we call a type two error. Because in this instance, we need to be committing a type one error because we actually want to reject the null hypothesis and make the alternative true. But because we not rejecting this null hypothesis, we are committing what we call a type two error. With statement number two, where this is what the researcher wants to prove. If we reject this null hypothesis on statement number two, we are committing what we call a type one error. And that brings me to this. A type one error is when we reject a true null hypothesis. If we reject what the researcher is claiming, and that is the case that they stated that they claiming that and we reject it, then we are committing a type one error. And that is the error that for most of the hypothesis testing, when we fail to or when we reject the null hypothesis, we will be committing it. It's a good error to use actually. Because it will give you the correct decisions that you have. And it will actually, it doesn't give you the correct decisions for all the measures. But in terms of decisions, you find that you will get very few people who are incorrectly placed within those categories as well. When we commit what we call a type two error, it is when we are not rejecting, we are failing to reject the false null hypothesis. Therefore, it means if we go back to this, this statement will say we do not reject the null hypothesis. So this one will say we do not reject the null hypothesis. Therefore, we are committing a type two error. It is when we fail to reject the false hypothesis, the false null hypothesis testing. I hope by now you understand a little bit about the hypothesis testing. Now, let's learn how we do the hypothesis testing, the actual hypothesis testing. So to do the hypothesis testing, remember we need to state the null hypothesis. Remember the four steps. First step is to state your null hypothesis and your alternative hypothesis. So your null hypothesis always contains the equality sign. So in the null hypothesis, it doesn't really matter what the sign says. It will always contain the equality sign, whether it is the equal or greater than or equal or less than or equal. The most important statement, it is what happens in your alternative hypothesis. The statement that comes in your alternative hypothesis and then alternative hypothesis, sometimes it's subscript one, it's h subscript one. And yeah, we can state if it is equal, we say it is not equal. Sometimes we use subscript eight, sometimes we use subscript one. So here are the statements for the null, for the alternative hypothesis. So for an alternative hypothesis, remember for the null hypothesis, we can either have any of those, but usually an equal sign surface. In your alternative hypothesis, you need to be very, very careful in terms of the sign that you place there. And please make sure that you are always muted. Elena, please make sure that you muted. And please switch off your videos. Can you please switch off your video as well? Okay, so in your alternative hypothesis, you go into state. If the claim was for equal, then you go into put not equal. And when you would say it is not equal, then we are creating what we call a non directional test or a two tail test, because here we will have two areas where we make decisions. And with hypothesis testing, we always use the small areas. So for example, the area for a two tail test, since this is a normal distribution, the area will be those areas for a two tail test. You go into make the decision based on those two areas. So anything that falls in this area or that area, because you only calculating your test statistic, which is this test statistic, you only calculated once, anything that falls within those two areas, you go into reject your null hypothesis. If it falls this side or that side, you reject the null hypothesis. That is for a two tail or what we call a non directional test, because this has two areas to make a decision for the less than or greater than for these two. Let's start with the one, the greater than. So for the greater than, then the researcher would have wanted to prove that the, the mean is less than. Therefore, in your alternative, you will put the mean is greater than the decision because this is a one tail test. And it is a directional. So always remember for the sign where it says the current that gives you the right, the angle like that, the greater than. So it looks like that. So therefore it means the area will be to the right. So then for the decision, when you go make the decision, once you have calculated your test statistic and when and found the previous and the critical values, you go into use one side of the test. So the area will be to this side or to the right side. And this is one direction or a directional or upper tail test for a less than for a less than. Therefore, the null hypothesis would have been that the researcher wants to prove a greater than or equal. Therefore, for a less than, the rejection area would have been on the left side. So this will be your rejection area. And this is also a one tail test or a one directional test or a left or lower tail test or a left test. You can take many definitions of it. When we do the hypothesis testing, you need to remember to state your null hypothesis and your alternative hypothesis because they will help you in terms of making a decision. You also need to state what you are given in terms of whether are you given the population me. Step number two is to state what you are given so that you know what test statistics you're going to be calculating. So in this instance, because we're using a Z, therefore the population standard deviation should be known. So it means they would have given us Sigma or they would have given you and told you that the population has a standard deviation of this much or they would have said the population standard deviation is. And then you can use your test statistics, which is the Z score or the Z value of your sampling distribution formula, which is your sample mean minus your population mean divided by the standard error, which is the standard deviation of a population. So is the population standard deviation divided by your sigma. And remember, this is for the mean when the population standard deviation is known. So we know that this is Sigma bar, which is our standard error, is the same as our Sigma divided by the square root of n. It doesn't mean when you're answering questions on hypothesis testing, they cannot ask you questions around the sampling distribution things that you know, which includes also the standard error. So you need to know that the standard error for the population mean, it is the value underneath the line there, which is your population standard deviation divided by the square root of n. And we're going to do some examples and then we can look at how we apply all this. And once you have your test statistics, then you can, you will know which test you are doing, then you can go and find the p-vane. So the next slide, I'm just explaining the same thing that I just did with you in terms of the region of rejection in terms of the one-tail test. A way it is greater than, less than on your alternative hypothesis and for a two-tail test in terms of where your region of rejection will be. So once we have calculated the Z test, remember we calculated that Z test, our Z statistic, then we need to go and make a decision. Making a decision, we need to use the p-vane, which is our probability value. So we're going to take the Z test. We're going to take the value we find on the Z formula once we have calculated, let's say our mean minus the population mean divided by the standard deviation over the square root of n and we calculated this and we found that it's 1.13. We're going to take this value. We're going to go to the table, the normal distribution table. We're going to look at the one, sorry, not the one, but the small portion, the small portion size. We're going to look at the small portion size. What is the value that we are looking for in terms of the p-value? So we'll read to 1.13 and go there and find the value. And once we have the value of our p-value on the one-sided test, then we can make a decision. If the value of our p-value is less than or equals to alpha, then we reject the null hypothesis. If the value of p-value, if it's less than or equals to alpha, then we're going to reject the null hypothesis. Otherwise, we do not have to reject the null hypothesis if it's greater than the alpha value. We do not reject. So it means if the p-value is less, we reject. If it's small, it must go. The smaller the p-value or equal, small or equal, it must go. We reject the null hypothesis. We reject the claim that the researcher is making. On the same aspect, maybe I should have put this before when we were still discussing the test statistic. We need to also remember that also in terms of the z-value or the test statistic, which is this test statistic that you calculated, you must always, always remember and bear in mind the effect size. The effect size is based on the value of your n, and this will tell you how big or small your z-score will be. This is what we also use to determine the sensitivity or the power of a statistical test. The bigger the sample size, the closer your test will be significant. And the smaller your sample size, the chances are your p-value significance will be very big, and you will end up not rejecting the null hypothesis. With that, we just covered when we are given the population standard deviation. The same method will still apply of the hypothesis testing, stating the null hypothesis and alternative hypothesis, stating what you are given because you want to know what test statistic you're going to be using and how you're going to make a decision. If the population standard deviation is not given and your sample size or your population is small because you don't know the value of your population, if your population is so small that you can't even calculate the standard deviation of that population, so therefore it means in the absence of the population standard deviation, they would have given you the sample standard deviation, therefore it means we're going to use the t-statistic formula. The same, the null hypothesis alternative will still work the same, but when we use the critical values, the challenge or the difference then becomes when we go and find the critical values, not the t-values. And in your module, you don't have to worry about the critical value areas at this point, but the same concepts will apply. So in terms of the p-value for your t-test, in your module they made it easy because it's very difficult to find the p-value for t-test, so they will give you the t-value, they will tell you that the p-value for t is this, but you just need to make sure that you know how to make a decision when you're basing it on the p-value as well. And similar, your null hypothesis and alternative will always stays the same. So what have we learned so far? What we've learned so far is if we do hypothesis testing for the mean, then there are two things that can happen. If the population standard deviation is known or given to us, or if the population standard deviation is unknown, therefore it's not given to us, we need to know that if the population standard deviation is known or given, then we're going to use the z alpha mean by divide by the standard error. If it's not known, then we're going to use t, which is alpha b minus the mean divided by the sample standard deviation divided by the square root of n, that we know now. The other thing that we need to know about the hypothesis testing in general is we need to know that we need to state the null hypothesis and the alternative hypothesis. The value in your null hypothesis, if it's equal, then it will be a non-directional hypothesis. If it's greater than or equal, therefore it will be a one directional. If it is less than or equal, then it will also be a one directional. We also know that based on the value of the z, where we can find the p value, if we find the p value, if our p value is less than alpha, we reject the null hypothesis. If we use the critical values, then we can find the region of rejections because we know that this area we reject, that area we reject for a two-tailed, for a one-tailed. If it's less than for a one-tailed, the area that we reject for the one-tailed will be that area, for a less than. If it's for a greater than, the area that we reject for a greater than will be that area. Those are the things that you always need to remember when you do hypothesis testing. Let's look at the example on applying all the steps of hypothesis testing and how we use the table to find the p value. We're going to come to this example that we have. Before I go to the example, do you have any questions? Don't ask me to go to the black screen, it's gone. We won't have that anymore. Any questions? If there are no questions, then you can look at the example. Before we look at the example, I'm going to check the chat. If there's anything there, nothing, no questions in the chat. Please make sure that you complete the register. Those who joined late, just reposted the register in the chat. Please make sure that you complete that. Let's look at this. We need to read the statement and understand what exactly we are given in this statement. Identify the key elements that we will need to assist us in making a decision. Then we will do the calculations and then make a decision. Mabato, the social scientist, took a random sample of 30 adults with autism spectrum disorder ASD and found their reading time to be normally distributed with the sample mean and the sample standard deviation of 90 with per minute and 18 WPM respectively. Putali and Mabato are collaborating to test the hypothesis that the mean reading of the adults with ASD is less than 100. That was supposed to be less than 100. Assume a 5% level of significance. Remember there are four steps of hypothesis testing. In your exams or your assignments, the four steps might be a question on their own. They might ask you that because you're writing multiple choice questions. The hypothesis testing will be a question on its own. A test statistic might be a question on its own. Decision making might be a question on its own. Standard data might be a question on its own. You just need to know the steps and know how to make a decision and how to get there in order for you to be able to answer all the questions. So let's start with the hypothesis testing. So we know that the first step is to state our null hypothesis and our alternative hypothesis. So reading the questions, what are we actually given in terms of to enable us to answer the hypothesis testing question? It says they need to test the hypothesis that the mean reading of adults with ASD is less than 100. So it means we need less than 100. Now here is the case. We know that the researcher wants to prove that it is less than, but we cannot put it in the null hypothesis. Where are we going to put it? We're going to put it in the alternative. So therefore it means our mean reading will be less than 100. Therefore, whatever I put in the null hypothesis does not matter because I can just say it is 100. Or I can use the correct sign and say it is greater than or equals to 100. So in the null hypothesis, you can leave your null hypothesis as equal or greater than. It will not change anything. The most important one or the most important sign is the one that is sitting or located in your alternative hypothesis. So what are we given? So we need to go back to the statements above and state what we are given. Given the sample size, so our n is equals to 30, we are told that found that the reading time is normally distributed with the sample mean and the sample standard deviation. So sample mean, which means x bar, sample mean is x bar. Remember, all the Greek letters will represent the population parameters. All the normal letters of your alphabet will represent your sample statistics. So our sample mean will be 90 because it says sample mean and sample standard deviation of 90 and 18 respectively. So the first one will be the sample mean, the second one will be the sample standard deviation. So since they give us the sample standard deviation and the sample statistic is s or the symbol is s, which is 18. Therefore, it means our population standard deviation is unknown and when the population standard deviation is unknown, then it means we need to find, let's go to step number three, we need to go find our test statistic, which is t, which will be your x bar minus your mean divided by s divided by the square root of l. Before I go calculate this, actually, we need to go back to the top one because what I didn't do there, I needed to also state what type of a test then will this be? This is a directional test and it's also a lower one-tail test or you can say it's a one-directional test or a one-tail test or a lower limit test and so forth. So now let's calculate this. I'm going to show you on my calculator, depending on what type of a calculator you have as well. So let's substitute the values. So here we have 90 minus the population mean, which is your mu, is always going to be stated in your hypothesis testing, which is 100 divided by your standard, sample standard deviation is 18 and your sample size is n divided by the square root of n is 8. So depending on the type of a calculator you have, so since I have a different calculator to use, I'm going to share my entire screen. I'll come back to the presentation shortly because I want to share my calculator. It's fine. So here we have our test statistic. So I'm going to use my calculator. This calculator is a Casio calculator. If you do have Casio which has a fractional things like this, if you don't own one, it doesn't matter. You can use any calculator that you have. As long as it's a scientific calculator and it has square root, always got the powers, you can use that. So on this calculator with the fraction, I can put the entire formula into the equation. So by using my fraction, by using the fraction, I do the first fraction, which is the top, which is x bar minus the mean divided by the standard error. So that will give me my x bar is 90 minus 100. And when I go down with the arrow, when I come to the block at the bottom, I need to do the same as what the block at the bottom, sorry, the values are at the bottom. So there is a fraction. So I'm going back to the fraction and create it as a fraction and say 18. And then take my arrow down, divide by the square root of theta. So on this calculator, I'm able to do that. On those calculators where you do not have a fraction, the first thing you need to do is say 18 divided by the square root of 30. Find the answer, then take the answer you get at the top, divide by the answer you get at the bottom. Do not try and do everything all at once on your calculator. It will not work. Say equal, and when I get the answer that looks like this, then I can just take it to... Sorry, my boy. Yes. I don't know if you're trying to show us your calculator, but it's coming up on the screen. Is the calculator not shared? No, no. Can you see this? There it is now, yeah. Oh, sorry. I thought I did share the entire script. Okay, so if I show you the formula, the formula is 90 minus 18 divided by 30. And because my answer is in this root function, I want it as a decimal, and that is my answer of minus 3,03. And I can leave it to two decimals. And yeah, the answer is minus 3,0. Let's go back for God's sake. 0,4. Okay, so because this is a T calculation, I cannot do the P value. I cannot go find the P value even if I go to your... Even if I go there, the only table you get is your normal distribution table. So we cannot find the T, the P value for a T distribution. You can only find it when you use a statistical app software. So I'm not going to do the P value on this question. I am also not going to use the critical values because you do not have the table to find the critical value. I'm going to leave it at that because I'm going to assume for your module, you do not get to use the critical values. But let's say, let's assume now, let's assume like they will do the assumptions for you. Let's assume that our P value, let's assume our P value in this instance is 0,03. Let's assume that our P value is 0,03. Now we need to make a decision. So number four, we're going to make a decision because we need to, if this was Z, we would have used the Z value to go to the Z table and go find the P value. So now we don't have the P value. Let's say this is our P value. So we need to make a decision. A decision, so the decision rule, remember the decision rule states that if the P value is less than alpha, we reject the null hypothesis. That is the rule. So let's make that decision now. What is our P value? Our P value is equals to 0,03. And what is our alpha? Our alpha is equals to, they gave us 0,05, 0,05. Is this bigger than, is this greater than or less than? So this will be less than because we know that 0,03 is less than our alpha of 0,05. Therefore we reject the null hypothesis. So we can say we reject the null hypothesis in this instance. So what you have learned from this is you should be able to know how to state your null hypothesis and alternative. What test is that? Is it the directional and undirectional? One of the things that you are given so that they can assist you identify what type of a test you're going to do. Is it a T test or a Z test? And do the calculation to find the test statistic. And once you have the test statistic, you need to go find the P value. In the instances where P value cannot be find automatically for the Z value, you should be able to find the P value. Because when you go to your table, sorry, I've got so many things open. When you come to the Z table, these are your Z values. So let's say our Z was 0, I just want to choose the one that is 0,15. If our Z is 0,15, so we just come to the Z, we're always going to use for hypothesis testing the smaller portion. Because remember, for whichever decision you're making, you're using the smaller side, whether it's the one tape, whether it's only this side or only that side, or it's a true tail where you apply both, we always rely on the smaller portion side. So therefore your P value in this instance will be your P value would have been 0.44. And you will make a decision based on that. So you will say your P value of 0.04, it's less than your, oh, sorry, it's not less than, it is greater than, therefore in this instance, we will not reject the null hypothesis. And that's how you will use the Z values to find your P value and make a decision. Let's look at another example. Before we look at how the questions are asked in your module as well. So here is another example. So in this example, they ask if a sample size of N20 is selected from a normal population, the sample mean of 58, the population standard deviation of 12, suppose that the E-Twitter wants to test the following hypothesis. Your hypothesis is mu is equals to 55 and the hypothesis mu is not equals to 55. So step number one done for us, the null hypothesis and alternative hypothesis are stated. So we do have our null hypothesis, the mean is 55, the null hypothesis or alternative hypothesis, the mean is not equals to 55. What type of test is this? Now, yeah, me and you, we're going to answer this together. What type of test is this? Is it a directional, a non-directional test? Yes, it's a non-directional test. Can I just give me a sec? When my screen is shared like this, it doesn't, I struggle to find my pen. So this is a non-directional test. Step number two, state what you are given. What are we given here? That will help us to answer the questions. You are given N of 20. We are given the sample mean of 58. We are given the population standard deviation, which is sigma of 12. So yeah, population standard deviation is known. Sigma is known and when sigma is known, step number three, it means we need to do the calculation for test statistic, which is the sample mean minus the population mean divided by the population standard deviation divided by the square root of N. Let me go back one step backwards. So sorry, I forgot to mention this here. So this at the bottom, it is the same, you know, because we have the sample standard deviation divided by the square root of N, it is our sampling error, which is s bar. Similar when we have the sigma over the square root of N, this is our sigma x bar, which is our standard error. So which is our standard error. So now let's substitute the values into the formula and calculate our x bar is 58 minus our mean always given in the null hypothesis and alternative, which is 55 divided by our standard error. Can I ask you to do the calculations and we can find the answer? So the standard deviation is 12 divided by the square root of N is 20, not N, but we need to substitute with the right value, which is 20. Can you do the calculation and give me the answer or do you want me to do the calculation? Do you have the answer? The answer here is 118, which will be 1112. So we're going to round it off to two decimals. It's 112 because our z table has only two values. And this is where it becomes very tricky because I need to move between two screens. Are you able to see the table? Yes, yes. So remember our z that we calculated, if I can show you from the calculator. So 58 minus 55 divided by 12 divided by the square root of 20 gives us 1 comma 118. Since we're going to use the z table and the z table contains only or it only has two decimals, so we need to round off our answer to two decimals, which is z is equals to 112. So we need to go to the smaller side. So first we need to find our z of 1. So you have to scroll to the one we're looking for 112, which is and we need to go to the larger portion, sorry the smaller portion, which is 0 comma 1314. Our p value is 0 comma 1314. Okay, so now we need to be very careful as well because this is not a one directional. So what we have found so far, we have found one side. We have found because this is two directional. So it means it has two regions. So even this will fall within two areas as well. So this will be 0 comma 1314 and 0 comma 1314. 14. We'll have to split it into two. So therefore this p value for when it is two sided, it is going to be twice the size of one sided p value. So it's going to include both sides. So therefore for a two side, our p value will be equals two. So we need to add both. So it will be 0.1314 multiplied by two because it's the same as adding both. So it will be 0 comma 268 to 8. Yes, so that will be 0 comma 268. So we need to make a decision now. So when we make a decision, we're not going to use this to make a decision, but we're going to use the p value to make a decision. So let's make the decision. We know what the decision rule says. I cannot find my pen. Let's see if I can find them. I just want to change the color. So the decision, we know that it says p value. If it's less than alpha, we reject the null hypothesis. So what our p value is, our p value is 0.26. I can just leave it at two decimal. Our level of significance, we just take the 5% divide by 100, which is 0 comma 05. So is this less than that? It is greater than. Therefore, we do not reject the null hypothesis or we can say we fail to reject the null hypothesis. We do not reject the null hypothesis. The other one will say we fail to reject the null hypothesis. And that's how you do hypothesis testing. So all the other questions that you will get either in the exam assignment, they will be related to how well you know your steps, how well you know how to unpack the question in order to identify the key things that you need to make the right decision. Like when your population standard deviation is given or is not given, whether you need to use a Z-test or a T-test, whether if it's a one directional test, you're only going, if it was a one directional test, let's say it was a less than, we could have just used 1, 0 comma 1, 3, 1, 4 and made decision based on that only. But when it's two directional, your p-value will be, because it's for both sides, we add or multiply. And that is why with the p-value, you need to know that for a one directional, it's a p-value, it's a two directional p-value divided by two. And if it's a one direction, if it's a two direction, it is one direction multiplied by two. And that's all what you need to know about hypothesis testing. So now for now, let's scan through some of the questions. If we get time, then we can go through some of your past exam papers, but the discussion should not stop here. We can have these discussions, because you are on WhatsApp, we can continue on WhatsApp. So I've picked on Tuesday when we were going to do that, I picked some of the questions and I put them on the slides as well from the exam papers. So we're not going to go through the exam papers. So I started from this one from the 2009, if we can find time, we can do the others. So I used most of the question, they come from this exam paper. You will see that also I just did a Qopien paste. So don't be surprised when you see the blocks. Okay, so let's look at the exercises. So this, this is where you need to discuss with me because then I do not, I have been speaking a lot and now is your time to, to speak, to speak, to speak. A researcher wants to test the hypothesis that the mean depressions go on a, oh sorry, the other thing before I continue. If it's an exam paper or anything, I do put there at the top to show you where did I get that question so that you do not have to ask me that at the latest stage. Where did I find this or where can I find this? So you should be able to identify where what is, where everything is at. A researcher wants to test the hypothesis that the mean depressions go on a depression scale for a patient diagnosed with clinical depression is greater than 120. The statistical hypothesis should be tested is this. They stated the statistical hypothesis. Bear in mind the sign you see on the alternative hypothesis, it's very important. She uses a random sample of n equals to 64 drawn from the population of diagnosed patients and finds that the mean, the sample mean equals 127 and the sample standard deviation s is equals to 24. Which of these values below is the closest to the correct value of s s bar? What is that? What is s s bar? Is your standard error, which is the sample standard deviation divided by the square root of n? So do that calculation and let me know. And if you have any questions, I just want to go back to the screen. What's the answer? Very easy to do because you have, oh sorry, I must fix my pen. You have all the facts in front of you, so you just substitute and calculate. What is your s is 24 divided by our square root of n, which is 64. And if you don't know how to use your calculator and do some of the calculations, please speak now. Do not be left behind. We know that a square root of 64 is 8, so this is 24 divided by 8. And what is 24 divided by 8? Do you have? Three. Is equals to three. And that will be option number two. Option number two. Yeah. The hypothesis testing, when they state that the alternative, the mean is less than 30 is a hypothesis and requires a statistical test. What is this? Directional or non-directional? Directional. So the mean is a directional and requires what type of a test? One tail. Or is it the one tail test? Tail. So which makes that the correct answer. When applying a statistical test, the p value represents the probability of obtaining the, does it represent the probability of obtaining the sample statistic under the alternative? Does it represent the population parameter under the null hypothesis? Does it represent the sample statistic under the null hypothesis? Sample statistic under the null hypothesis. In the null hypothesis, we always use population. We always use population parameter in the null hypothesis. So every way they have the sample statistic, it will not be correct because your null hypothesis always uses, or we always state the null hypothesis using the population parameter. Even though when we do the calculations, we use the sample statistics to assist us to find the value or make decision, but we are facing the area underneath the curve will be for the population parameter because we use the population mean. We know that for a normal distribution, the area underneath the curve is distributed with the mean of zero and the standard deviation of one. It's always, always the population parameter. I know that we didn't cover that now in class, but I just hope that it can also give you another information in terms of when you answer this question as well. So the only option which is correct is number two. Type one error or case when? When do we commit a type one error? When do we commit a type one error? Hi, we committed when the alternative hypothesis is wrongly rejected. You are half right half. I say it's the first one when the null hypothesis is wrongly rejected. Yes, it is the first one. We commit type one error when we reject the true null hypothesis. So it means when we reject the hypothesis wrongly, but it's not purposely, but it just it means that we are rejecting the claim that the researcher is making. This says when we're not rejecting it, so if we're not rejecting it, then it's fine. If we reject the false hypothesis, then we're committing a type two error. So which also you need to also bear in mind that we don't even when we commit a type one error type two error, we don't always also refer to the alternative because when we make decisions, we conclude with or we use the null hypothesis. So when you answer questions like this, do a process of elimination in terms of things that you already know? For example, this should have been an automatic one and then you are left with two statements. Then you have a 50-50 chance of selecting the correct statement. Now I'm bringing you to the probability questions chapter. So then you have the two. But this is because of that not makes it incorrect. So that is not correct. That is not correct. Okay, so next. So this question has a couple of questions linked to it as well. So without reading the whole sentence in a home, we can identify things. So the sample size of 25, she measures the IQ of each using the SAW AIS and the IQ scores of this test are normally distributed for this population with the mean of 100 and the standard deviation of that amount. So suppose she finds that the mean IQ of her sample is this and her standard deviation is that. Which one must she use to do the test statistic? Which is the appropriate test statistics to calculate? And remember today's session is about one sample grugne or one sample size. Remember to do a process of elimination. Which test statistic? All I can say is ignore what is written here in the bracket because that is just additional information to confuse you because it's inside the bracket. It means it's just a it's not to be taken into consideration. It's just yeah. It's just to help with other statements going forward but not for these papers. So how do we answer this question? The first one it says it's for independent two groups. It is the difference between two groups just by reading that. That one's out. This one will have been out automatically. Then we are left with two. We know that we're dealing with one sample size because they only gave us the sample size of 25. Which test statistic? What are we given? Let's start with what are we given? What sample standard deviation or are we given the population standard deviation? That is if we ignore what is inside the bracket. That's why I'm highlighting it like that. Suppose have sample is the mean. Suppose the mean of a sample is this and the standard deviation is that. So it means the sample standard deviation and the mean and the sample mean comes from that sample. So since we are given the sample standard deviation what type of a test are we doing? Miss Boy I went with the Z test because the standard deviation is known. It's just 17. It's given to us. No. What standard deviation is given to us? You need to be very careful. What standard deviation? The sample mean is that oh suppose she finds that the mean of her sample is this and the standard deviation is that. So because this comes from the sample, from the sample therefore we are given S. We are not given the population standard deviation. So this is unknown but we are given S. So when the population standard deviation is unknown we use a T test. So therefore only number two would have been the correct one. Like I said the same statement is going to run multiple. So this is the second one. What are the requirements with regards to the type of statistical test that may be required to interpret the results? So yeah they're asking you whether is this going to be a one-tail test or a two-tail test or there won't be any statistical test required. Read the first sentence. Read the first sentence. That will give you the key in terms of what you need to how are you going to state your hypothesis testing. Are you done reading it? I think it will be a two-tailed statistical test because we are comparing young adults to their peers. Okay. Do you see where I've put the blogging? The list then means from? Yes. Are you still sticking with your two-tailed? I'll say one-tailed because it's less than that less word for me is the signal that we're going to use the one-tailed. That should be the key for you to know whether are you going to do a one-tailed or a two-tailed. So it says less than. So those others are less than their peers. So that tells you that you're doing a one-tailed test. Yeah. So you need those key weights that will help you. If they would have said a research hypothesis that babies born prematurely will somehow be different to their peers, then it would have been a two-tailed. If they say they are more likely, more than their peers, then it would have been a one-tailed with but a, it would have been a one-directional, but a left or sorry, not left right or upper tail area or tail test, something like that. So this patch tells you that it's a one-directional. Okay. So moving to the next question, we left with five minutes and I think let me just before you answer that question, let's see. Oh, so I only had two questions, but if we're not done, doesn't matter. We have the WhatsApp group. You can always ask questions there. We can always carry on the discussion on the WhatsApp group. A researcher wants to test the following hypothesis and they give you the null hypothesis and the alternative hypothesis. On the basis of the data provided, the output from the program indicates that the t-value of t equals to 1.72 was found and they calculated the p-value. So you see here, they used a computer program. They are able to calculate the p-value for a two-tailed test is given as such. So they already, so yeah, they already combined the two values. So it's a two-tailed test p-value. What should the researcher do to evaluate the result of the level of significance at this? Does she have to, the first one, two, three statement is saying she needs to manipulate this. The last one says she needs to make a decision. So what does the researcher needs to do? When I have my p-value and my alpha, what do I need? Step number four. What do we do in step number four of the hypothesis testing? We make decisions. So the question is asking you the same thing. What should the researcher do to evaluate the results at the level of significance alpha of 0.05? So are you saying the number four? Yes. Yes. That's what you need to do. If I have my p-value and my level of significance at this point, there's nothing I need to do. All I need to do here is to compare the value of my p-value and my level of significance and make a decision. I don't have to do the division and multiplication and calculation. I would have done the multiplication division, all those, if they gave me a two-tailed, but the hypothesis was a one-directional test. So if the test here was a less than, let's say, for argument sake, which option you would have chosen? Option number two. You would have chosen option number, nope, it wouldn't have been option number two. You would have chosen option number one, because before you do anything, you would have evaluated, to evaluate this, you need to take the two-tailed test value divided by two, because this is a one-directional and then make a decision. So probably they wouldn't have had to compare the value of p-value, because this is a two-tailed test. You would have taken this two-tailed test in order to create a one-tailed test, you will take your two-tailed test divide by two to find your p-value of a one-tailed test. In order to find a two-tailed test, you take a one-tailed, this is equal, one-tailed p-value and you multiply it by two to find the two-tailed. So those are the things that you need to remember at the beg of your mind when you answer questions like this, because they can be a little bit tricky, but for these papers and for this question alone, they gave you a two-tailed, because it's a two-tailed, they give you a two-tailed test p-value, all you just need to do is take the p-value and the alpha and compare them and make a decision. I'm not going to do this, so you can go and answer this on your own and if you're still lost, you can chat on WhatsApp. Let's continue the discussions there. Just to recap, because I need to jump off this meeting and go to another meeting, which started right now at half past. Just to recap, we learned the hypothesis thesis that you need to know how to state your null hypothesis and alternative. The alternative hypothesis is very important because the sign you put there will tell you what type of a test you're going to run. We learned that how you make a decision and we know that we make a decision based on the null hypothesis, we either can reject or not reject the null hypothesis. We learned how to make a decision based on the p-value that we can calculate the test statistics and find the p-value and make decisions based on that p-value and if the p-value is less than the alpha, we reject the null hypothesis. You also learned that in order for you to calculate that p-value, you needed to calculate the test statistic, but for the t-test, you will not be asked to calculate the p-value, but for the z-test, you can be asked to go find the p-value on the table or they can ask you questions about the p-value. You just take the test statistic, you go to the table, you look for the z-value and you go to the smaller portion to find the p-value. You need to know that if it's a directional test, you just use the p-value as you find it on the table. If it's a non-directional, which is a two-tailed, then you're going to do the value you find on the table, you're going to multiply it by two or you're going to add it to itself and that will give you your p-value and then you make a decision and that's all what you have learned today. So next week or not next week, because next that other week on the 20th, we will look at how we do hypothesis testing for independent groups or when we have two groups. With that, enjoy the rest of the weekend. I apologize, I'm not going to be able to answer questions at this point, because I need to go to another. I've got a tutorial class right now. Thank you, Ms. Bowie. Thank you very much. Appreciate it. Thank you so much. Those who joined late, please remember the register. I'm going to repost it in the chart. Remember the notes are uploaded as well. I will upload the notes for the other sessions so that you can also see what other topics we're going to discuss for the future as well. Otherwise, enjoy the rest of your weekend. Thank you. Bye. Thank you. Bye-bye. Thank you. Bye-bye. Oh, connection. But it was inside. Do you put anything inside? It was on inside. I'm sure we got the ones that were very stopped. It's not record. Yes, it's not record. I've come back here. I'm trying to look for more notes. Where can we find the notes?