 Welcome students to today's presentation on hypothesis testing, and it's, as you understand, one of the most important parts of statistical research and quantitative research. And we have some idea about how to perform these quantitative tests. So this is a kind of repetition of what we've done already. So let's begin this presentation with talking about the research basics first. And these are things that we already know about these nine points of research, which begins from a topic selection. Basically, we decide what is the topic that I want to work on. And then we do a systematic literature review of that and have a conceptual theoretical framework. And finally, decide on either the research questions or a hypothesis clarification. And that leads to research design, data collection, analysis, conclusions and results. So as we understand that hypothesis clarification or research questions are a central part of our research basics. So it's important that we have a fair idea about how to do hypothesis testing in our quantitative tests. We also have some idea about what are independent variables and what are dependent variables. Dependent variables are the ones from which we get data. And independent variables are the ones which are thought to play a causative role in a cause-effect relationship. So they are either manipulated or different kinds of independent variables or different elements of independent variable are changed to see the effect on the dependent variable. So that is why often we use the term predictor variable and outcome variable in place of dependent and independent variable. Of course, the influence of moderating variables, the mediating variables and the control variables must also be known. This is just for reference. And once we have those data with us, we have to find out about the normality or abnormality of the distributions. So this is just to repeat what we already know. One of these is a normal distribution, the well-known bell curve and the frequency of the scores right at the middle is the highest. It basically clusters around the mean. So when we have a normal distribution, the majority of the observations would be spread around the mean. And in today's discussion on hypothesis testing, I will just talk about how far they are distributed so on and so forth. When it's a skewed distribution, it could be left skewed or it could be high skewed. Higher scores would have greater frequency or lower scores have less frequency. So that is how we get a skewed population. In many of the parametric tests that we will talk about today and in many of the hypothesis assumptions that we'll make, normal distribution is a very important requirement in statistics. And we'll talk about those requirements of parametric statistics in today's discussion as well. So this is a very, very important distribution, the normal distribution and we must know the very basics of this normal distribution before we get ahead. And this is just the standardized scores and when we talk of standardized scores, we subtract the observation from the mean and divided by the standard deviation. So this is the probability density function of this normal distribution. So there are two or three very, very important things that we have to understand here. And that is why these normal distributions are important and that is why we can make inferences from statistical distributions. We can make inferences from these observations. So I will give you some live examples also, but before that let's just talk about what this means. So if I can just zoom it a little further to explain it even better. So if you see that the z score here is zero because we have subtracted the observed values from the mean. So this is where it is centering around. So it could be any other score as well. Important thing to note here are these two points, minus 1.96 and plus 1.96. So if you have a mean and if you have this distribution, then 95% of your population or 95% of your observations will be between minus 1.96 and plus 1.96 standard deviations on either side of the mean. So if I subtract minus 1.96 times the standard deviation and if I add 1.96 or we can even round it off to 2. So between minus 2 standard deviation and plus 2 standard deviation, 95% of my observations will lie. Or the probability of all the observations lying between minus 2 and plus 2 is 95% or 0.95 if I can say that. So that is why this is such an important function as you can understand the probability of the small thing right at the end. It's just 0.025 and 0.025 on the either end. And in today's discussion I will explain that as well to make assumptions or to draw inferences about our hypothesis. So when we have a normal curve, two very important things to know. First of all, it will be spread around the mean and then this minus 1.96 and plus 1.96 standard deviation is an important function because these are the boundaries for 95% of the observations to lie. So this could be for example, you know, the height of people. So if I say the average height is 5 feet 10 inches of minute being all then if you just from 5 feet 10 inches. If you subtract, say for example, the standard deviation is 2 inches. For example, I'm just giving a hypothetical figure. So if 5.10 inches is the mean, then, you know, majority of the people will be around the mean and two standard deviation would be 5 feet 10 inches minus 4 inches. That would be 5 feet 6 inches. And, you know, two on the other side will be 6 feet 2 inches. Between 5 feet 6 inches and 6 feet 2 inches will be the 95% of the population. The outliers people who are below 5 feet 6 or above 6.6 feet 2 will be much less. So this is just from an everyday example. So when we talk of these normal distributions and when we talk of these probabilities, these are related to everyday life. These are not things which are unreal or these are not abstract concepts. These are very real concepts and we must be very clear about these concepts. And to take this rule further, this is known as the 6895 and 99.7 rule. So if I take one standard deviation away from the mean and one standard deviation further from the mean, then 68% of the population will be between minus 1 and plus 1 standard deviation. Keep on referring to these terms, mean and standard deviation. In another class, I've already explained what is mean and standard deviation. So I'm not going to get into the details of that. But these are two parameters which are so very important for any kind of a distribution. And if we know these parameters, we can have these parametric tests or the hypothesis testing that I'm going to talk about today. So 68% of the population will be between minus 1 and plus 1 of the mean. So if I have to talk about the earlier example I just gave you that the mean height is 5 feet 10 inches and the standard deviation is 2 inches, that means that between 5 feet 8 inches and 6 feet will be 68% of the population. Similarly, if I have to take the 95% of the population, that will be between minus 2 standard deviation and plus 2 standard deviation. If I have to talk about the 99.7% of the population, that will be between minus 3 standard deviation and plus 3 standard deviation. So as you can understand, this is such an important thing to know about these normal distributions. So this is not just about the shape of the curve and all that, but it gives us such an important input that if I know the mean and if I know the standard deviation and if I know that it's a normal curve, then with certainty I can say that 99.7% of the observations will be between this and this value. So this is such an important inference to draw from just two or three parameters as I just spoke of. I have spoken earlier just to give you an illustration of what standard error means, although in hypothesis testing we are not going to talk about a lot about standard error. Standard error basically means that if we have a population, say for example which has a mean of 3, and if I draw samples out of that, all these samples will have different means. So it's not possible that when I draw from a sample, so if it doesn't say for example a sample of people in Hadov, for example, and if I take samples of different kinds, then the mean of those samples will be different. And at times it will be very similar to the population mean. At times it will be different from the population mean. So the population mean is 3, at times the sample mean. So just try and understand the difference between the population and the sample. So if the population mean is 3 and the sample mean is 3, means they are similar. But at times they might be less or more. At times when I draw samples from the population, their means might be less than the population mean, or it can be higher than the population mean. But if I draw them on a curve or if I just plot them, then I will get a normal distribution. And the standard deviation of these sample means, the standard deviation of the sample means is known as the standard error. So that basically tells us how much the, how much is the range in the population. So we are looking for standard, smaller standard error. So error does not mean the literal meaning of there being some systematic error in that. It's just a definition of standard deviation of the sample means. So this is what I've just explained. So when we talk of these hypothesis testing, we are assuming them to be parametric basically. And parametric tests assume statistical distributions. And they also assume three or four other conditions which I'm going to talk about. So when you have a fairly large population whose parameters are known, and they satisfy the conditions which I'm just going to talk about, then we will do the parametric test which I will talk in a moment. Non-parametric tests like the chi-square test I did some time back, they do not rely on an indistribution and they can work even on smaller samples. So even if the conditions of validity are not met, we can do those non-parametric tests. There are these four assumptions for parametric tests. First of all, they are independent. So independent means that every observation or every item has an equal probability of being chosen. So they are independent, normal. I've just spoken of those normal distributions. Homogeneity of variance means the variance in the population or the variance in the observations is homogeneous across the sample. It is not kind of clustered at one end and it's not thinner at the other end. And most importantly, the measurement has to be an interval scale. We know about the three or four different kind of measurements. We know that there are nominal scales where we just talk of categories. It could be categories of states or categories of people or it could even be binary categories like male-female or those kind of categories. We have the nominal scale. We have the ordinal scale where we talk in terms of ranks, second highest, third highest, fourth highest or whatever. And then we talk of these continuous scales where they are basically numbers. So the numbers can be interval or ratio. In ratio, we are looking for an absolute zero, but we don't have to get into that. The first part is that it should be independent, normal. There must be homogeneity of variance and the scale must be interval. So it should not be a nominal scale. It should not be ordinal scale. It must be about numbers. So it could be about your salary, your age, your marks, how much you can jump or your IQ or even the Likert scales are regarded as numerical interval scales. So these are the assumptions that we make in parametric tests. Hypothesis, we have to distinguish between what is my statistical hypothesis and what is my research hypothesis. So one example which I love giving these days is about, my hypothesis could be that the time you spend on online classes has some impact on the marks you get. So if you spend better time on online classes, you get better marks. Or if you spend higher time on online classes, you might get higher marks. So that could be one of my research hypothesis. So this is a premise or a claim that we want to test. Or it's a prediction that we want to test. And we are suggesting that, okay, if a student spends more time on online classes, he or she will get more marks in their examination. So this is my research hypothesis and this is what I want to prove. So that is the research hypothesis. In statistics, we have to start with a different thing. In statistics, we have to start with the null hypothesis because that is something that can be proved or disproved. So very, very important to understand the difference between my research hypothesis, which is a prediction or a claim or a premise that I want to test. And the statistical hypothesis where we are beginning with a null hypothesis. Because null hypothesis is something that can be proved or disproved. So very, very important to understand the distinction between two hypotheses. It's very different from the research hypothesis that we will talk about or which we generally talk about. So what is a null hypothesis? Null hypothesis assumes that there is no effect or it starts with the assumption that there is no effect. So it will assume that doing online classes or not doing online classes has no effect on your marks. But even if you do a lot of online classes, it has no effect on the marks you get or you do very little online classes, it does not have any effect on the marks you get. So this is what we start off with. Null hypothesis is an important tool because that is something that can be proved. That is something we can prove as we go along. Or with the mathematical examples that I just spoke of using a normal curve or whatever, these are the things that can be proved. So we can begin with or we have to begin with null hypothesis during our statistical tests. So we begin with the hypothesis that there is no effect or whatever that I'm going to prove in my research hypothesis, we have to begin with the contention that there is no effect. So it could be about anything. It could be about whether exercise has any impact on your weight loss. So you start with a null hypothesis that exercise has no impact on losing weight or it could be about watching too much of television does not make me dumber. So there is no effect of watching too much of television. So eating 500 grams of chocolates has no effect on my health or on my dental structure or whatever. So we always start off with null hypothesis because that is something which can be proved with the data we have. So we have to have data and based on that data we can prove or disprove the null hypothesis. So we always, always, always begin with null hypothesis because that is something that can be proved with the observations. You can just have those independent and dependent variables and the observations and based on that after performing some very simple statistical tests using Microsoft Excel or SPSS or manually you can prove or disprove your null hypothesis. So I again want to emphasize that research hypothesis is very different from statistical hypothesis. In statistical hypothesis, we begin with null hypothesis. So basically what we do in null hypothesis is that basically what we do in statistical hypothesis testing is that first of all we formulate a null hypothesis. Null hypothesis suggests that there is no impact. That there is absolutely no impact between there is no impact of the research that we are trying to study. Then we formulate an alternative hypothesis and the alternative hypothesis is what my research hypothesis is. So if I want to suggest that there is no impact of online classes on marks the alternative hypothesis will be that there is an impact of online classes on marks. Whether it is positive or negative we'll just see in one-tailed and two-tailed tests in a moment's time. But important to understand that this is the alternative hypothesis is strengthened or my conviction in alternative hypothesis is strengthened when the null hypothesis can be proved to be wrong. So I'm looking to prove the null hypothesis wrong. So that is what my idea is, proving the null hypothesis wrong. And for that we have to start off with a test statistic. So it could be t-test, I will talk about that. And we have to find out whether our value that we are getting is higher or lower than that critical value. And then we'll have to specify an alpha level. I will just talk about those alpha levels, the p-values in a moment's time. And then the assumptions that we make and we have to calculate those test statistics. And then finally we reach the conclusion. So it's important that we know about these three things that are happening. So if I have to just put it very simply. So first of all we have to have a null hypothesis and the alternative hypothesis. Then we have to have the alpha level and test statistics. I will just talk in a moment's time about what is alpha level and what is p-value. And it's very important for students of quantitative methods to know these things. And finally we have to calculate those test statistics and we have to draw inferences from there. So these are the basic three things that we are doing in the null hypothesis significance testing. So we have to now understand the most important part of hypothesis testing which is about p-values or the probability values. And we have to give some time to understand all this. So let's just talk about the probability. So as I said we have to prove or disprove the null hypothesis. What is the null hypothesis? Null hypothesis means that there is no effect. There is no effect of online classes. There is no effect of exercise. There is no effect of eating chocolate. There is no effect of watching television. So we are trying to prove that the null hypothesis is wrong. And to prove null hypothesis wrong we have to find out a probability value. Probability means how probable that is. So we generally want to put that probability to very less. And generally that value is taken as 5%. So please try and remember that these are about probability of the null hypothesis being true. The probability that there is no effect of these variables. So we are beginning with this p-value which is the value for the probability of the null hypothesis being true. And we keep that at maybe 5% or at times we keep at 1%. So the usual criterion is 5% would mean 0.05 or it would also mean 0.01 at times. So if the probability is very small, if the probability of the null hypothesis is very small, then we reject the null hypothesis. If the probability of the null hypothesis is very small, very small means less than 5% or 1% whatever value you keep. So what I am trying to emphasize is that this value is decided by us, by the researcher. That it could be 5% or it could be 1%. So if the probability is very small, then we reject the null hypothesis. And if we reject the null hypothesis, we gain confidence in the alternate hypothesis. But as you can understand that we are keeping that value point as 5% or 1%. So that means that there will be some error. There are times when the null hypothesis will be true but we will be rejecting it. Because we have put in a value, we can never be certain in statistics. It can never be absolutely certain. There will be some value however small of the probability of certain other thing being true. So there will be errors. Why? Because we have put that value. So if we put that value at 5%, the error will be more as you can understand. If I put that at 1%, then the error will be less. Or if I put that at 0.01%, then it will be much lesser. So it depends on the kind of test you are doing. In the case of clinical examinations, that has to be very less. The error has to be kept much, much less. Because type 1 error means false positive. What does false positive mean? It means that I am rejecting the null hypothesis. When in case the null hypothesis is true. So I am sure that it should not confuse you. Because researchers do often get confused between type 1 and type 2 errors. But we must be very, very clear about what these means right at the outset. That type 1 error means false positive. Means we are rejecting the null hypothesis when the null hypothesis is true. So that depends on your value that you have decided beforehand. As I said, it depends on the discipline. In social science, we often take it as 5% or 1%. In behavioral science and psychology etc. as well. And then there is the type 2 error of a false negative. That we are accepting the null hypothesis when actually the null hypothesis is false. So these errors have their own repercussions. And we will talk about those repercussions in a moment's time. So important to note that the p-value that I am deciding, whether it is 5% or 1%, this is going to decide how important or this is going to decide what my errors will be. But that value, if I get my probability less than these values, then I will reject the null hypothesis. So that is why this null hypothesis significance testing is dependent on these p-values. But there are problems which we will see in a moment's time. So this is just again what I have just discussed right now that when the null hypothesis is not true and we reject null hypothesis, then that is what we are doing correctly. But when the null hypothesis is true, that means there is no relation. But even then we reject the null hypothesis, that is the type 1 error. And type 2 error when the null hypothesis is false actually, but we accept null hypothesis. So that is a false negative. So very important to understand what is type 1 error and what is type 2 error, because that has some impact on a lot of the other tests we do. Important to understand that the distinction between one-tailed and two-tailed test. So I will just give you an example of these one-tailed and two-tailed tests. So this one-tailed test means we are just talking in one particular direction. We are talking about the mean being greater or lesser than the null hypothesis. So that is the alternate hypothesis. And the other, so this is one-tailed. One-tailed means it is saying that it is greater. If you can see right at the top it is saying greater. If we just say it is different than the mean, then it will be two-tailed. And we will talk about that in a moment's time. But if we are saying that the mean of the sample is greater than the population mean, then we are talking of this one-tailed test. What is the other hypothesis? This one is again a one-tailed test. So this is suggesting that the mean of my sample is less than the population mean. So one-tailed test is directional. It is telling you either this is greater or it is lesser than the sample mean or a particular value. So if I want to suggest that more time spent on online classes causes an increase in marks, this is one-tailed because this is telling you that the marks will increase. Or the other way that if I spend more time on entertainment on these online channels, then it will lead to decrease in marks. So that could be one-tailed. So very important to understand what is one-tailed and two-tailed. If you see this, this is two-tailed. This is just suggesting if you see right at the top that my sample mean is not equal to the population mean. So not equal means it can be either less or more. So there is a very important difference between these one-tailed and two-tailed tests. In one-tailed tests, we are saying either it is less than or greater than the mean. And in the two-tailed test, we are saying that it is just different. So why are we interested in one-tailed and two-tailed tests? Very, very important to understand. Because one thing which is related to the p-value is also the critical value. So they are basically the same. When we are disproving the null hypothesis, we have to either suggest that the p-value that we are getting is less than a particular value or the critical value. It could be a p-value or your f-value or your z-value. The critical value in the present sample is more than that critical value. So if I'm having these one thing, the critical value for this is 1.65. So if I'm getting a critical value, more than 1.645, I will reject the null hypothesis. So as you can see this very importantly, that when I'm having these two-tailed tests and one-tailed tests, so these one-tailed tests are more stringent than these two-tailed tests. We have to be very clear about three or four things as we go along. One of it is known as the statistical power. So just p-value is not enough because there are lots and lots of problems with p-value. So the ability to find an effect in my sample is known as the statistical power. So very important to know what is statistical power. And statistical power is basically 1 minus beta. Beta is the type two error that I have just spoken about. So it's important to reject the null. So statistical power is the effect to reject null hypothesis when it is false. So that's a very, very important thing. So we also have to provide confidence intervals and effect sizes. And it's important that there are times when there might be some kind of correlation between two variables, but the effect size could be very small. And we have seen correlation and we know that the correlation coefficient could be very small. So there could be an effect size which is present, but it could be very, very small. So just p-value, just p-value significance testing is not very enough most of the times. And one of the tests we do in the statistical test is to do the t-test, which is a very good example of one kind of hypothesis testing. And what we actually do there is we try and find out whether there is a difference in mean between two groups. So is there a difference between marks of male students and female students? So my null hypothesis will be that there is no difference. And if I get that there is no difference in marks between male students and female students. But if I get my p-value as less than 0.05, then I will reject the null hypothesis and I will say that there is a difference in marks between male students and female students. So when I'm comparing means between two groups, I'm doing that kind of a t-test. So that again is a very, very simple way of doing hypothesis testing. So hypothesis testing is a very important kind of testing where we have to understand the limitations of the p-value and also know the impact of these normal curves and how much of it is more or less or whatever. So these are very, very important things to understand and remember. So that's all. Let me finish this right here.