Now, 75 coronet monarchs installed by the past administration in Nogun State, the Bikula Mosul, have applauded Governor Dakbo Abiodon for giving independence to judiciary in the state. Addressing journalists in Abel Kutah, the general secretary of the coronet, Obas Oba Tani Moor, for Larry said they will give their unflinching support to the present administration in the state. He however commanded Governor Abiodon for the completion of abundant projects by his predecessor. There are many projects in Nogun State that are not being completed. We are really happy that His Excellency has started working and completed some and up to present is still working on them. We are very much happy to be associated with His Excellency. Find out about fresh news updates.


to Hawaii Together on the Think Tech Hawaii Broadcast Network. I'm your host, Tili Akinah, president of the Grassroot Institute. Hawaii lawmakers are poised to hike the minimum wage, but some say that could actually hurt people it's supposed to help. Now there are many arguments by those who have opposing points of view, but the research is in. The minimum wage mostly hurts the people it's intended to help. That's what we found, and that's the finding of David Newmark, economist and co-director of the Center for Population, Inequality and Policy at the University of California, Irvine, and he's also the co-author of Minimum Wages published by the MIT Breasts. On today's program, I'm going to talk with Newmark about his comprehensive review of evidence on the effect of minimum wages on employment, skills, wages, and income distributions, and also on the longer-term labor market outcomes. But most notably, we're going to take a look at his conclusion that ultimately, minimum wage laws aren't a good policy tool. Please welcome to the program David Newmark. David, thank you so much for joining us today all the way from California. Thanks, Kili. Well, I'm so glad to have you. I've been looking forward to chatting with you. Tell me a little bit first about your own background and how you got into the field of researching minimum wages. It's a long story. It's not a long story. It goes back a long way. It's a little funny though. I was a grad student. I finished in 87, my Ph.D. at Harvard, and sometime must have around 84 when we first started taking our courses in our fields. I actually remember my advisor coming in and saying on the first lecture was kind of, well, what do labor economists do? What might you do research on? And he said, let me tell you one thing you shouldn't waste any time on is the minimum wage. And he plopped down on the table, and it's going to be generation specific, whether you know what I'm talking about, what looked like something as thick as the New York phone book, which used to be like five euros. And it was about that high. It was the comprehensive report by the U.S. Department of Labor by something called the Minimum Wage Study Commission. And they had commissioned papers by everyone under the sun who was at that time probably eight or nine years older than me or more who worked in labor. And he said, look, we know everything. Forget about it. And then my first job was at the Federal Reserve Board. And one of the things that happened when you work there is, you know, they obviously keep you very up on the news, right? Because you're working at the Fed. And we would get these, this thing called the Daily Labor Report, which was a compendium of state laws, court rulings, anything that might affect labor policy. And my co-author, Bill Washer and I noticed that a bunch of states started raising the minimum wages. Previously, we had pretty much only had a national minimum wage. Well, David, you mentioned Bill Washer. He's the co-author of your book, Minimum Wages. How extensively have you researched this topic compared to other people across the country? I'm pretty sure I've written more papers than anybody else. That's right. So we really appreciate that. And we're glad to be able to pick your brain today. Overall, just a banner question I have for you. What are the effects of minimum wage laws? We'll dive into it a little more deeply as we go on, but just in general. Sure. So first of all, let me say, you know, I think my view is a little more nuanced than your introduction. All right. Because this is the point of my research, but something that is often missed by advocates. Well, as frankly, I would say by sort of the blanket opponents. And that is that the effects are heterogeneous. Okay, so obviously the first thing minimum wage laws do, wages for some workers. I mean, that's what the laws do. And, you know, we tend to comply with these things in the US. So wages are pushed up now. As far as the advocates are concerned, you know, end of story. So that's all good. Right. It's a free lunch, apparently. And people who make low wages now earn more. But what the research shows, I think most of the research shows is that there are other consequences. There is some job loss that stems from the minimum wage going up, which doesn't mean that everybody is hurt, right? Because obviously some people still keep their jobs and earn higher wages.  But there are offsetting effects and there are some tradeoffs. So my point is always based on the research that you got to think about these tradeoffs and you can't pretend they don't exist. The other key point, I think that is slightly more subtle, but we can get into it in more detail is that the minimum wage is really blunt, by which I mean not precise, a really blunt tool to try to affect the underlying problem. The underlying problem is low family income. That's what we care about. I don't care if I don't know if you have a teenage son or not, but I don't really care if your teenage son works at a low-wage job. That is not a policy problem that any business owner or taxpayer should be worried about. What I am worried about is a single mother who works at a very low-wage job and how many resources she has available for her and her kids between what she earns at work and what she earns and what she gets from other government programs. But that's a problem of poverty, which we define or low income, which we define at the family level. And the simple fact is, and then I'll stop, is that while obviously being a low-wage worker and being in a low income family are related, they're not very related. And as a result, when you try to increase incomes of low income families by raising the minimum wage, you give a lot of benefits to people who aren't low income families, like the upper middle class teenager. And you don't do anything for the people who don't work very much because you don't get anything for the minimum wage unless you work. I think you've explained it very well that the minimum wage is a very blunt tool and probably the wrong tool for trying to solve the problem of overall low wages. But you spoke about the fact that the effects of minimum wage laws are not homogenous. They're different for different people in different categories. So specifically, how do minimum wage laws affect the poor? So let's just go back to heterogeneous effects for a minute. So in the aggregate, some people's wages are pushed up, and then there's some job loss and some hours reduction as well. We don't actually have great evidence on how the employment effects or the hours effects differ across the income distribution that is for a low wage work in a poor versus non-poor family, which is sort of mystifying because I think we should. It just turns out, and this is a reflection of my point, that low wage work and low income families are by no means the same thing, that when you start slicing the data really finely to divide it by both wage level, it was really affected by a high minimum wage, and very low income families. Unless you have really huge samples, you don't have many people left. But what you can do is say, what does a high minimum wage do to the poverty rate or some other measure of the instance of low income? The poverty rate is one we use. It's not necessarily the ideal one, but it's sort of an agreed upon one. And the answer there is, say, there's been more research since my book, but essentially you can't find any strong evidence, I think, that minimum wages reduce poverty. It's probably largely awash, whereas obviously a policy that is intended to reduce property, the instance of low income, ought to have the effect. I mean, look, if we take TANF, the welfare, the current incarnation of our welfare program in the U.S., in Hawaii as well, and raise the amount of income we gave to families that were eligible for TANF, we would clearly reduce poverty and increase incomes of low income families. Why? Because the program is targeted to low income families. You don't get it if you're not a low income family. If you do kind of simple back of the envelope calculations, I know it's recent one of these I've seen as a grad student of mine from a few years ago, and you say, let's imagine we raise the minimum wage. I don't have a calculation for Hawaii, for Hawaii, or for an $18 minimum, but he was considering sort of a $15 minimum for the U.S. as a whole. And you say, let's assume there's no response. Everyone keeps their job. No one's hours fall really simple. But so your wage simply gets swept up to the minimum. There's a big increase in the wage bill, right? Employers are paying low wage to workers more. And now you simply say, okay, all those extra dollars going to workers who are now making more because of the higher minimum, how are they distributed across families at different parts of the income distribution? And a very small share goes to poor families. Even a smaller share goes to what we call extremely poor families, families half the property line. It's it varies slightly, but about 15%, you know, varies a bit than what you would use data. And in contrast, about 40% goes to families with above median income, right? Median is the income level, which half are below, half are above. So median income is not high income, it's still a fairly low income. But the point is more, you know, nearly half the benefits go to families in the top half of the income distribution. And it's exactly because there's a lot of young workers, age, just because they're younger, they don't know anything yet. They're not going to be poor later. They might be an MIT engineer later on, but they have to have some job in high school. And as I said, you know, so they get the benefits when you raise the minimum, employers have to pay them more. It's not clear we're accomplishing anything from a policy perspective. And it's even conceivable that yet another sort of side of the equation is that the people paying the higher prices to pay those, you know, higher income teenagers, higher minimum wages have lower incomes than they do. David, your research findings are based on having reviewed a vast amount of studies that is out there now. And there's tremendous research. My question is this, why is there still so much economic debate about the minimum wage? Why do people come to so many different conclusions about it? What do you think is going on here? Well, let me just first to tell us the two premises. Is there a lot of continuing research on the minimum wage? Yes, there is. We'll come back to the wine a second. You said, why do people come to so many different conclusions? Actually, that's the surprising thing. So I have a recent student of mine who finished Pete, very recently, Pete Shirley and I. We wrote a paper which just got accepted from publication. And we were asking, it seems like a very simple question for a researcher to ask, which is what all the other papers actually say? Because you often get this summary of the literature. Oh, half the evidence finds job loss and half doesn't or no one can agree or estimates are all over the map. And I was kind of interested in actually systematically tabulating that all up and seeing if that's true or not. Which you might think I should be able to do because I study this stuff all the time, but you know, someone sends me a paper or someone else doesn't send me a paper. Who knows what I'm seeing. So we did this very systematic survey and we actually surveyed the authors of all the US minimum wage papers, pretty much all of them published since 1992. And I don't have the picture with me. I didn't think to prepare for the talk for this for this discussion. But we kind of plot the distribution of estimates, almost all negative, right? There's some outliers. Sure. This is not laboratory science where, you know, you replicate something under identical conditions, you get the same answer. It's social science. People study minimum wage effects differently. But actually, there's a very strong preponderance of the evidence that says there is job loss from a higher minimum. And again, that doesn't mean you shouldn't do it. It just means there's a trade-off and you should think about the cost and benefits and not just the benefits. I like the fact that you point out that it's a trade-off that it's not either or but what if you could summarize for us or some of the fundamental downsides of imposed minimum wages. Right. So the fundamental, okay, so this is, I think it's most useful to go back to the economic model, you know, how do we teach, how do we think about it? And how do I just finish teaching my undergrad labor course? How do we teach this? So the standard story, which I think is largely worn out in the data, is the following. The minimum wage goes up, right? Government raised the minimum wage, federal government, state government, city government now. That, of course, forces up the wage of people who were previously making less. That makes that input, call it low-skilled labor, relatively more expensive compared to the other inputs firms use, which can be materials, physical capital, and perhaps most importantly, more skilled labor. Labor whose price isn't affected directly by the minimum wage. So that leads firms to substitute away from what the labor that's now got more expensive, just like if we tax cigarettes, you know, people smoke less, right? Same idea. That inevitably raises the cost of production for firms, which raises prices. And it's that effect, prices go up, consumers demand less, that leads to an additional source of employment reduction. So the fundamental downside is, you know, we have a type of worker, namely the low-skilled labor, the ones who are affected by minimum wage laws. And we are trying to help them, although I would argue we should be trying to help more families, and some will have low wage work, some won't. But we're trying to help low wage workers. What a minimum wage does is tax the use of low-skilled labor. That's all, that's, you know, from the point of view of business, for a business, it's no different if I say, you know, Killy, you've got to pay that guy $7 more, or you've got to pay me $7 tax when you hire that guy. Absolutely no different, right? And we know taxes tend to discourage the use of the things you're taxing, unless it's something you absolutely can't do without. So as long as there are substitutes for lower-skilled workers, a higher minimum wage effectively will, you know, the fundamental source of the downside is, it makes using that labor more expensive. And that's what creates the trade off of you. That makes sense, David. We're going to take a quick break right now, and I say to our viewers, we will be right back, so don't go away. We're with David Newark, we're talking about minimum wage. We're on the Hawaii Together program on the Think Tech Hawaii broadcast network. We'll be right back. On April 1st, at 10 a.m. Hawaii time, Think Tech will be presenting a 90-minute webinar panel program called Burning Global Issues. This will be an examination of six continents by thought and community leaders living in or expert in those continents, discussing burning issues affecting each of them, how they relate to the prospects for functioning democracy, and what we can learn from all of that. The moderator for the program is Pamela Spratland, a 30-year Foreign Service Veteran who has served as U.S. Ambassador and Consular Official in a number of overseas posts. The panel is comprised of Carl Baker, Senior Advisor of Pacific Forum on China and Asia, Rupamati Khandekar, Director of Global Relations Forum on India, Elsa Jhark Hadyan, a consultant with Project Expedite Justice on the Middle East, Ilbert Nuagira, an Economist in Kampala, Uganda, on East Africa, Carl Akrami, of the Social Studies Faculty at Punahou School on Eastern Europe, and Juan Telo, a Business Attorney in Bogota, Colombia, on Latin America. The program is sponsored by Project Expedite Justice. We hope you will attend, and that this program will help you better understand these important global issues. Please go to our website, thinktechhawaii.com, and register. Mahalo. Thanks for coming back. I'm Kili Akina on Hawaii Together, on the ThinkTech Hawaii Broadcast Network. My guest, David Newmark, is the co-author of Minimum Wage, as published by MIT Press. We're talking about some of his findings he has produced in the last several years on Minimum Wage. We're going to go right back to it. David, I've got a question for you about one of the major studies that used to be cited all the time. It was produced in the 1990s by David Card and Alan Kruger, and it supposedly showed that the Minimum Wage doesn't always cause job loss. What has your research found about this quite often cited study? Well, first of all, it is often cited, and it's not just that it used to be cited. It's still cited a lot. It was written in 1992, so it's not as old as me, but it's getting old. I want to say a couple of things. First of all, what people often forget, and I'm not always sure it's unintentional on the part of advocates, is the study didn't just show that the Minimum Wage didn't reduce employment. It actually found whopping large positive effects on employment. People often forget that, but I'm not cherry-picking here. Go back to the paper. We argue in the literature about disemployment effects, about what we call elasticities of around 0.2, let's say, minus 0.2, which means that the Minimum Wage goes up by 10 percent, low-skilled employment falls by 2 percent. They had elasticities of about positive 0.7, which means you raise the Minimum Wage 10 percent, you raise employment, in their case, fast food restaurants they were studying, by 7 percent, which let me not mince words, patently absurd, it's not possible. There's essentially no research that finds positive. That was written in 1992. It sort of took labor economics and the policy world by storm. I never quite understood why, without regard to the quality, it was a study of one Minimum Wage increase in one industry, fast food, in one pair of states, New Jersey, Pennsylvania. It was never clear to me why you would draw really strong policy conclusions from one paper, especially when there's lots of other papers. You might have had differing opinions on their quality, but there are lots of other good papers that found very different effects. That doesn't mean we should ignore new studies. It doesn't mean we should ignore new evidence, but we should proceed slowly. When a study is surprising, there's often a reason it's surprising. Sometimes it's right and we were all wrong before, but we should always give some weight to the alternative that it's surprising because it's wrong. In this case, Washer and I went back to not the data because we wanted to collect better data. They had surveyed managers at restaurants and asked them a very vague question, which literally, I don't know the exact quote, but pretty close, was how many people work in your store? And they did that twice before and after the increase in the same period in the state, Pennsylvania, where the minimum wage didn't increase. That's a really strange question to ask the fast food restaurant manager. You know, at the New Jersey Turnpike where McDonald's opened 24 hours a day, seven days a week, and I asked you how many people work in your store? Do you say, oh, there's 130 workers on the payroll to cover our 24-7 shifts? Or do you say, eight, because there's eight people here right now? I have no idea. And then you come back eight months later and ask a different manager, whoever answers the phone, the same question, and who knows what they tell you that time. So we actually went back and got the payroll records from the restaurants back in the day of computer paper printout. You'll remember this, you know, with the holes down on the side and they mailed it all to us. I actually have it in my file cabinet, believe it or not. Went through the data and found modest small effects, which is kind of what you get in most of the research literature. So the study was based on flawed data. And I think most researchers, if they find a really surprising result that really overturns a lot or may overturn a lot of other studies, they sort of say, this is really interesting. We should think a lot more about that. I think the problem with this study was, and it wasn't so much the authors, but others, you know, with skin in the game, just said, this is the be all and end all. And forget everything that came before it. And frankly, everything that came after it. And that's really why your approach works so well in terms of doing a comprehensive review of the studies that are out there. Let me bring this home to us here in Hawaii. Our legislature is in session now. It's looking at a couple bills. HB 2510 would increase our minimum wage in Hawaii to $12 per hour this year and gradually increase it to $18 by 2026. So what do you think the downsides are that could occur with such a policy? And do you think that delaying the $18 amount until 2026 reduces the harm that could be caused, particularly due to inflation? Okay, a lot of questions. I want to come back. One thing is just before you ask a question, you know, you talked about me doing reviews of the evidence. I mean, most of my work is original papers on minimum wage effects. The book has a lot of review and course. And my studies, they don't all find strong evidence of negative effects. Sometimes they do different approaches weaker, stronger. You know, I think the consensus, the evidence is is you reduce jobs. So to the to the Hawaii question, first caveat is I've been to Hawaii and I love it. I don't know much about its economy other than whatever is true of other economies. So I don't have any strong reason to think that minimum wage effects are very different in Hawaii than elsewhere. I will say, you know, I think there's one potential qualification of that. And that is remember what I said before that that kind of what gets the ball rolling on the job loss is that the higher minimum wage pushes up prices and that reduces demand. And that's one of the reasons employers use less labor. The other is you can substitute away from low skilled labor, right? Now, so what that means, there's two implications that one is one implication to two questions. If minimum wage effects will be less severe in terms of job loss for the low skilled, where two things are true, one is there's not as much scope to substitute towards low skilled labor. And the other is where consumers are not sensitive to prices. Okay, so it's conceivable. I'm just just to raise the possibility that that, you know, I know the tourism industry, I know that's not the whole Hawaiian economy, but I know it's a it's obviously disproportionate share road for other places, for obvious reasons, right? It's beautiful. You know, the tourism industry may be an industry where there's fewer substitution possibilities where I can walk you to a McDonald's, there's a lot of capital back there that has replaced workers, a very capital intensive production of food.  But you can't reduce it, maybe you can't reduce it nearly as much, you know, you can't automate. Yeah, you can have self check in and this that and the other, but there may be, you know, some limited scope for that. The other is people may be somewhat insensitive to the price for the same reason that I'll pay a lot more for a beer and an airport than I will anywhere else because I'm in the airport, I'm stuck, I can't go anywhere. You know, to the extent that Hawaii has a unique idiosyncratic product, which to some extent it does, there aren't a lot of great substitutes. You know, demand may be somewhat less sensitive tourism industry is one where one might expect somewhat weaker effect, the direction is the same. I'm just saying there they may not be as pronounced as they are for all the other parts of the economy. And obviously, there's a lot of other people in Hawaii doing a lot of things other than tourism. And I think those sectors of the economy are probably no different from anywhere else. If I'm running a restaurant, you know, for locals, I might try to economize. If I'm running a Starbucks, I might make the line longer, I might automate. If I'm running a factory, obviously, more scope for it. David, many people point to the fact that employment in Hawaii increased, or so they say, after the state minimum wage was increased in 2015. What are your thoughts about that? Well, I don't know the facts. I'm not sure why I mean, I don't we probably don't need to so they say on whether employment increased or not. We must know that. So let's take it as true. That's not how you estimate the effects of policy. I actually, I have a question like that on my Labor demand midterm of a month. An example like you cite and, you know, and somebody says, ah, that proves the minimum wage doesn't reduce employment, good or bad argument. And the reason it's a bad argument and bad evidence is kind of let's go back to the card and Kruger paper. The premise of the card and Kruger paper, the idea was a good one, right? If you just look at one economy, call it Hawaii or call it New Jersey's fast food industry, and just see the minimum wage change and say, ah, what happened to employment, you don't learn anything because a lot of other stuff happens to the economy. The minimum wage is not the cause of the business cycle. There are far, far bigger influences on the economy than the minimum wage. So if Hawaii was booming in 2015 and a lot of states were booming, why also was, and we happen to raise minimum wage, then you might have seen employment go up. The question is, what would happen absent the minimum wage? Now, of course, I can't see Hawaii in 2015 within, without the minimum wage. I can only observe one, whatever, whatever was actually the case. So the beauty of the card and Kruger paper, it's not just on people have been doing this for decades before they wrote their paper, is to compare two economies where we think developments are similar, right? One in which the minimum wage goes up and one in which it doesn't. So card and Kruger did New Jersey and kind of Eastern Pennsylvania that bought it. There may not be a good, we call this the counterfactual, right? What would have happened absent the policy change? We can only estimate that in a real experiment, you know, a vaccine experiment, I can have a treatment and a program. So I, you know, and there I know exactly what would happen if you got the vaccine and if you didn't. In social science, we don't have that. So one would have to find a comparable economy to Hawaii exist and say what would have happened otherwise. So I think you're stuck with the evidence from the other states, which are a lot more similar. Yeah, David, we've got just a quick minute. So I want to come up for air and go back to the original question about helping the poor. What's a better way to help the poor than raising minimum wages? I'd say two things. One is actually create a program that targets benefits based on being poor or low income. You might have someone who's your definition then or the second is, and this is my own perspective, and maybe it's more partly philosophical, partly economic, think about something that actually encourages work rather than discourages work because we know that people's earnings and wages grow with labor market experience. I am a huge fan of the Earned Income Tax Credit. We have a federal Earned Income Tax Credit. Many states do, I think Hawaii does, but I should have looked that up before this interview. Well, on that note, it subsidizes work and it targets benefits very well. Okay, very good. And on that note, maybe we'll have you back some day and talk about earning income tax credit. But thank you, David, for being on the program today. Appreciate it. Everyone you've been watching Hawaii Together on the Think Tech Hawaii Broadcast Network. I'm Kaley Ikeena with the Grassroot Institute. Until next time, we bid you all aloha.


Vice President Professor Yemi Oshimbajo said that it would be an injustice if it did not contest the 2023 election with his vast experience in the last seven years in office. Oshimbajo, who said this when he visited Governor Uluwarut Semihakreidulu of Undostate in Akure, held President Mahmoud Bawari's open-mindedness in delegating to him serious tasks that had aided his experience. He said that God puts people in power but noted that it would be a betrayal to his country for him to retire with all his wealth of experience and exposure in government. Get notified about fresh news updates.


Welcome to this week's session. I'm not going to cover a lot because today's session, it's a combination of two things. But it builds up on the session we had the previous time we met. We're still going to look at hypothesis testing. When we compare into population or two samples. By the end of the session, you should be able to know how to do your hypothesis testing for two independent population. Also for two related population and we're doing hypothesis testing of the mean. For all the examples of today. I do have a little bit of activities, but because the session might take long. So you won't have any activities at the end of the session. And I think we might repeat some of the activities the next time we meet or we will build on some of the activities. Okay, so but today I just wanted to make sure that you understand the concepts and you are able to. When you come across some of the questions relating to hypothesis testing for the mean when you have two population. You are able to identify that you have two independent groups or you have two related groups. And we're going to touch on those two scenarios just like just charging my laptop. Okay, so in terms of the two sample tests that we're going to be looking at, like I already explained, you need to be able to identify whether you have two independent samples, which means you have two groups. It can be two variables, boys and gals, males and females, single and married, South Africans in Barbway, things like that. So those two are independent. One has no bearing on what the others is doing. So they are not related to one another. Or when you have two sample tests that you are doing and you have related samples. So it means these samples come from the same thing. So it's situations where you have the before and the after you the control and the treatment. Yes, control and treatment.  So it's the before and the after type of a test that we're going to be looking at. So it means one has influence on what happens to the other. One is related to the other one. One is dependent on another thing happening as well. So you need to be able to look at the question and identify those. Are they related or are they independent? So in terms of the hypothesis testing, there are several things as well that you also need to identify in the statement and in the question whether those are assumptions that are made during the hypothesis testing. So remember with every hypothesis testing, the goal is to test if there are differences between the groups. So if there are two population groups, then we are testing whether there is some differences between population group one and population group two. The other thing you need to make sure that you understand the assumptions. Given the population standard deviation is known, the population standard deviation of those two populations. Or are they also equal? If they are known and they are equal, you need to be able to know what kind of a test you're going to be doing. If they are unknown and you assume them not to be equal or type of a test you need to be doing. There are different tests that you can apply. So not the same for everything. And when we look at the differences between the two populations, always remember your point estimates which are the values that you're going to be calculating when you calculate the test statistic. We use the sample statistics in the hypothesis testing when we state the statement in your non-hypothesis and the alternative hypothesis you're going to use your population parameters. When you are calculating your test statistic, you're going to use your point estimate which will also tell whether there are differences between the test statistics or the sample statistics that you are calculating. So what do we mean by independent samples? So independent samples are samples that are unrelated to one another. They are independent from each other. One does not influence the other one. One has no bearing on what happens to the other one. Now when you do the hypothesis testing like I already explained in the beginning, there are some assumptions that needs to be met before you can carry out your tests. Number one, the sample needs to be randomly and independently drawn for you to be able to do a hypothesis testing for the independent samples. Your population needs to be normally distributed. If it's not normally distributed, then there are other tests. We call them non-parametric tests that you can do, which is not part of your syllabus. If they are also not normally distributed, but at least your population needs to be greater than 30. Your sample size, sorry, your sample size needs to be at least large. So you need to work with a large sample size. Your population variances, if they are unknown, but they should be assumed equal, then there will be some type of analysis that you do. So let's look at that. If your population standard deviation is unknown, but they are assumed to be equal, then we're going to use what we call a pooled variance, which is denoted by an SP. You will see how we calculate the pooled variance t-test. If your population standard deviation is unknown, but it's not assumed to be equal. So it means that the population standard deviation of the two populations that you are working with, they are assumed not to be equal. Then we use what we call a separate variance t-test. So now, usually in your studies, you are going to be using the pooled variance t-test. You're not going to use the separated variance t-test as often as possible. You will see when we do a lot of other activities as well. So hypothesis testing for the mean, whether population standard deviation is unknown and not assumed to be equal, like we said, also the assumptions are the same. For that one, samples needs to be drawn independently. The population needs to be normally distributed, or the sample needs to be bigger than 30. And your population variance might be unknown, but cannot be assumed to be equal. And that will also be one of the tests that you can do. It's just a slight disreputing what I've just said. Now, how do we test, how do we state the hypothesis testing statements? Remember, the previous time we looked at the six steps of hypothesis, we looked at how do we state the non-hypothesis, the alternative hypothesis. Those same concepts are still applicable here. Now, when you read the statement, sometimes in the statement, they might give you some hint in terms of the type of a sign that you need to allocate in your hypothesis. Whether you're doing a two-tailed test or a one-tailed test, what do you call a directional test or a non-directional test, which is a two-tailed test. Where does such S decrease? If in the sentences, in the statement that they gave you, they say another population is decreasing, then you know that we're talking about the lower test. So it means your non-hypothesis will state that the population mean one is greater than or it's equal. Remember, in your non-hypothesis, there's always an equal sign. So it will state that the population is greater than or equal the population two. Your alternative will state that your population one is less than the population two. Alternatively, you can write the statement as such. Your non-hypothesis will say there is a difference or the difference between the population one and population two mean it's greater than zero. And your alternative will state that the difference between population one and population two is less than zero. And that is for decrease, decline, less than, below all those kind of weights. So immediately when you hear weights like that, know that in your alternative statement, you will have a less than. Remember also, the sign that you put in your alternative hypothesis will help you with understanding how you make a decision, where to find the critical value and how to make a conclusion and also how to find your Z critical values or your true critical values and so on. For an upper tail, statements will read increase, more than, greater than all those statements. They will refer to where you are testing for the one directional upper tail test and the statements you can see there. For a two tail test, the statement might read there is a difference, there is a change, they are the same, things like that. So when you find weights like difference, change, same, not the same, not equal, then you know that you are doing a two tail, which is a non directional test, because it's got two regions of rejection as well. Your non hypothesis will say the mean are equal, the alternative will say the mean are not equal, or you can say the difference between the two means are equal to zero, and the alternative will say the difference between the two means is not equals to zero. How we're going to make a decision will be based on the alternative side. So if it's a decreasing, remember, lower side, less than decreasing, then your region of rejection will always be on the left hand side, and it will be on the negative left hand side, or it can also be on the positive left hand side, but the region of rejection will always be on the left hand side. And to make a decision, you will reject your non hypothesis, if your test statistic, which you will calculate your T test statistic, if it's less than the critical value in the negative side, you will reject the non hypothesis. For the upper tail way, it is increasing, it's greater than, if your test statistic is greater than your critical value, you go into reject. For a two tail test, you have two regions of rejection. If your test statistic value you find falls in either one of the areas of rejection, you reject your non hypothesis, otherwise you do not reject your non hypothesis. Remember the six steps? Always, always, you need to follow the six steps because in your, in your exam or your assignment, usually the options that they're asking you, they might be asking you about the one of the steps of the hypothesis. So you need to know the other because they build on each other. The first one is you need to know how to state your non hypothesis and your alternative hypothesis. Always remember, when you state your non hypothesis and your alternative, we use the population parameter. Number two, always remember that in your non hypothesis, it's always an equal, there is an equality sign. So it means it's less than or equal, greater than or equal or equal or sometimes it's just an equal. Your alternative does, will never ever have an equal sign to it. It will either be less than, greater than or not equal. It will never have a less than or equal or greater than or equal or equal in it. You need to know how to make your decision. Baking your decision is based on several things. You need to know your level of significance, which is your alpha value. You need to know your critical value. You need to know whether you're doing a Z test or a T test. All those things are very, very important. And now remember, when you're looking at unknown standard deviation, we use the T test. For known standard deviation, we use the Z test. You always need to also remember that. Step number three, you gather and calculate your test statistics. If we're doing a T test statistic, you also need to know how to calculate the test statistic. You need to substitute the values correctly. Remember, we use the difference because we're talking about independent samples. The difference between the sample. So remember, we use the point estimate or the statistics, which are the difference between your sample means divided by the standard. Here we're going to use the proved variance. And we're going to look at that formula just now, but you need to know how to calculate that. Then lastly, you need to know how to make a decision. And making a decision, remember, based on your region of rejection, which is based on the critical value and your test statistic. And always when you make a conclusion, you refer back or you refer back to your null hypothesis in state of your alternative statement. So always your conclusion is made from your statement made in your null hypothesis. Let's look at how we do a hypothesis test for independent sample where we do a T test. So the first step always remember the six steps or the first steps, the first step to state the null hypothesis. So your null hypothesis will say there is no difference. That is your null hypothesis. So your null hypothesis always say there is no difference. It means the two means are equal. So remember, you can write it as such or you can write it as your H naught. Your mean treatment minus your mean control is equals to zero. You can write it like that. Doesn't have to be mean equal mean. Your alternative will say there is a difference. So your alternative always say there is a difference because we want to prove that phenomenon that we said there is a difference. There is no difference. So the alternative, the mean treatment is greater than the mean control. And here is if we are doing the before and the after. I'm just using this as an example. Number two, define the decision method. We know that we're going to be using the T test statistics. So that will be how we're going to make a decision. We're going to go to the critical values of T. We're going to calculate the test statistics of T. And when we go find the critical values of T, we're going to use your degrees of freedom and your level of significance. But you don't have to worry about knowing how to get those points. Because usually they don't give you the T test table. Then you need to gather and compute. So it means you need to identify the values that you need or the facts that you will need to use to calculate. So the test statistics says the mean, the sample mean minus the sample mean one minus the sample mean two minus the difference of the population mean. Now, usually in the formula, this is always equals to zero. So we don't even have to write it. So divide by the square root of your sample variance. Divide sample variance of one divided by the sample size one plus your sample variance two divided by the sample size two. So usually the test statistics, if we're using this will be T stat is equals to the mean one minus mean two divided by the square root of S21 divided by N1 plus S22 divided by N2. That's how we write it because the hypothesis testing said the mean difference is always going to be equals to zero. So you don't even have to write the formula in this manner. Okay, making a decision. We know we use the region of rejection, which is the critical value will tell us if it's a one thing. If it's an in the lower side, it will be on the left hand side. If it's on the upper side, it would be on the right hand side. If it's a two thing, an undirectional test, it will have two regions of rejection. If the test statistics that we calculated in step number three, if it falls within those two reject regions of rejection, we reject the null hypothesis. Let's look at an example. We are interested in whether the type of movie someone sees at the theater affects the mood when they leave. We decided to ask people about their mood as they leave one of the movies. So the people looked at the first group looked at the comedy and the second group looked at the horror. So group one had 20 sample size and group two that looked at the horror also had 20 sample size of people. So both of them had 2020. Our data was recorded so that the highest goal is indicated a more positive mood. So it means after they have watched the movie, they ranked the movie so that they can score based on how they feel after the movie. So the higher the score indicate a positive mood. This group one sample size, we already made connection there. There are 20, there were 20 people. We calculated some sample mean of the scores and we found that group one, the score was 10.6 and group two, the score was 6.15. The standard deviation, which means the variation between the scores of group one was 3.20, the variation between the scores for group two was 3.18. We have a good reason to believe that people living the comedy will be in a better mood. So we use a one-tail test at alpha of 0,05 to test this hypothesis. So there are a couple of things that they have given us they given us the facts or the stats which we're going to be using. They also told us that we need to be using a one-tail test. So already they are telling us we're doing a one-tail test and yeah, they're talking about better mood. So better mood might be greater than. So that is also another thing that they are giving us here as a test or as some of the facts given in the statement that we can also gather. They also giving us the level of significance which is very important when we go find the critical values and so on. So let's test this hypothesis that people living a comedy will be in a better mood than people in a horror movie. So we're going to test that. So the first step of everything is to state your null hypothesis and your alternative hypothesis. So we state the null hypothesis. The population mean of those from comedy is not different to those that are in horror. So there is no difference between the population mean of comedy and the horror. The alternative will state that there is a difference between but because remember there we said, oh, sorry, let's go back. Better mood means greater than. So how we state that statement going to say there is the mean of the comedy is greater than the mean of the population. Of those from the horror movie. So we're going to use a greater than sign instead of the not equal. So we're not going to just say it is there is a difference. So we're going to say it's greater than the other one as well. So you must pay the difference. If it was not equal, we would have just said that the difference between the two groups, the mean of the two groups is different. But because it's not just different, it's greater than we say the mean, the difference between the mean of the comedy to those that watch the horror, it's greater than zero. So. We also can go to step number three or step number two and determine what kind of a test we're going to be doing going to be doing a tea test. And we also identify the facts we write the facts we go went and found the critical value remember like I said, in your studies, you don't have to worry about finding the critical value they can either give it to you. They will not ask you to go and find it because they don't give you the tables to go find the critical value. So we have our critical value which helps us to define our region of rejection. And because of this sign, so our region of rejection because of the sign, it's greater than then it will be on the positive side. So it means one comma six, eight, six is our region of rejection. Anything that falls on the right side, we're going to reject the non hypothesis. We went on we calculated the test statistic, which is the sample mean of comedy minus the sample mean of horror. We know that it was 10.65 minus 6.15 divide by the. Remember we use the pooled variance right so they. The variance. Which is the standard deviation of 3.0 squared, which gives you the variance divide of the sample one divide by the sample size of sample one, plus the variance of sample two divide by the sample size of sample two. And when you calculate that, you get 4.6 4.461. So after we have calculated the test statistics, then now we are ready to move on to making a decision, which is step number four to make a decision. We draw our picture. We write where our critical value is anything that falls on the red shaded area. We're going to reject an alpha of zero comma zero five anything on the white area. We do not reject the null hypothesis. Our four comma four six one falls on the shaded area. Therefore we can conclude and we're going to make a decision and reject the null hypothesis at alpha of zero comma zero five and conclude that the average mood after a comedy is better than the mood after the horror movie. And that's how you make the hypothesis testing. Process. Okay, so. Are they any question. Before I go to the previous. Sometimes. Sometimes you can also because on this one we use only the critical value and the test statistics. Sometimes you can use what we call the P value. So when you use the P value remember always the decision says if your P value is less than your level of significance, which is your alpha value, you reject the null hypothesis. And whether you're making a decision based on the critical value and the test statistic or on the P value and the alpha value. You need to get to the same conclusion. You make a decision based on the P value. So some statistical software they will go in. They are able to calculate the P value and give it to you. For example, Excel is one of the simplest tool that you can find and you can use to do this kind of a test. But you can plug in the values and calculate the P value and your P value of 0,0007. The rule says if the P value is less than alpha, we reject the null hypothesis. So now our P value is 0,0005 our alpha value was 0,005. Therefore, we can reach the same conclusion. We reject the null hypothesis and 0,005 and conclude that the average mood after the committee is better than the mood after the horror movie. So you can still reach the same conclusion. And with the P value, you need to be very careful as well.   You fight. Remember, this is where you have your statement of alternative. If you're dealing with a non-directional, which is a two-tail test, then your P value will be two times your P value. Therefore, what do they mean? If, for example, they tell you that you have a two-tail test, you have a two-tail test and the P value is equals to 0,25. I'm just going to say that that is your P value. And they say, what is a one-tail test P value? If the question says, we've done a test and the P value for a two-tail test is 0.25, what will be the one-tail test P value? That is easy. You just say, because a two-tail test, which is a non-directional test, is two times the P value, therefore it means if this is your P value for a two-tail test, therefore a one-directional has two of them. So we're going to divide this. It's made up of two of them.  Remember, because for a non-directional, therefore it means this area and that area, this one is 0,125 and this area is 0,125. But both of them, they make up a P value of 0.25. Always remember that they like to treat you with questions like this in their exam. Be mindful of that. Okay. I'm going to go back to my presentation. Now let's look at this activity. Because we're going to run out of time next on Tuesday, we're going to do hypothesis testing for related samples so that we have more time for that. So let's look at the following exercises. Consider the following statistics regarding post-training attitude score. We have two groups, group one and group two, and they recorded their mean average and their standard deviation. The first thing you need to notice in the statement is what they said right there. They said consider the following statistics regarding the post-training attitude. Please give me one second. I need to open for my daughter. She just came back from school. Just give me one second, one second, one second. Apologies for that. You say the hint here is the weight statistic. Because on here they're not talking about the population or anything, just give you the hint of what to follow. So they say the following statistic. So what you need to remember from our first session that we had that the measures that comes from a sample are called statistics and the measures from a population called parameters. Remember that, right? And remember statistics are also called point estimates. Those are the things that you need to always constantly remind yourself with because they can use them interchangeably. So they gave you the mean and the standard deviation. The question here is what are the values in the table called? Are they called population parameters? Are they called sample statistics? Are they called test statistics? Is it option one, two or three? You see how tricky they can make the questions look like? That's a question for you. Is it option one, two or three? No one wants to try. Okay, so based on the information I just gave you, Karen. Who would the answer be two? The answer will be two because they told you the following statistics regarding post-training attitudes goes as follows. And then they are just asking you what are they? So you just need to look at the mean, the standard deviation. These are the measures. Remember, these are what we call measures. These are measures of central location and measure of deviation. Remember, the mean is a measure of central location or central tendency. The standard deviation is your measure of dispersion or variation. So the measures that comes from a sample are called statistics. So sample statistics are those measures or the values in the table. Exercise two, we're also still following the same question, right? It says, consider the following statistics regarding the post-training attitudes go from group one and group two. You have the mean and the standard deviation. Group one has the sample size of 20 and group two has the sample size of 20 as well. And they say, what is the value of your test statistic? So always remember what the test statistic is. The test statistic is that formula that you need to calculate. Your t-stat, which is your sample mean. Remember, this is also denoted by the sample mean and this is denoted by an s. Also remember the symbols. The sample mean for one minus the sample mean for two divided by the square root of your variance. Now, your variance is your s squared, right? Is your standard deviation squared divided by n or I must put the subscript n1 plus the variance two. So we just substitute. Remember, this will be your x1, x1, x bar 1, x bar 2. This will be s1. This will be s2, sorry. s2. That's all because this is for group one. This is for group two. Always remember that. Now for mean is 21.65 minus 20.40 divided by the square root of 2.99 squared divided by 20 because they told us n is 24 group one and 3.05 squared divided by 20. Now, usually I have a cashier calculator on my laptop to show you, but if you don't have cashier calculator, you have to calculate this step by step. So what I mean by step by step is you're going to calculate the top patch and get the answer and put your answer for the top part and then calculate the bottom patch. Do the 2.99 squared divided by 20 answer plus the 3.05 get the answers, the squared get the answer divided by 20 and then add them together, get the answer and then take the square root. When you do that process, do not round off. Keep all the digits as you write all the numbers down and then once you have the number for the square root, take the top number divided by the number from the square root. Don't try and do everything all at once. If you want to do everything all at once, use brackets. Unfortunately today, I will check maybe on Tuesday, I will show you how to use your cashier calculator if you didn't know how to. Or if you are using another calculator that is not a fraction calculator for cashier. But for today, I will answer this question for you. It's 21.65 minus 20.40 because I'm using a cashier with a fraction. Functionality makes my life easier. 2.99 squared divided by 20. And I'm going to show you so I will also advise you to maybe if you've never bought a calculator, try and find a cashier calculator. They sell them at pick and pay, at shop right, at checkers, at books, at CNN, all the bookshops. Don't buy a too expensive one. At shop right, I think it's 272.325 now, but sometimes they have it on special. And if you are not sure about it, you can even ask before you go and buy. And my answer from my calculator. If, for example, I'm not sure if you will be able to see it, it might not be visible. It's 1,3. I'm going to keep only three decimal 1,309. I'm just going to keep the three decimal because I'm already at the end. I'm already at the end. So looking at the information I have in front of me, the answer is option or I can keep one decimal because I could also look at the options. What do they give me? So the option is 1,3. So even if I round this to 1,31 or to 1,3, the answer would be just option number three. But you need to know how to calculate this because these are some of the questions that comes from your past exam paper, your tutorial letter. So they will expect you to know how to do these calculations, but we can work through them together as well. We are left with two minutes before the end of the session, like I said. Let's look at the last question I think is the last question and the rest of the other activities will do them on Tuesday just to seal off the hypothesis testing. Suppose the two-tailed p-value for the test, t-test of the differences between two means in the previous question is 0,19. Now remember, we get this type of an example previously. The key word, two-tailed p-value. So let's remember for a two-tailed p-value, that value has two regions of rejections within it combined. So they say the p-value there was 0,19. If alpha is set as 0,10, what is the decision regarding the null hypothesis? So on this one, they are asking you to make a decision. Now always go back to what does the rule says? What is the decision rule? So the decision, the decision rule always says if the p-value is less than alpha, reject the null hypothesis. Now knowing that you know that this is your p-value. Okay. If your p-value is 0,19, your alpha value is 0,10. Which one is which? Is the p-value greater than or is it less than? So your p-value of 0,19 is greater than your alpha value of 0,10. So knowing that, therefore we do not reject the null hypothesis. That is the decision. Not the rule, the decision. This will be the decision. So now, which option will be the right option? Number one says do not reject the null hypothesis because 0,19 is greater than 1,0. Number two says accept an alternative because 0,19 is greater than 0,1. Number three says reject the null hypothesis because 0,095 is less than 0,1. Always remember that we make the decision based on the null hypothesis, never on the alternative hypothesis. So this will be with the process of elimination, that will not be even correct. Even though it looks like it's correct, but because they use the alternative to state the decision, then it makes it not acceptable. And this one you can see that using a different p-value, therefore the only correct answer is number one, but also is the same thing that I just said when I looked at the decision. I could have put do not reject the null hypothesis because 0,19 is greater than 0,10. But I started first by looking at the p-value comparing it to the alpha v and then make my decision. Easy, isn't it? Easy, easy, easy. Easy stuff, easy stuff, easy, easy stuff. Are there any questions? We are at the end of an hour, even though we started late, but I don't want to start a new section when we left with few seconds. And I hope with all this, at least it opened up some of the things that you were unsure of, you are now aware of what is happening. Are there any questions? Any comments? If there are no questions or comments, then we can conclude today's session. For those who, I will make two scrolls, scrolls, scrolls, scrolls, until I get to the end. For those who are not aware, I am not part of UNISA, I come from Pambili Analytics. My role is just to support you within your studies by offering you these free services or free online sessions. At Pambili Analytics, our mission is to bridge the gap where it comes to data literacy, statistics literacy, numeracy, but more specifically analytical skills. On the other hand, we offer all these sessions, but we don't only do that, we offer a range of services, including consulting, as well as the skills development. If you want to have one-on-one discussions, you want assistance, we do offer instructor-led one-on-one training, either virtually or face-to-face. If you're running a special, you can sign up. It's 150 per hour. By June, we're going to revert back to the normal price. I think we've been running the special for almost three months now. If you are interested in programming, learning how to do research, we also have self-led training sessions. On the research methodology, research design, guided learning, you can take those courses and equip yourself. Remember, you can subscribe to my YouTube channel. I do have a YouTube channel. You can subscribe, like, share with others. There are plenty of videos which are free of charge that you can watch and binge watch on them and use them as you prepare for your assignments or your exam. Otherwise, if you want the recording of these free online sessions, the recordings you can only get access to them if you sign up for a membership on YouTube. And those are the membership. And I would like to end the session by giving shout-out to our lawyer list for the last month, which are Bambo Koji and Izal. I will say thank you for being part of the session today. If you need to get hold of us, these are our content details. Other than that, adios. Enjoy the rest of your weekend and see you Tuesday. Bye. Thank you. Goodbye.


Laura, so very, very glad to see you looking safe and well in the hideous times that Israel's going through. I literally can't imagine it. Please tell me what you know about the latest happenings because it's just tragic from this side. Yeah, thanks Malia. We're very fortunate as to just because of where we live in the country. Others have not been that fortunate. Those living close to the border and stuff. But I was awoken yesterday by my brother-in-law seven o'clock in the morning saying close the mamad room. It was probably an overkill from just because of where we live. But I was like, what's happened? We're at war, was the answer. A mamad room, it means a bomb shelter room. So every apartment block that's newly built is built with one of these. So it's almost like a building within a building that's reinforced. So and it's unpenetrable. Once you close the steel beam, it's completely unpenetrable in the door. So this is a safe room. So it's normally a normal window, but actually there's a steel window here in between. So once that is shut, then you've got a smoke detector. Stand here. So you can't burn this room or anything like that. And then you've got the steel door. Steel door. So once that is completely sealed, then you're completely in a safe room. So in the event of anything, fire, missiles, anything, if you're in the safe room, it's like a building within a building. It's totally unpenetrable. So technically the whole building can collapse, but everybody who's in the safe room, that will remain intact around it. So that's how it works. I mean, if the Twin Towers had a safe room, you would have the safe room still standing is the point of it. Well, we certainly are all our hearts are with you, Laura. And as I said, we hope you and the family can get out and just watch from afar instead of from your safe room. Yeah. Yeah. Thank you very much.