Lean Six Sigma Green Belt – Six Sigma Analyze Phase Part 5

10. Shortlist Critical Inputs – Hypothesis Testing

And here is the next step, the third step. Oh, sorry, my bad. This is the second step shortlist to identify the critical inputs. Out of the umpteen number of inputs which we have found out, we have now identified all the potential inputs. We will have to validate whether the potential inputs are critical or not using some quantitative tools. And we would be using hypothesis testing, which is a quantitative tool, by the way. We are going to prepare the data collection plan.

We are going to collect the data according to the data collection plan, and then we are going to perform hypothesis testing if required. It’s not mandatory that we do it always if required. And here are a few other steps which we need to perform for hypothesis testing, by the way. We need to define the practical problem. We need to convert practical problem to statistical problem. We need to carry out an interpret statistical analysis. We need to convert statistical to practical solution, that is, we need to convert the statistical solution which we have at hand to a practical solution.

That is what we have to do here. So these are the major steps that we would be doing as part of hypothesis testing. Let us look into few high level things about hypothesis testing and which are the various areas where hypothesis testing has found its presence and usefulness, basically. So the first thing is we need to come up with a business problem. What is the problem which I am facing, right? I need to identify what are the inputs and what are my outputs, what is my Y measure and what is my x measure. I need to find out that as part of mathematical case, I need to come up with a mathematical inference. I need to use the actual test, hypothesis test.

To do that, I would be using minitime to draw this mathematical inference because calculating it manually is extremely cumbersome. And finally we would arrive at the business inference. What is the impact which you have created to the bottom line? How many dollars did you save, right? Or what did you impact? Basically, that is extremely important in this case. Let me tell you this hypothesis testing has found its roots in most of the interesting arenas, interesting work areas. For example, angelina Julie Acres, right? She got her double breast max dust to me done. That is removal of her breast organ. Because mathematicians and statisticians have proven using hypothesis testing that probably sometime, as in how she grows old, she would be getting breast cancer, right?

 And based on that decision, she is the first celebrity who has taken an action with respect to her health. It has also found its roots deep inside the legal system. Lawyer, right? Judge. There was a judge who has convicted a nurse for X number of years behind the bars because supposedly under her supervision, there were a lot of children who have died. Mathematicians once again pitched in and told that hey, if you have imprisoned her, then there are here a bunch of hundreds of other nurses who should be imprisoned because of a similar case, right? Because of that similar reason. So they have brought in a reasoning which says that that nurse actually was innocent, children died because of various other reasons. It wasn’t just because of her.

So ultimately she was released from the prison. See how powerful hypothesis testing can become, right? Let us look into a small example and what are the various components of hypothesis testing? Here are the two components you will have to first develop the hypothesis for population. You always develop hypothesis for the population and make statistical inference by determining the acceptance of hypothesis using sample data. So I always have access to the sample data. I do my testing on sample data. But when I make the final statements, I make my final statements about my entire population. That’s the good part. I always have a small sample to which I would have access to. Based on that, I am going to make inferences about my population.

And within hypothesis you have two hypothesis. One is called as null hypothesis and another is called as alternative hypothesis. Null hypothesis is always represented using h zero. It is all about the argument which has been made until now. It is a hypothesis which says there is no change or there is no difference. The existing system remains as is.

Look at this example here. There’s a new car engine and that is newly developed by a German company which provides mileage of 10 miles per gallon more than existing car engine. So 10 miles per gallon is the excess mileage which I get using the new car engine. So if I want to write down the null and alternate hypothesis for that, this is how I would write down h not it’s status quo, no change, no difference. It says new car engine and the existing car engine provide the same mileage. There is no difference between these two. And this is what your null hypothesis says.

Think about alternative hypothesis. It can be represented using h one Ha or h capital A as subscript. It is a new argument. That is a hypothesis that you want to prove with solid ground obtained from sample. It needs a lot of evidence, right? And look at this example. New car engine provides mileage of 10 miles per gallon more than the existing car engine. So your null hypothesis says both of the car engines are the same and it provides the same mileage. But your alternative hypothesis says that the new car engine provides more mileage in comparison to your existing car engine. And in order to prove your alternate hypothesis, you need a lot of evidence. You need a lot of evidence. We will understand this example or understand null and alternate hypothesis using another case study which would be extremely easy for us. But that is coming. It’s on the way. All right. Here is an example of the same case study which we are discussing. Quality Manager claims that he has brought in a significant improvement to the input variable and this has increased the yield of the process. Yield data are collected before and after improvement. So which process is doing a good job? Process A or process B? Before improvement data, is it significantly different from after improvement? Was there a real improvement after quality manager has brought in some improvement, some process tweaking? Right? We need to find out that. Look at this example of our previous case study.

Random samples are drawn from yield data from existing car engine and the new car engine. So you have found out miles per gallon for your existing car engine and also you have done that for the new car engine. By looking at this data, can you comment and tell me whether there is a significant improvement by bringing in a new car engine? Yes or no? How do you do that? By looking at with the naked eye I feel that the values are more or less the same. But is there a real difference between process A and process B? Is there a real difference between existing car engine and the new car engine? What do you think? So, all right. Hypothesis testing continue. So this is what it says, right? Can we say that the yield of improved process B is greater than old process A? Can I say that the mileage provided by new car engine is greater than the existing car engine? Can I say that right? Maybe may not be. But how do we prove it? Look at this.

There is descriptive statistics information available which says the mileage of existing car engine and the new car engine wherein we have ten data points each. For the existing car engine, there is a mean of 34. 19 and the new car engine has a mean of 35. 54. What does this mean? It says that on an average the existing car engine provides a mileage of 34. 19 miles per gallon. On the other side, the new car engine provides on an average a mileage of 35. 54 miles per gallon. And here is a standard deviation. By looking at these values, can you say that there is a statistically significant difference between the existing car engine and the new car engine and the mileage it provides? We cannot see.

So here is the statistical question. Is there a statistically significant difference between mean of existing car and the mean of the new car? Or is this difference in mean just due to chance? Why do you think so? What do you feel is the right answer to this? If you have these kind of situations, that is when hypothesis comes in handy, you’ll be easily able to identify whether the petrol or the miles per gallon has increased or not. You’ll be easily able to identify whether the yield of a process has increased or not. Right, here we go. How to develop a hypothesis?

My objective is to determine whether the mileage of existing and new car engine are different or not. And I have to determine this using a sample. But the inferences that I’m going to draw would be about the entire population, right? Let us look into null and alternate hypothesis. Here is the statistical interpretation. Population mean of mileage of existing and new car engine are the same. Is what my null hypothesis sees. What is practical interpretation of that? There is no difference in mileage between the two car engines. That is my practical interpretation. Let us look into the alternative hypothesis.

11. Hypothesis Testing_Part 2

What is a statistical interpretation of this population mean of mileage of existing and new car engine are different. What is a practical interpretation of this? It says average mileage of existing and new car engine are different and not the same. Null hypothesis says mileage provided by both the car engines is DC. That means there is no difference between these two and your null or alternate hypothesis says it is the same. I confuse you guys, right? Null hypothesis says the mileage between the two car engines is the same. Alternate hypothesis says it is different, right? Same and different. Let me put that here. Same, different, do not worry, do not buckle. Things would become extremely clear, as in how we move on. Here is a hypothesis testing, right? You use some kind of statistical, statistical, some test statistical statistic that is used as criteria for selecting null and alternate hypothesis.

You might end up using ZTEST or Ttest or F distribution, depending on the context, depending on the data which is provided to you. Two errors in hypothesis testing. This is extremely important. Let us understand this using this case study. Suppose someone has accused you, right? And put a case against you. What would happen? Cops would arrest you and present you before the judge, right? Someone has accused you, the cops have arrested you and they have put you before the judge, right? This is what happens, right? Whenever a judge starts off with a case, he assumes that the accused is innocent, right? And if the judge says that, or if the judge gives a judgment that hey, you are innocent. And if in reality you are innocent, that is the right decision. So judge has taken no action on you. He’s saying, hey, you are innocent, and in reality also you are innocent.

That’s the right decision. But assume you are innocent. But the judge has given an evidence against you. He has given a judgment against you. He says that you have to serve a sentence of five years because I found you guilty based on the evidence provided to me. So what is happening here? Though you are innocent, though it’s not, is true. You have been sentenced, right? To imprisonment. That is an error. That is a mistake that has happened. And this mistake or error is called as type one error or alpha error, right? This is the probability of committing the type one error. And when you call it as type one error error that you reject. Null hypothesis. Although null hypothesis is true, you are innocent. But judges rejecting the fact that you are innocent, errors do happen. That is called a step one error, by the way. All right? The next thing is alternate hypothesis. Suppose you are accused, cops have arrested you and presented you before the judge. If in reality you have committed the crime, and if the judge sentences you to imprisonment for five years, then that’s the right decision.

You have made a mistake. You have committed a crime and you are put behind bars. That’s the right decision. By the way. However, think about the situation. You have committed a crime. But judge says that there is no lot of evidence for me to prove that you have committed the crime. Release you. I’ll say you are innocent and release you. That’s a mistake. That’s a grave mistake. You have committed a murder. Suppose and the judge is saying that I do not have sufficient evidence, so I’m releasing you. That is called as beta error or type two error. Type two error is you accept the null hypothesis.

 Although null hypothesis is false and the probability of committing type two error is the beta error or the beta arrest, right? And one more thing. If I have really committed a crime and if I were to be sentenced to some imprisonment, then there’s a lot of evidence that I require, right? There’s a lot of evidence which I require. So as to how I get more and more evidence, I keep throwing away from my mind that you are innocent. During the start of the investigation or before the start of the case, the judge has filled his mind with the thoughts that you are innocent. And for those thoughts to go away, there has to be a lot of evidence against you that you have committed a murder.

More and more evidence you provide, higher are the chances that you might be imprisoned. And there is one more very interesting thing called as p value. It is the probability of you making a type one error. And in most cases, this alpha value is selected as 5%. That means there is a 5% chance that you might go wrong with the decision. And there is a 95% chance that you might always be right. So you are taking that 5% chance. This is a management decision. In few organizations such as nuclear plant, rocket launching station, missile launching station and all that, or healthcare related products, your management might say that, hey, I do not have the provision to give you 5% chance of making a mistake. So I will bring it down to one person. I will say that there is only one person chance that you might make a mistake. 99% of the times you should be right. There’s a management call, right? But otherwise you stick to this five person. Most business decisions acts of this 5% of mistakes that might happen. You are a human, right? You cannot be right.

100% of the times you might go wrong with your decisions. But how much percentage is a management willing to give you? How much leeway is a management willing to give you? Right? And that is called as this alpha value probability of committing a type one error. All right, let us move on. Types of Errors we have already discussed this. There’s a type one error and there’s a type two error. Type one error is also called as alpha risk of the producer’s risk. It is the risk of rejecting the null hypothesis when you should have accepted it, action is taken when there should have been no action, you should have been left scot free. You should have been termed as innocent and left alone. But they have imprisoned you. Though you are innocent, they have imprisoned you, right? That’s a type one error. What is type two error? It is also called as beta risk or consumer risk. The risk of accepting the now when you should have rejected it. No action is taken when there should have been action. Hey, you have committed a murder. You should have been put behind bars. But you were released scot free.

Terming that you are innocent, right? Though you have committed a murder, you are released as innocent. That is type two error. Now you’ll have to balance that out, right? Type one error is determined upfront by a management or you pick up that value called five person. It is called as alpha value. You have chosen one minus alpha will give you the confidence level. If I’m saying there’s a 5% chance that you might go wrong, I’m okay with that. As a management person, then one minus alpha which is 5% is 95% and this is called as my confidence level. Type two error is determined from the circumstances of the situation. Remember this. If the alpha is made very small, then the beta risk increases. That’s okay. That is acceptable in most cases.

So there has to be a balancing act which you will have to perform. But that is acceptable. Think about this. Any law, any legal system says that if there is a person who has actually committed a murder, if because of lack of evidence you release him, that is fine. But you have to be extremely, extremely relevant. You have to be extremely cautious when you’re sentencing a person to imprisonment. Because you should never ever imprison a person who is innocent, right? That is what the legal system sees.

So I’m going to give you very less chance of say 5% that you imprison an innocent person. But here probably I’ll give you a 10% leeway of releasing and actually convict. So type two error requires overwhelming evidence to reject the null hypothesis, right? Increasing the chances of the type two error to minimize the beta while holding alpha constant requires you to increase the sample size. The amount of data that you have has to increase significantly. One minus alpha is called as confidence level and one minus beta is called as power of test. It is a probability of rejecting the null hypothesis when it was false. Fine. That’s a good case. Basically.