Are you concerned that some Democrats may not support this? I would hope that all members would support a resolution that condemns terrorism, the brutal attacks that were perpetrated against the Israeli people that were killed. We have 218 hostages. They took 222. Someone who votes against this, I would think doesn't have a soul. You just saw comments made by corrupt Congresswoman Debbie Wasserman Schultz on October 25th ahead of the House vote for a resolution in support of Israel. Now, as you just saw, she preemptively attacked her Democratic colleagues who would vote against it. Now, at the time, she didn't know specifically who wouldn't support her resolution, but we all had a pretty good idea, as did she, women of color, like Rashida Tlaib on Omar, Cory Bush, and others. But she claimed that anyone who didn't support this didn't have a soul, which is an incredibly ironic statement to make because she's Debbie Wasserman Schultz. This is the same person who was forced to resign from her position as DNC chair in 2016 after leaked emails confirmed that she was covertly sabotaging Bernie Sanders presidential campaign in violation of the DNC's own charter. And on top of that, she was given the nickname Debt Trap Debbie for lobbying against regulations on predatory payday lenders after they donated tens of thousands of dollars to her and campaign contributions. And fast forward to today and Debbie is still just as corrupt as ever. Now, after taking more than $100,000 from the pro-Israel lobby, she sponsored a resolution on their behalf saying that the US stands with Israel even as they carry out war crimes with bombs that we gave them. And the resolution that she's referring to says that the US stands with Israel as it defends itself against the barbaric war launched by Hamas and other terrorists. The resolution makes no mention whatsoever of proportionality and it doesn't mourn innocent Palestinians murdered indiscriminately by Israeli bombs. And furthermore, it commits to additional aid in Israel and calls for sanctions against Iran. But she's saying if you don't support that, then you have no soul. Now, leftists did not support it because Israel's right to self-defense should not include collective punishment, the use of white phosphorus against civilian populations, and in total nine progressives voted against it. Bowman, Carson, Bush, Green, Lee, Ocasio-Cortez, Omar, Ramirez, and Tali. They were also joined by Thomas Massey, and additionally, six Democrats voted present. Kesar, Castro, Velazquez, Presley, Garcia, and Jayapal. Now, the question is why are we still talking about this resolution? And it's because that resolution caused a lot more friction within the Democratic Party than we initially thought. And we're learning now that Debbie Wasserman-Schultz's dumbass comments there, her saying that her colleagues of color don't have souls if they don't go along with her genocidal support, that really pissed them off, apparently. Surprising, right? And this is according to Manu Raju, but he explains what happened after she made that dumbass remark. And that comment sort of prompted outrage among key Democrats, and particularly the 15 who voted against that measure. Some of them, all of them minorities, some of them Muslim Americans, and some of them bringing this directly to the Democratic leader, Hakeem Jeffries, who has started to tamp down these tensions internally, tried to get his members on the same page as this divide has been growing. We're told that one of the members even talked to Debbie Wasserman-Schultz and was concerned that that is a racial trope that would be used calling someone soulless, something that could be interpreted very negatively among Afro-African-American members in particular. Wasserman-Schultz, we were told, indicated she was not aware of that in where she made that comment here. But Dana, all showing the divide that is just persisting as this work carries on, the debate within the Democratic Party that is now spilling out into public view, but very much a private debate as well. So needless to say, she pissed them off so much that one of them actually confronted her about that. Good. I'm glad that these Democrats are standing up for themselves because they're right, and she's wrong. They're right about that resolution, and they're right about the need for a ceasefire. Now the fact that this resolution created tension behind the scenes isn't surprising at all considering the fact that a lot of this tension actually spilled over into the public. For example, Josh Gothheimer, who took nearly $300,000 from the Israeli lobby, condemned his Democratic colleagues on Twitter for not supporting the resolution, writing last night, 15 of my Democratic colleagues voted against standing with our ally Israel and condemning Hamas terrorists who brutally murdered, raped, and kidnapped babies, children, men, women, and the elderly, including Americans. They are despicable and do not speak for our party.  I think he's not active in the role as a member of Congress. I think he's shown himself to be very emotional. Like most cowardly people, when you confront them, they're afraid. I'm unafraid of the guy.  And he should remember why the people sent him here. And if he wants to play some kind of tough guy or gangsta, we can handle it like gentlemen and we can get into something else. I'm pretty strong words that you really don't hear from a member of one, the same party going out, even members of opposing parties saying that I did add reach out to Godheimer himself. And I asked him about Mr. Carson's comments. Godheimer said, said, I'll sit down with Mr. Carson anytime to talk about how we can bring hostages home, including all Americans, provide immediate humanitarian aid to Palestinian civilians being used as human shields and crush Amos and all terrorists seeking to do as harm, trying to de-escalate things. I'm also told that they'll Carson and Godheimer are likely to meet according to a democratic leadership source. And by now, I'm assuming that they've already met.  Love it. Democrats like Godheimer and Wasserman Schultz are genuinely spineless. And they may actually be right about the fact that they represent elected members of their party. But let's be clear, they do not represent voters. The 15 Democrats who they're attacking constantly, those are the ones who are representing voters. Those are the individuals who are representing 80% of democratic voters who support a ceasefire in Gaza. And these same cowards, Debbie Wasserman Schultz, Josh Godheimer, predictably voted with Republicans to censure Rashida Tlaib along with individuals like Richie Torres and others. And I'm sure that you'll be shocked to learn that all 22 Democrats that voted to censure Tlaib received donations from APAC. What a surprise there. Now, we did a whole video on this, but APAC is a pro-Israel interest group that lobbies on behalf of the Israeli government, not the Israeli people or Jewish people. And they also donate to anti-LGBTQ plus and insurrectionist Republicans. But these Democrats took money from this far right racist organization. And now they're condemning one of their own colleagues because that's what this organization would want them to do. Now, like the vote on the resolution, the vote to censure also caused a lot more attention within the Democratic Party. And here's some of the immediate responses after the vote took place. Phrases like from the river to the sea. You're not simply advocating for the creation of a Palestinian state, you're advocating for the destruction of Israel as a Jewish state. And that crosses a line that no member of Congress should ever cross. It's hate speech and Congress has a right to condemn it. It is outrageous. I am embarrassed for those Democrats who voted to censure their own colleague, who voted against free speech. It is an embarrassment. Listen, I've been critical of Jayapal before, but good on her for calling them out and shame on Richie Torres. You know, as the chair of the Congressional Progressive Caucus, Pramila Jayapal could vote to remove Torres from that caucus. And I think that his support for a far-right ethno-nationalist war criminal government is grounds for his removal, considering that that doesn't really scream progressive to me.  And they're done biting their tongues. And it's really nice to see them finally fight back and defend themselves. For example, AOC responded to Tlaib Sancher saying, It is not lost on anyone how many offensive, violent, and racist things people regularly hear members of Congress say, yet virtually the only one that gets censured for her political speech also happens to be the only Palestinian American. It does not reflect well at all. Now, Cory Bush added, It's outrageous that my colleagues are blatantly attempting to silence the only Palestinian American representative in Congress. But it's not surprising. This is the same house that upholds bigotry and racism every day. Solidarity with representative Rashida, your voice belongs here. And to be clear, it's not just over at bigotry and racism from Republicans. It's casual racism that they also experience from Democrats as well. Democrats like Debbie Wasserman Schultz and Josh Gothheimer, who are quick to scold Muslims and black and brown people within their own party for not falling in line. But yet, they don't have that same energy for Republicans like Max Miller or Brian Mass who are literally calling for a genocide against Palestinians. No, they spend their time condemning the one Palestinian Congress who's saying, Hey, maybe we shouldn't allow Israel to indiscriminately murder Palestinians. But yet, people like Debbie Wasserman Schultz have the audacity to say that people like Rashida Tlaib and Cory Bush are the ones with no souls.   So Rashida Tlaib is courageous, as are the other Democrats who are doing the right thing here and challenging the status quo. And the last thing I'll say is support Rashida Tlaib. I'm going to leave a donation link to her in the description box. She is going to be primaried for this. And there's going to be a lot of Israel lobby money against her because she is speaking truth to power. And she's persuading people. So that's a threat to the status quo. And she's going to be crucified because of it. So I think that it's incumbent on us to support her during these really difficult times. So support Rashida Tlaib because she is somebody who has a soul, which is why she is acknowledging the humanity of Israelis and Palestinians and individuals like Debbie Wasserman Schultz don't even pretend to care about Gazans.


Hello and welcome back in today's lectures we will be looking at hypothesis testing. So we will be defining different types of hypothesis and how to handle different problem statements post the questions raised in the problem statements in form of suitable hypothesis and then define a test statistic use the appropriate statistical distribution find the probability and based on the probability we make certain conclusions. So the references for this topic are the books written by Ramachandran and so course mathematical statistics with the applications from academic press and the prescribed textbook by Montgomery and Runger applied statistics and probability for engineers fifth edition from Wiley India. So where do we apply this hypothesis testing procedures it is an important tool in decision making the person making a decision may be the manager it may be the owner of a company or it may be even a researcher what decision does a researcher make he performs experiments finds the response as a function of the variables he changed during the course of experimentation and the decision he makes are which of the variables are important and which of them do not have a bearing on the response of the process. So there can be different types of decision makers. So statistical inference involves hypothesis testing I really have not told you what is meant by a hypothesis I will do so shortly and parameter estimation a random sample is taken from the population and suitable statistics are obtained again we are going to use the random sample since the random sample is being used in so many places in so many different ways it is important that we ensure that the sample we have taken is indeed random and the sample elements are independent of one another and follow the same probability distribution again we have to emphasize that we do not know the parameters of the population we do not even know the nature of the population whether it is having a normal distribution or it is having a skewed distribution the random sample is used to make decisions on the sample distributions of the means the sample distribution of the variances and so on we have seen that the normal distribution the t distribution may be used in the tests carried out on the mean the enquiries made on the population parameter mu we use the chi squared and the f distributions to make enquiries regarding the variances of the population and also the ratio of variances. So the sample statistics in the form of sample mean and sample variance have a lot of significance another important thing is even though we are using the sample information we are always querying about the population parameters. So the sample information is used to make decisions on the parameters of an unknown population we do not know the mu and sigma squared and so we use the sample information as point estimates of the population parameters and then we make certain decisions how we make the decisions we will see shortly. Obviously this impractical to use the entire population for decision making. So we are forced to resort to sampling and use a sample information to arrive at a decision usually only one sample is taken from a given population. So I was just introducing the term hypothesis testing what is really meant by a hypothesis. If you look at the dictionary some of the synonyms of hypothesis are guess assumption speculation suggestion and initial conjecture. Basically a hypothesis expresses a statement it concerns with the probability distribution of a random variable and its associated parameters. So the hypothesis deals with the population parameters either mu or sigma squared. Again I am emphasizing when you are writing down a hypothesis or a statement you are involving the population mean or the population variance you are not using the sample mean and sample variance when writing down the hypothesis. Roughly you can see that hypothesis will involve mu or sigma squared it will not involve x bar and s squared in the hypothesis statement. Even though further analysis will be done with the sample mean and the sample variance the x bar and s squared do not find a mention in the hypothesis statements. So I said hypothesis statement and then I am saying hypothesis statements the reason for that is there are two hypothesis statements okay and neither of them involve x bar or s squared both of them involve either mu or sigma squared right. So doubt that may be in everybody's mind is we know alright hypothesis is a statement or a guess or a speculation but it does not mean that all the people would make the same kind of hypothesis statements. Well the hypothesis statement is usually binary guilty or not guilty. It is having a population mean value of 50 units say 50% marks or it is either greater than 50% or less than 50%. So the hypothesis statements are rather simple. So you can go ahead and make a hypothesis statement without bothering too much and if the sample data does not support your initial speculation or initial hypothesis then the hypothesis statement stands to be refuted or rejected. Then you can always go back and look at the problem history and problem data and come up with a new hypothesis. The preliminary hypothesis test you carried out will give you valuable clue as to what should have been the proper hypothesis statement. So the hypothesis testing involves procedures which help us to decide whether the original hypothesis made may be accepted or rejected based on the information provided by the sample. So you are using the information given or available in the sample to make decision whether the original hypothesis statement is correct or not correct. So we look at the difference between our assumptions and the evidence from the sample and then to decide whether the differences are statistically significant to warrant the rejection of the original hypothesis. For example you may be thinking that the nuclear reactor supplied by a company is going to give you a mean power output of 2.3 gigawatts. But the plant is actually giving a mean power output of only 2 gigawatts okay. So you are having 2.3 gigawatts as the population parameter mu whereas X bar is telling you it is only 2 gigawatts. So the difference is – 0.3 gigawatts. And whether this difference is statistically significant enough to conclusively say whether the mean output from the reactor is 2.3 gigawatts is an incorrect statement or it is a correct statement. So coming again we look at the difference between the X bar and mu. X bar is 2 gigawatts and mu is 2.3 gigawatts. So the difference is – 0.3 gigawatts. So we can see whether this difference of – 0.3 gigawatts is statistically significant enough through hypothesis testing procedures. If it is statistically significant enough then we can say that okay look this reactor which the company has supplied cannot give a mean power output of 2.3 gigawatts as originally claimed. However, if the test shows it is statistically insignificant then what we say is based on the data which is available to me I cannot really conclude that the reactor supplied by this company XYZ company is not supplying average power output of 2.3 gigawatts as claimed okay. There is insufficient evidence for me to actually reject the company's claim that the mean power output is 2.3 gigawatts. Again I will like to emphasize that hypothesis testing concerns with the parameters of the probability distribution of the population and not with the sample. However, it relies on the data from the sample from the population of interest. Always please remember that the hypothesis testing is all is subject to some uncertainty. Unless we examine the entire population we cannot conclude anything about the population with 100% accuracy or certainty. Since we are using samples to make inference upon the probability distribution of the population we cannot be certain about our conclusions regarding the population parameters made using hypothesis testing. There is a small typo here I will correct it okay. Since we are using samples to make inference upon the probability distribution of the population we cannot be certain about our conclusions regarding the population parameters made during hypothesis testing. We are speculating on the parameters of the population. Can the mean value be this much? Can the variance be this much? And these are the population parameter values. So after we take the information or evidence provided by the sample we do some test procedures and then we make a conclusion. Based on the evidence provided by the sample I cannot conclude that this population parameter mean can be 50 units or I can say that right based on the evidence provided by the sample I can conclude or accept the original problem statement that the mean is indeed 50. So you are making a decision based on the evidence provided by the sample but this decision cannot be considered to be conclusive or 100% fail proof. Hence we agree that there is a certain uncertainty in our hypothesis testing and we quantify it. So whenever we are speculating on the population parameter the mean of the population can be this much or the variance of the population can be this much. On what basis we are doing the speculation? We make these estimates or assumptions regarding the population parameters from prior knowledge or experience, from prior experiments and now we check whether the process parameter has changed. Suppose we are having a machine which is operating at a certain power other settings of the machine are also fixed. It is giving a average particle size of 50 millimeters. So we are using the machine for a long time and it has been working very well and so the average is very very close to 50 millimeters. So we can say that mean is equal to 50 millimeters. Suppose some modifications have been made to the machine maybe it went for maintenance check or some part was replaced then again the machine is grinding the raw materials and providing particle sizes. Now we want to know whether the previous mean value of 50 millimeters is again met by the machine or because of the process modifications it is producing particles which are lower than the original mean value or greater in size than the original mean value. So we know from prior knowledge or experience from prior experiments we have collected or accumulated large amount of data and after some modifications we want to see whether the process parameter has changed. Also the parameter may be set based on a mathematical theory or a model and we want to verify whether the data agrees with the model predictions. A parameter is proposed from engineering or design specifications or contractual applications and we have to find whether the current data supports the parameter. For example the XYZ company was guaranteeing 2.3 gigawatts from its nuclear reactors as the average power output. So that is based on contractual applications or design specifications and then we actually monitor the reactors performance and take random sample and find the mean power output we can check whether the mean power output of the reactor is actually 2.3 gigawatts or lower. So what is the procedure involved in hypothesis testing? We first take a random sample from the population of interest. Again this is a most crucial step. Sometimes its importance is underestimated. You cannot blame the final decision or the final outcome if the sampling was done properly. Compute the relevant test statistic from the sample. The relevant test statistic we are familiar with so far are the sample mean and the sample variance. We really have not looked into any other test statistic but the sample mean and sample variance would do just fine for us including our analysis of design of experiments. Well you must be curious what are we doing here now? I mean where is it going to tie up with design of experiments. I am eager to learn design of experiments. I would suggest you to be a bit more patient. We will be definitely looking at design of experiments very shortly and you will really appreciate all the concepts we have learnt so far when you see them being applied in design of experiments. Use the test statistic to make a decision about the original or null hypothesis. So I have introduced the new term original or null hypothesis. Let us now get into the defining of original or null hypothesis. So many real life problems encountered in engineering may be formulated in terms of hypothesis testing problems. Hypothesis testing forms the foundation of more advanced experimental design techniques. That is what I told you just a short while back that whatever we are learning now hypothesis testing the t-distribution, chi-square distribution, the f-distribution will find immediate applications in the design of experiments. In design of experiments as well as in linear regression tools we will be extensively seeing the application of the t-test, the f-test and also the 95% confidence intervals. So hypothesis testing and confidence interval estimation of parameters are fundamental tools applied at the data analysis stage in comparative experimentation. There are two types of hypothesis. One cannot exist without the other. The first hypothesis which is usually made or initially made is the null hypothesis and the second hypothesis is its contradiction the alternate hypothesis. In defining two hypothesis we imply that the rejection of the null means automatic acceptance of its alternate. So why do we make the two hypothesis statements? So any null original or initial statement or hypothesis is a speculation. So we cannot claim that it is 100% accurate to allow for the existence of an alternative to the original statement we have made. We propose or state the alternate hypothesis. There are different ways of disagreeing. The one way is a kind of neutral manner saying that I do not agree with you. The second way of saying is what you are saying is not correct. Actually the performance is better than what you are saying and the third way of rejecting the original statement or rebutting the original statement is the performance is lower than what you are claiming. So there are different ways to disagree. One is simply saying it is different or being bit more specific saying it is greater than or less than. So normally we are not talking here of philosophical arguments. We are talking about entities that may be quantified into numbers and mathematical formulae. So there are two hypothesis. One is the initial original or the null hypothesis and the second is the alternate hypothesis which is so stated that it contradicts the null hypothesis. How do you propose the null hypothesis? This is a very interesting issue here. You are going to a company after finishing your B-tech or M-tech. You are very enthusiastic and bubbling with energy and you find a process being done in the industry and you feel look this process is not being done correctly. If I make this modification the process will run much more smoothly or much more efficiently. Then you go and tell it to the management. The management of course welcomes new ideas but that would mean also a lot of commitment from their side in terms of manpower, time, money and they also do not know how the customers or the client would react to it. So the management would be rather inclined to let things continue as they were probably not as efficiently as you are considering but there is no immediate shutting down of the plant or revamping of the process. So things will continue to run as usual. So you have to provide a strong convincing evidence for them to stop everything and make the process modification. Even when you are doing experiments in the laboratory for your MS or PhD research program you have to be skeptical. You have to assume that none of the variables you are investigating is going to affect your process response. Well this is sort of in contradiction or opposition to our aim of doing the experiments. We want to clearly show that some of the variables or all the variables we are investigating is impacting the process in a strong manner but a true researcher will always be a skeptic. He will have to say initially that there is no change or status quo is being maintained. Even if you change the variables it is not going to affect the response. Of course this may not be true because obviously when you change something there is going to be a effect on the response. Judge sitting in the court may have to make the most important decision of all whether to allow a person to continue to live or to be sent to the gallows because he has apparently committed a crime. So the judges initial attitude will be look this person is innocent. The prosecution has to provide evidence beyond a reasonable doubt so that person may be convicted. So the null hypothesis is sort of a stabilizing one. It keeps us grounded and is based on the concept that there is no change or status quo is being maintained. And guideline for choosing a null hypothesis is when a new claim is being made the nullification of the claim is taken as the null hypothesis. According to Ramachandran and Sokars the nullification or the contradiction of the claim is taken as the null hypothesis. So the claim is I am going to increase the plant performance by 10% thereby making you a profit of 1 million dollars per annum. So that is a new claim being made. The management is cool to the idea and say that look your claim is not going to make a difference the plant will continue to run as before. So null hypothesis is always the nullification or the neutralization of the claim that is being made. Anyway we cannot live in speculation we have to try out different ideas and then we have to come to a conclusion. After hypothesis testing we conclude that the null hypothesis cannot be true then we reject it in favor of the alternate hypothesis. If the alternate hypothesis is not true then it means that we cannot reject the original hypothesis. We are not making an absolute statement that the null hypothesis is correct. We have to say perhaps in legal terminology that enough evidence was not provided to establish beyond reasonable doubt that the null hypothesis is wrong. I am repeating enough evidence that could establish beyond reasonable doubt that the null hypothesis is wrong was not provided. Hence there is no sufficient evidence to reject the null hypothesis. We are not claiming or we are not stating that the null hypothesis is true. We are only saying that sufficient evidence was not provided to reject the null hypothesis. We use the sample data and identify a test statistic which is a function of the sample measurements using which we try to establish the null hypothesis or its alternate and subsequently make a decision. So far we have seen what is meant by a null hypothesis and what may be the forms that may be taken by the alternate hypothesis. Now we have to see how to actually go about carrying out the hypothesis test procedure. So when you formulate the hypothesis statements we can take an example from the legal side. So as I said previously the null hypothesis is the person is innocent. The prosecution vehemently wants to prove the case and send the charged person to jail. They have to come up with suitable and clinching evidence. So they are working towards the rejection of the null hypothesis. The management is making a null hypothesis that the process modification is not going to make a difference. Whereas you the originator of the idea will be actively looking for suitable evidence to indeed prove that your concept is correct and the null hypothesis can be rejected. So if the prosecution is unable to provide evidence beyond reasonable doubt that the person is guilty then the person is released on the grounds of insufficient evidence to prove his guilt rather than the firm statement of his innocence. It is indeed probable that the person was guilty but the prosecution simply could not establish his guilt. So there can be different types of alternate hypothesis. If the alternate hypothesis contradicts the null hypothesis by saying that the parameter under test is not equal to what was proposed in the null hypothesis. It is called as a two-sided alternative hypothesis. The null hypothesis proposes a statement that the population parameter is equal to a certain value. The alternate hypothesis considers the possibility that based on the evidence provided the population parameter may be less than the proposed value or greater than the proposed value. Because we are talking about random phenomena when there is a negative deviation on one occasion. On another occasion there may be a positive deviation. So in order to account for such kind of deviations we are using the alternative hypothesis in terms of not equal to. The original null hypothesis may say that the population mean value is 50. The alternate hypothesis may be the population mean value mu is not equal to 50. Under such situations we allow for the possibility that the population mean may be greater than 50 or less than 50. So we are allowing for the two-sided possibility. So it is called as a two-sided alternative hypothesis. If the alternate hypothesis claims that the parameter is either greater than or lesser than what was claimed in the original hypothesis it is termed as a one-sided alternative hypothesis. For example in the nuclear reactor case you are having a null hypothesis of mu is equal to 2.3 gigawatts. The sample mean is showing you a value of 2 gigawatts. So the company or the industry which is using that reactor sees red and claims that no no no the parameter value of mu of 2.3 gigawatts is not correct. The actual average power output from the reactor is less than 2.3 gigawatts. It will not even consider the possibility that the mu average power output can be greater than 2.3 gigawatts. It will say it will be less than 2.3 gigawatts. So this is a one-sided alternative hypothesis using the less than sign in the alternate hypothesis statement. Alright. What about the alternate hypothesis involving greater than? For example you are monitoring the air quality in a particular place where recently there has been lot of industrial activity. So the null hypothesis which the new industrial companies will be interested in is that there is no change in the pollution levels. The average pollution levels are the same as they were originally before the industrial activities started. I am using the advanced or state of the art pollution control measures. So I am not letting out really any toxic gases to the atmosphere. So the mean population levels of the pollutants in the ambient air is the same. It is unchanged. But the pollution monitoring agency may be more interested in proving that the mean value cannot be what was originally it has indeed gone up. In which case you are going to use the greater than sign in the alternate hypothesis. So the null hypothesis will be mu is equal to mu not the original or the baseline value. The alternate hypothesis for the industrial pollution problem is mu greater than mu not. So after the industrial activity has started there has been an increase in the pollution levels. That is what the environmental pollution agency will be looking for. It is really not going to be interested in the possibility mu less than mu not. It is interested only in mu greater than mu not. So again we will be going in for a one sided alternative hypothesis and that would involve the sign of greater than in the present case. So after the judge has passed a decision regarding the innocence of the accused either way he may have found him not guilty or may have found him guilty. So there will be always question did I release the person wrongly or did I send an innocent person to the prison. So these kind of doubts will always be there. So whenever we have made a decision with the best of intentions we still may have the doubt that our decision was not correct. So how to quantify these kind of errors in decision making. From a common sense point of view we make a decision after giving a suitable margin so that we do not make the wrong decision. The margin may be quite lenient. For example if students are writing an exam our expectation would be the student should get at least 60% marks to pass the course. For example the exam is so easy in my opinion they should get at least 60% if they have really understood something from this course and then pass it. But we set a pass mark of 40% just to be on the safe side so that no person who has reasonably understood the course is failed. So the errors in decision making are wrongly accepting the alternate hypothesis. So the null hypothesis in fact was correct but you have accepted the alternate hypothesis wrongly. The second one may be failing to reject the null hypothesis by rejecting its alternate. Which of these two is a more serious error? Let us again take the court case and the first error wrongly accepting the alternate hypothesis would mean the person was in fact innocent but the judge based on the evidence presented to him wrongly accepted the prosecution's arguments and send the innocent person to jail. That is a very serious issue or the management got carried away by your technical presentation and accepted your idea spent lot of money and unfortunately the idea did not really work okay. That was not any significant or noticeable improvement in the process after the modifications were carried out okay. The second error is to fail to reject the null hypothesis by rejecting its alternate. So what it really means is the person was indeed guilty but the judge released him okay. So what really happened was the guilty person got away free. When you compare the two types of errors the first one is pretty serious okay. The second one is also an error but it is not as serious as the first error. The first error an innocent person was sent to jail or he was hanged or whatever and in the second case the guilty person got away. So which of these two is more serious you yourself will be able to answer. So our decisions are associated with uncertainty. There are also probabilities associated with the wrong decisions. The probability of rejecting the null hypothesis when it is in fact true is called as the type 1 error and it is termed as alpha. We have seen alpha earlier when we were constructing the confidence intervals. The probability of accepting H0 when it is in fact false is the type 2 error and it is termed as beta. So probability of rejecting H0 when it is true is called as the type 1 error and it is denoted by alpha. The probability of accepting H0 when it is in fact false is called as the type 2 error and it is termed as beta. Maybe I should write the probability of failing to reject H0 when it is in fact false is called as type 2 error. So when we come back after a small break we will be looking at this table and we will see after a few minutes.