 Hi, welcome to today's session. Today, we're going to be looking at hypothesis testing for the mean, when we're looking at one sample or one population. By the end of the session today, we should be able to cover the basic concepts of hypothesis testing that you should be aware of and how to use hypothesis testing to make a decision. There will be some activities from your past exam papers or tutorials, as well to guide us in terms of the types of questions that you get in your assignment or your exam. When we talk about hypothesis testing, I think when we started with these discussions, we started introducing the types of branches of statistics that you need to be aware of that we said there is the descriptive way you just describe and summarize your data, and there is the inferential statistic where you infer the discussion or you make decision from the sample about the population of study. One of the techniques that you use in inferential statistics is hypothesis testing. Hypothesis testing is more about testing your assumptions as a researcher. With hypothesis testing, it is an inferential procedure that uses a sample data to evaluate the credibility of your hypothesis about the population. We always use the population, remember the measures that we calculate from the population, those we call them the population parameters. If it's the measure that we calculate from a sample, we call those statistics or point estimators of the population. A population characteristics can be also a combination of multiple parameters, or it can be about when we are testing the hypothesis, it can be about the form of an entire population distribution, probability distribution that we want to test, but more especially it's more about having the research question and testing those assumptions that you made as a researcher to prove or disprove those assumptions. There are steps that you need to follow when you do hypothesis testing, and here I'm going to show you the four steps. You can break this into multiple steps. It doesn't have to be the four steps, but we're only going to concentrate on these four steps because there are general steps that you need to follow when doing a hypothesis testing. The first one, based on that assumption, you need to be able to state the claim in terms of a non-hypothesis and an alternative hypothesis. This will always use the population parameters like a mu, your sigma, and because today we're dealing with the mean, so we're always going to use the mu to test the hypothesis. Step number two, you need to be able to define which method based on the characteristics or the information that you are given in your question or in your research, you need to be able to identify. Remember the first session that we had, we had a decision tree. You need to ask yourself, am I given one sample? Is my population known? Is it unknown? Different method for different thing. Am I using two population? Am I having two categorical data? Or am I having numerical data? Things like that. You need to ask yourself questions in order for you to determine the type of method that you're going to be using. But not only that because at this point you also need to ask yourself what kind of a decision you're going to arrive at. Depending on the type of test that you are doing, you also need to know what kind of a test are you going to be doing a Z test, or T test, or a chi test, and you also need to know whether is it going to be on one side, or two sides, or is there a direction to it or non-directional? So is it directional or non-directional? Those kind of decision methods that you need to encapsulate into this step so that you are able to make a decision at the end. You need to form them there. Once you know what kind of a method you're going to be using, whether it's a Z test, or a T test, finding the critical values, then you need to do the calculation of those test statistics. When you are doing a one-tail test statistic or a two-tail test statistic, you need to be able to know how to calculate the test statistic. And once you have your test statistic, then you can make your decision based on the methods that you already have made or made aware of. You can use that with your test statistic to make a decision, whether you also use the P value, which is the probability value, or you're going to use the critical value and the test statistic to make your decision. And when you make that decision, it needs to refer back to your hypothesis testing or your statement that you are making about the population. When making a decision, there are certain elements that you also need to be aware of. So making a decision is also called a body of proof. And that is also placed directly on whether you believe the alternative is true. Remember at the beginning, when you are stating your hypothesis, you state your null hypothesis and you state your alternative hypothesis. When you make a decision, we always look at the alternative hypothesis and the body of proof lies there. Whether you are doing a one-tail or a two-tail, it's based on that. In the testing of the statistical hypothesis, the problem will be formulated so that one claim will be favored. So because there are two statements, always remember there are two statements. There is the true statement and the false statement. There is two sides to the coin. Always remember that. Initially, as a researcher, your claim that you are making is set at the null hypothesis. The null hypothesis is what the researcher wants to claim. But not always that your null hypothesis will be what the researcher wants to claim because there are certain things as well that you need to be aware of. In your null hypothesis, they can never be a greater than or a less than, there is always an equal sign true your null hypothesis. So if your assumption was that males perform better than female, you cannot put that assertion in your null hypothesis. You will put it on your alternative. Now it changes what the researcher wants to claim and later on we're going to talk about having to make errors when you are making a decision. And those are based on those little things that I just mentioned now. So remember, your null hypothesis is what the researcher claim and it can either be rejected or not be rejected in favor of the alternative as well. If the sample does not contradict your null hypothesis, we will continue to believe that the null hypothesis is true. So there can be two possible outcome that you will get. You will reject the null hypothesis or fail to reject the null hypothesis. Always remember those two. Reject the null hypothesis or fail to reject the null hypothesis now. We were speaking about null hypothesis and alternative hypothesis as well. I need to also emphasize on one of the two things. Your null hypothesis always has an H subscript zero. Your alternative, it can interchange. Sometimes some books, they have it as a compliment. You have H subscript A or H subscript one which refers to alternative hypothesis. You always need to remember that. And when you make a decision, we always remember go back to the null hypothesis instead of the alternative hypothesis. But when we are making a decision, we consider the sign and everything that is on the alternative hypothesis or the alternative statement. And that's the two things that you need to aware of. I've mentioned that sometimes you might get some errors when you make a decision, but there are possibilities of those errors happening. So what are those errors? There are two errors in statistics or in your hypothesis testing that you can get. You can get what we call type one error or type two error. A type one error is when you reject the null hypothesis when it's true. When you reject the null hypothesis, when it's true, you are committing a type one error. And that is mostly most of the time that's what you will do because the claim that you are making as a researcher will always be in your null hypothesis. But there will be cases where you also commit what we call a type two error. And a type two error is not rejecting your null hypothesis when it's not true. When you fail to reject the null hypothesis, when it is not true, you are committing what we call a type two error. So now, how do I identify the two scenarios? A type one error, that's what I said, you are going to be making type one error every time you are doing your hypothesis because as a researcher, you assume, you make your assumption and you say, I believe that the prices in shop right equal to the prices in, oh, the people who buy at check us, they prefer, they see the prices being the same as the prices at pick and pay, right? So if I look at the null hypothesis, there is an equality sign to it. So I'm saying they are the same, the prices are the same, the prices are equal. If I'm going to reject that statement that I'm making, I am committing a type one error, that's what. Let's assume the other one. So maybe let me not use pick and pay and shop right. Let me say the height. So I'm going to say as a researcher, I'm saying the height of boys in this class less than, how do we measure height? 1.82, that is my assumption that I'm making. The height is less than. Now, because I'm saying it is less than, it cannot be the claim that I put on the null hypothesis because on my null hypothesis, it must always have an equal sign. It must be greater than or equal or less than or equal. So the minute I put a greater than or equal in state of a less than, so my null hypothesis will have the mean height is greater than or equal, and my alternative states that the mean height is less than because that's the, so now it means my claim is in the alternative is not in my null hypothesis. Therefore, I am creating a false null hypothesis. So now, if I'm going to accept, if I'm not rejecting my null hypothesis, then I am committing a type two error because my null hypothesis is a false null hypothesis. Do you understand the difference between the two? The first one I said, let's say the prices are equals to 90, let's say, let's use the same. It can pay your books of, let's use mayonnaise. Let's say a bottle of mayonnaise costs about 36 rent. So we will say in your null hypothesis, can you mute yourself? Someone is not muted, and there are people talking in the background. So let's say the mean price of a bottle of mayonnaise at this door is equals to 36 rent. That is my claim, I can put it there. My alternative claim will state that the mean is not equals to 36 rent. So if I reject this null hypothesis, I'm going to be committing a type two error, a type one error, sorry. If I'm rejecting this null hypothesis, which is the claim of the researcher, I'm committing a type one. As a researcher, my claim is less than on the second one, but my null hypothesis is a false one because I have to create the null hypothesis statement with an equality sign to it. Therefore I'm creating a false null hypothesis. If I'm not rejecting this null hypothesis statement and I'm accepting it, then I'm committing a type two error because I am not rejecting the false null hypothesis. I hope you understand the two scenarios. The two scenarios so that you are able to interpret it when you get questions in the exam. Okay, so when testing the hypothesis, like I said, you need two statements. So you need to state the mean is equal or the mean is equals to a thing. We will always use the population parameter. Never use X bar, which is the sample statistic. So we always use the mean. So we state the null hypothesis and because you also need to ask yourself, is the population standard deviation given? Is it known? If your population standard deviation is known, usually they give it with a sigma or they tell you the normal population standard deviation is known of a normal population with a known standard deviation. That's what is missing there. With a known standard deviation, then we're going to be using a Z test statistic. That is what you're going to determine at the beginning. Now the other thing that you need to take into consideration when you are at that step number two for the method of the decision method, you also need to ask yourself, what is the statement on my alternative? Remember your null hypothesis always has an equal sign. It can be equal, greater than, less than your alternative has the following sign. So it has a greater than or a less than or not equal. If it is a, if it has a greater than sign without an equal sign at the bottom, then we call this a one-tail test or we can also call it a directional one-tail test. And the rejection area or the method that we're going to use to reject this will be in the upper side of the table. And usually I like to draw a graph and draw it like this and say this will be my rejection area because it's in the greater than. You will see when we do the example, how we manage this. If it's less than, it's also on the lower side. So the rejection area will be on the lower side. Anything on this side, it is less than. Therefore it means it also a one-tail test. It's a directional test because when it's less than or greater than it has a direction, it tells you the direction. If it is not equal, therefore you have two regions of rejection because there is no direction in there, then we call it a non-directional or a two-tail test. So where it is not equal, it is a two-tail test. It is a non-directional because we have two regions of rejection. If your Z test statistics falls within those areas for each one of them, you make a decision to reject the null hypothesis. So for any value at that point, we're going to reject the null hypothesis. You reject the null hypothesis if it falls there. You reject the null hypothesis if it falls within those two areas. I've already explained this. The other thing that you need to take into consideration is finding your critical veil. And this I wouldn't call the critical veil. For a one-tail test, when it is greater than in the upper tape or when it is less than on the less than tape, your critical value you find it by going to the table and finding Z alpha by using Z of alpha. And alpha is your level of significance. We're going to get to that in a moment. When it's a two-tail test because it's two areas, we're going to divide our alpha by two and split the areas into two portion. And I'm going to show you just now how to do that. When we make a decision, remember here, we're making a decision based on the critical veil. I must also write it here. Critical veil. So you always remember that when we're making decisions, you can either use your Z-test statistic and the critical veil, or you can use a P-value. A P-value is just the probability on the table that you get after you have calculated your Z-test statistics, you go to the normal distribution table, you go and find the Z-value. We dealt with this session last time when we met, so you should be able to know how to read the table. All right, do you remember? It's everything that we did on Tuesday. We're building up on that with the hypothesis testing. The P-value, this probability is calculated, assuming that your null hypothesis is true. You need to also be aware that this probability is between the values of zero and one. It can never be more than zero or one. It takes up a decimal number. And the P-value is not the probability that the null hypothesis is true, nor it is an error of a probability. The P-value just gives you a probability of observing your test statistic. So in order for us to find or to make a decision, we need to go back. Once you have your P-value, you also need the level of significance. Your level of significance and the P-value, when you look at both of them, you compare the values, you then make a decision. And the decision rule says, if your P-value is less than or equals to your alpha-value, if your P-value is less than or equals to your alpha-value, you reject the null hypothesis. If it's greater than, you do not reject. We say we fail to reject the null hypothesis. And we can look at examples, both examples later on. There is also what we call an effect size, especially when you do hypothesis testing. And this is a determinant of sensitivity or power of a statistical test. Usually how you interpret this, it's either you say it has a larger effect or a smaller effect. And the reason for a larger effect is always, you always get, for example, let's leave the reason behind. If you do your hypothesis and there are some errors within, because not all values are equal in your data set as well. So therefore there will be some error, little bit of errors when you are creating those hypothesis testing, because also of those percentages that you set, your level of significance that you are setting at the beginning, because you can say either you want a 95% or you want a 90% or a 99%. So if you have a larger sample size, if you have a larger sample size, the less you will get those errors or the less are the variants of errors that will okay when you do your testing. And this is also based on the law of larger numbers. The larger the data set you have, the smaller the errors will be, right? Because your analysis will be closer to your original values, because then there are no huge variances or huge gaps within your data set. When your sample sizes are large, even the sample event or effect that seems to be significant can produce a smaller p-value leading to the failure of your null hypothesis as well. And you need to be watchful of those effect sizes, especially when you're doing your hypothesis testing. So with the hypothesis testing for the mean, if for example, you have a smaller data set or a smaller sample size, right? Then we're going to use what we call a T test or you're going to go to the T table. So it means instead of using the Z test for larger sample, we're going to use a T test for a smaller sample. But other assumption is that your population's done a deviation because it's a smaller sample, then it means you might not know your population's done a deviation. So when your population's done a deviation is unknown and your sample size is small, usually small they refer to less than thatch, then you're going to use a T test statistic. And the same will still apply when you are making a decision for a T test. You will still apply a one-tail test when it's a greater than or a less than in an alternative statement. You will still have two-sided tests when you have a not equal. But now using the T test. Now, to find the critical value, so here we're using the critical value. The critical value for T is different to the critical value for Z. The critical value for Z, you find it on the Z table usually they give it to you in the past exam papers they always have the critical values of Z in a smaller table called critical values. But for a T critical value, you need to go to the T test because you will need your level of significance which is your alpha value. You also need your degrees of freedom which is N minus one, your sample size minus one. And we can look at the example now. Mabatu, the social scientists took a random sample of 30 adults with autism and found their reading time to be normally distributed with the sample mean and the sample standard deviation of 90. 90 weight per minute and 18 weight per minute respectively. So our sample mean is 90, our standard deviation is 80. But clearly and Mabatu are collaborating to test the hypothesis that the mean reading time of adults with ASD is less than 100 as room at 5% level of significance. So we need to test this hypothesis. Now we need to ask ourselves a couple of questions. What is my sample size? The state, that's my N. What other information it is normally distributed with the sample mean? So 90 is my sample mean and standard deviation of 18, which is IS. And I must assume that the hypothesis that the mean reading time of ASD is less than 100. Right? So this be 100. Assume at 5% level of significance. So we are given also our alpha there. So what we need to do is ask ourselves step number one, can we state our null hypothesis and then our alternative hypothesis? Our null hypothesis, the issue with the whole statement is they gave us the sample mean and the sample mean and the sample mean. So this one is sample mean. I'm going to change this. I think this is an error on this. This should be a population mean. With the population mean, so we need to use the population mean and using the wrong population mean. Sorry, this is correct. It's the sample mean. This is our population mean because that's what we need to be testing. With the mean regarding time. So this is our population mean. Sorry, my bad. So that's the assumption that the researcher is making. He's saying butchali and Mamato are collaborating to test. So butchali and Mamato are the researchers are collaborating to test the hypothesis that the mean reading time of adults with ASD is less than. So they are making that assumption. So because it's less than, it cannot go in the plane in our null hypothesis. So it will go in the alternative hypothesis as the mean is less than 100. And in our null hypothesis, we can just say the mean is equals to 100. That is step number one. Step number two, we need to make a decision find out what decision method would be. We are given the sample statistic and standard deviation statistic, which is 18. So because we have S of 18, therefore we are going to use T test. And because we are going to use T test and because the alternative, what is this? It's a directional because of the sign of less than. Therefore it is a directional one tail test and it is in the lower site. It helps to make this decision because it will help us understand where it will be. So it means in terms of us finding the critical value, it will be on this site. So it means we're using a T. So it will be T critical value of alpha and the degrees of freedom. So our critical value, we still need to find that because we're not done yet. In step number two, we have to do a lot of things. So our critical value, we find it by using T alpha and the degrees of freedom, which we know that it's N minus one. So our T alpha, alpha value is 5%. So it's T of zero comma zero five. 5% is five divided by a hundred and our degrees of freedom is N minus one. Our N is 30, so it will be 30 minus one. Therefore our critical value that we're going to look for will be where T of zero comma zero five and alpha of zero comma zero five and the degrees of freedom of 29. Now, the other thing that you need to be aware of is you need to go to your tables. I'm gonna open the stop sharing and go to the past exam paper. Let's see that today's session is taking longer but we go to the tables and let's see if they do give you a T table. Sorry, I'm trying to open the past exam papers. They don't give you the critical values but I can tell you what the critical value would be so that we can use that. Usually they will give it to you in the exam so you don't have to worry about finding it. Okay, it's fine. I can go back to my presentation instead of showing you the critical value. I'll just get it from the statistical tables and just continue. Yes, I can. I'm opening it in the background so I can get the right values and we can move on. Okay, so critical value of T looking for 29 and zero comma zero five and because we're doing only one test. So zero comma zero five and 29, the critical value is one comma nine nine one. So the critical value here will be one comma six nine nine. Nine nine one. So we're going to make use of that critical value when we make a decision. So moving on to step number three, which is calculating the T test, which we know the formula is the sample mean minus the population mean divided by the standard error which is your sample standard deviation divided by the square root of your sample size. So substituting the values, the mean, the sample mean was 90. The population mean you get it in the first statement that you stated, which is 100 divided by the standard deviation, which is 18 divided by the square root of T. And the answer we get, I'm going to use my calculator quickly. It's 90 minus 100 divided by into bracket 18 divided by the square root of T. Close bracket. And the anti-get is minus, minus 3.04. Minus 3.04. I'm going to leave it in two decimal, it's fine. That's step number three. Step number four says we need to make a decision. So let's make a decision. So step number four, we make a decision based on the critical value. So I'm going to draw this and I'm going to replace my T alpha with the real value, my critical value. Since it's on the left side, you can see it's on the left side. So I'm going to put here minus, it's going to be a negative, 1.6991, that will be my critical value. So from that critical value, I look at my value of the T's test statistic. It is minus 3.04. And if you look at it, it will be on some way, on this side. Therefore, it falls in their rejection area. And since it falls in a rejection area, so anything on this side, you reject the null hypothesis. Therefore, we can conclude by saying, since there's, oh, not the Z, this is the T test, the T's stat is less than your T critical value. Probably I should have put the values as well. Test statistic of minus 3.04, it's less than negative 1.991. Since the test statistic is equals to negative 0.04, it's less than the critical value of negative 1.69, we reject the null hypothesis. Oh, I don't even have to repeat in with the null hypothesis and conclude that the mean is different from 100. The mean reading time of adults with ASD is not equals to 100, who is different from 100. And that's how you make your conclusion. We reject the null hypothesis. Any question, any comments? Are there any questions? Are we good? Let's see, there are no questions. So let's move to the next slide. I was gonna do this one, or maybe we can do this one because this one talks to two-sided tests. So let's do this one quickly. So in a sample of N is equals to 20, selected from a population, the sample mean of 58 and the population standard deviation of 12. Suppose that an E-Twitter wants to test the following hypothesis. The null hypothesis states that the mean is equals to 55 and the alternative states that the mean is not equals to 55, at 5% level of significance. So now on this one makes it easy. We can also, they already identify some of the things for us. So our N is 20, sample mean, which is x bar is 58, the population standard deviation, which is sigma, therefore the population standard deviation is known and the hypothesis and then our alpha. So step number one already stated there. I'm not going to state it again because the null hypothesis and the alternative hypothesis are stated. So they already stated step number one for us with those statements. Step number two, we're doing a two-tailed none, a two-tailed test and this is a non-directional. It's a non-directional two-tailed test that we are doing. Therefore it means we're going to have two regions of rejections. So our decision will be based on two sides and because our population standard deviation is known, therefore we're going to use z alpha divided by two, z alpha divided by two, so because of that. So we can go and find the critical value. So the critical value for z alpha of 0, 0 2, z alpha, which is z of 0, 0 5 divided by two, which is z of 0, 0 2 5 is equals to 1, 9 6. When you go to the z table and you go and look for 0, go and look for 0, 2 5, you will find that when you go in, so you will have to go inside the table and look at the smaller portion and look for a value of 0, 0 2 5 and you will see that it corresponds to the critical value of, maybe I must show you how to find it if in case you need to find it on your tables. So how to find the critical value, remember your tables, so you go to this table, you look at the smaller portion, you look inside the smaller portion for a value of 0, 0 2 5, 0 probably inside this table and then we're going to look for a z value that corresponds with that. So when you scroll down and 0, 0 2 5 is that value and the z value that corresponds with it is 1, 9 6 and that's how you find the critical value on the table. I hope you were able to see that. We can go back to our presentation. So we have our critical value, so therefore we have our negative, negative 1, 9 6 on this side and negative 1, or positive 1, 9 6 on this side because we have two regions of rejection. Now how do we go to number 3? How do we calculate the test statistic? Is z that is given by the sample mean minus the population mean divided by your population standard deviation divided by the square root of n. Our sample mean was 58 given to us minus our population mean is always in the hypothesis statement which is 55 divided by our population standard deviation was 12 divided by the square root of n of 20 and the answer we get is 58 minus 55 divided by open bracket 12 divided by the square root of 20 closed bracket and that gives us 1, 1, 2. I'm just going to leave it in 2 decimal 1, 1, 2. Now we get to the last step making a decision already. We know that our critical value 1, 9, 6 and the site negative 1, 9, 6 where is our 1, 1, 2? Our 1, 1, 2 falls somewhere. Yeah. It falls 1, 1, 2. Let's assume that it falls somewhere here. And since it falls here it's in the white area not in the shaded area then what we do is we fail to reject the null hypothesis. That's what we do. Remember? We do not reject the null hypothesis because it falls in there. If it fell in the shaded area we would have said we reject the null hypothesis. Fail to reject the null hypothesis and state that the mean is equals to the population mean is equals to 55 and that's how you look at hypothesis testing questions. I've just given you at a high level the steps of hypothesis. We left with 5 minutes. I just want to show you the types of questions that you will get in your assignment relating to these steps. Remember you need to know and remember that all these steps they can be part of the questions that they ask you. They can ask you to calculate the test statistic. They can ask you to identify which test statistic you are doing. They can ask you to make a decision. They can ask you how you make a decision. For example, what I haven't used here is the p-value. How do I then find the p-value and make a decision? With the p-value you will have the null hypothesis and the alternative. You will do your directional and non-directional so you will determine that because you will have to state whether how are you going to find that p-value. You will calculate your z-test statistic and once you have calculated your z-test statistic you're going to use this as a z score and go find the probability on the table. Depending because this is on both sides right? When it's on both sides the value we get on the test statistic we're going to add both of them onto each other. So you're going to use what we call the small area on them to find your value on the table. So you're going to go into the table and on the small value smaller portion you're going to find two probability one probability and then you're going to say your p-value because it's on both sides you will say your p-value will be equals to 0,1314. That's the value you will find on the table but you need to multiply that value by 2 because there are two sides to it which then it means your p-value would then become so when it is two sided you multiply your p-value and your p-value will be 0, 26 28. That will be your p-value and then from that p-value then you're going to make a decision. So your step 4 using the p-value would be if the value of your p-value is less than your alpha value p-value is equals to 0, 26 28 which is less than your p-value of 0 oh it's greater than not less than which is greater than your p-value of 0,05 therefore you fail to reject. If it was less than we will reject and the decision that you take on the p-value and the decision you take using the critical value you should come to the same conclusion as well and that's how you use the p-value if it was only one area then you will use the value you find on the smaller portion you will use that value that's how you do the decision. I had some exercise or activities that I also wanted to share with you bear with me to give me 2 or 3 minutes of your time then we will be done the hypothesis the alternative hypothesis of the mean less than 30 is a mmm hypothesis and requires a mmm test now look at this value there the subscript should tell you what it is the sign tells you what type of test statistics you are doing so also the because that is the alternative also look at the sign because it's on the alternative so this helps to determine whether this is non-hypothesis or alternative so this is important to state whether what type of a hypothesis is this is it a directional hypothesis or a non-directional one type of a statistical test is it a one tail test statistic or is it a two tail test statistic that's what you need to find out so when you have a less than it's a directional and because it is a less than will be a one tail test so which one will that be it will be option C so you just need to think about everything that we discuss today and see how you apply it to answer the questions as you get them in your assignment when applying a statistical test P value represent the probability of obtaining the mmm number A says is the sample statistic under the alternative B is the population parameter under the non-hypothesis and C is the sample statistic under the non-hypothesis all what you need to always remember is that on the when you state the non-hypothesis and alternative hypothesis we never use a sample statistic always remember that right so if we are not using the sample statistics then we can cross that out and we will know that the correct answer will be you need to always remember in your non-hypothesis and alternative hypothesis we always state those statements by using the population parameters now when it comes to the types the errors you also need to remember and know the types of errors what type 1 error means and what type 2 error means type 1 error remember it is when you reject the non-hypothesis when it's true or type 2 error will be when you fail to reject the non-hypothesis when it's false so if we know those 2 statements how can we answer this so type 1 error will be that the non-hypothesis is wrongly rejected or the alternative is wrongly rejected remember I spoke about this earlier I've never mentioned anything about alternative and when we make a decision we never use the alternative so statement number 1 statement number c will be out always when you do your type 1 error or type 2 error it's when you are making the decision and when you are making the decision you always use the non-hypothesis so when do we do type 1 error when we reject the true non-hypothesis so it means a will be correct because a says the non-hypothesis is wrongly rejected because if you are rejecting the true non-hypothesis you are wrongly rejecting the non-hypothesis and that concludes today's session I'm just going to give you these 2 questions as your homework you can go and look at them you can see this question as well they are asking you to calculate certain things and looking at this question which of the values below is the closest to the correct value of x x bar which is your standard error which one of these values which is something that we didn't discuss a standard error what is a standard error a standard error is your standard deviation divide your standard deviation by the square root of the sample size that is your standard error so this value represents your s divide by the square root of n that's what you just do you have your s you have your n you substitute into that formula and you answer the question that's all what you need to do other questions they might ask you to make a decision as you can see here they have the value of your alternative as greater than which means it's a one tail directional test and they say the research should test again the null hypothesis against the alternative if there you need to choose the right value here now based on that this statement they want you to say whether what statement will go in your null hypothesis or how you are going to reject your statements so you should test your null hypothesis always remember in your null hypothesis there is always an equal sign but also this statement is asking you to make a decision this is more or less about the decision so if this is greater so it cannot be that it cannot be that and the sample mean it's larger than it cannot be remember also the null hypothesis the alternative hypothesis you must always remember those things so you will test it if your p value is less than the level of significance that will be how you will reject the null hypothesis and test for this which makes it only option number four will be the correct answer for this question some of the questions are trickier but it needs you to to understand the basic concepts that you would have learned about hypothesis testing and that concludes today's session always remember with hypothesis testing to state your null hypothesis and alternative to find your decision method by looking at the level of alpha your critical value whether you're doing a non-directional or a directional if you are using a p-value if the value of your p-value is less than the value of your alpha you reject the null hypothesis and so on you need to always constantly remember this especially when you're doing your test your hypothesis test and that concludes today's session for those who are joining us for the first time just to introduce myself I'm from Pambili analytics the company that cares about successes of everyone else we deal with literacies data literacy, analytical literacy we try to close the gaps in relation to those type of literacies we want to create data literate population especially in South Africa and in the rest of the world we do offer a variety of services where skills development is one of the areas but we do mostly consulting we have different 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