Okay, So what we have in this question is that we have an experimenter who has prepared a drug does is level that she claims will induce sleep for around 80% of people who suffer from insomnia. We examined the dosage, and we feel that her claims are a little inflated. So in an attempt to disprove her claim, we administered her prescribed dosage to around 20 insomniacs and found the number of people for whom the drug does was actually affected. Now what we wanna do you wanna ask the hypothesis there, then the life Ah, thesis is that p is equal appointed where she's actually saying that truth. Her claims are correct. And the alternative hypothesis is we feel that it is inflated. Right? So this is going to be less than 0.8. All right, now, one of the questions, okay. And we also see the rejection Region. The rejection region, in this case is by less than equal to 12. This is going to be our rejection region. All right? Now, the first one is in terms of the problem. What is type one error? First of all, what is type whenever type one error is when you should not have rejected the null hypothesis. What? You end up rejecting it. Now, let's say that this line over here in the graph, uh, corresponds to my critical statistic. Okay, my critical, sir Value. Now, if I get a value of that, that is beyond you know, further to the right than this, I will end up rejecting man alive prosthesis. And what happens if my observation is over here to the left? In this case, I actually should not reject my inner light apotheosis. But if I end up doing so, I'm committing a type one error. So what will be a type winner type monitor will be if I conclude that her does it is actually not effective, or her dosage is actually 80% effective when eyes actually not 80% effective when it actually is. Okay, so let me just frame it nicely. My error type One error in this case would be if I say that her dosage is not 80% effective then In reality, it is 80% effective, which means that my null hypothesis is actually true, but I end up rejecting it. This is going to be the answer to part a moving on the party. But visas find Alfa find Alfano. What exactly will be Alfa? My rejection region is by less than equal to 12. Okay, so how will I actually like this? This will be submission. This is going to be submission where my vibe will go from zero to 12. Okay, I have 20 insomniacs. 20 c wide. What is my probability in this case? I'm assuming that my Nell is true. Okay, Because we have to do that in order to find Alfa. So this is going to be zero point eight raise to white and then over here it'll be 0.2. It is to the remaining stuff, right? It will be what? 20 minus. Why? I guess 20 minus one. Okay, eso if I expand this, this is going to be first of all, it is going to be 20 c zero. This is going to be 0.8. Raise 20 in 20 point to raise to 20. The next one is going to be 20 c one. This is going to be 0.8 raised to one in 20 point to raise to nine team. Yes, Yes. This will go up all the way to 20 C 12, 0.8. It is to 12. Zero point to raise two. What is this going to be? This is going to be eat. All right, so this is going to be the value off Alfa. So what I'm going to do right now is I'm going to take the help off a calculator and online binomial distribution calculator. Where I have my n is equal to 20 X is 12 p is pointed, and I'm going to find the probability that my capital X is less than equal to small X. So this is coming out to be 120.3214 This is my Alfa Alfa is 0.3 to one Food. This is going to be my Alfa. All right, we have done with part B. Let us go to parts. See what is parts you have to say, but sees in terms of this problem, what is going to be typed to it now? Type two error is a very interesting concert like Twitter means when I should have rejected my null hypothesis. But I don't do so For example, if this is my normal distribution, if this is my critical value Okay, let's say that this is the start. This is my critical body. Now I get an observation over here to the right off that star. This is my sad statistic that I calculate now, as I can see from the diagram, I should actually reject this value, right, because it is in the rejection region. But I don't do so and this is an error. And this is a type two event. I should reject something. Let me just write this. This is very important. Type two errors. Then I should reject h not, but I fail, but I fail to do so. Okay, this is typed Twitter. So what would be type two error in this case? My final hypothesis is that my P is equal to zero point it, which means he has that it is effective on 80% of the people. But I two error would be if this actually is not effective on 80% of the people. But I end up saying that yes, it is effective. Okay. When in reality it is not. This is going to be my type. Two error. Now, how exactly do we calculate the probability of type Twitter? It is given by Vita, right? This is the probability off type. Well, the probability off type to edit. Okay, now moving on the part d we have to find beat A and P is equal 2.6. Okay, so I'm going to find Vita. This is my part day. This is the right. Yeah. When p is equal 2.6 when p is equal 2.6 now my rejection region is why is less than equal to 12. Okay. Which means that for any value off, why that I get that is greater than 12 for example, 13 or 14 or all the way up to 20. I will not reject minor hypothesis. All right, so in this case, how do I find it? On my vita will be, as we already know, the formula. This is going to be submission off. Why? From 13, 2 20. This is going to be 20 c by now. What is the probability Probability 0.6. So this is raised to let me just it is this one This is raised to buy And then I have 0.4. It is to one minus way again. I'm going to take the help off the binomial distribution calculator. So my ex is 12. My probability is 120.6 is point six and I get my P Vally as point. Sorry, My probability is 0.58411 My probability is 0.5. Wait for 11 But this is not my answer. My probability for X less than equal to 12 is 0.58411 point favorite 411 Yes, but I have tow pay attention that this is my probability for X Less than equal to tell. What do I want? I want for greater than 12. So I will do one minus 0.58411 If I take my calculator to find this Just a moment. Okay, this is one minus 0.58411 This is 0.4159 This is 0.4 159 and this is going to be my Peter. Okay, this is going to be my beater for this case. What is this? This is submission. If you are confused about what this is. This is a mission. How do I write this? If I write this first time, this is going to be 20. See? 13, 0.6. Raise to 13 in 20.4. Raise too. What is? Okay. This This should be 20 minus y. This should be 20 minus way. This thing over here is 20 minus y. This should be seven, right? So that I'm will do some. All of these are all the vital 20 c 20 0.6. Raise to 20. And then I will also have a 0.4. It leads to zero. Okay. And this is going to give me my answers. 0.58411 And I have to subtract it from one in order to get the value of my beetle, which is point for one 59 Yeah, all right. Now, what is going to be the last thing? The last thing I wanna find beat up on my P is equal to point for in the last bit. That is a bit e. I want to find my Byetta when P is equal 2.4. So what is going to be my solution this time. This time again, This is going to be submission off. Why? From 13. All the way to 20 20. See why this time I have 0.4. Raise two y 0.6. Raise too. Two indeed. Minus white, Reese to 20 minus. Why? All right. So again, let us just take the help off a binomial distribution calculator. This is point for my end is 20. My ex is 12 my p iss 120.4 and I can see over here I'm finding the probability that X is less than equal to 12 in this case, at this 120.97897 So this is 0.978 97 But remember, this is my probability that X is less than equal to 12. Okay, What is going to be my beat up? My vita is going to be one minus 10.97897 And if I do this when minus 0.97897 this is point 0 to 103 This is my Vita in this case is 0.210 three. And this is how we go about doing this question a Langley question, but very simple. If you understand the concept, this is how we go about doing this question.