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Question 63 PtsHin!ATo test the effectivenes; Of ( new doug designed to relieve pain, patients were randomly selected and divided into two equal groups, Groups and ...

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Question 63 PtsHin!ATo test the effectivenes; Of ( new doug designed to relieve pain, patients were randomly selected and divided into two equal groups, Groups and Group inchuded 80 patients who were given @ pill coritaining thie drug: of these 62 felt pain relief. Group included 80 patients who were given @ piV cuntaining placebo; of tnese, 54 fel pain relief The researcher wishes detunin f there i5 sufficient cvidence that the true proportion experiencing pdin relief from the drug mote than

Question 6 3 Pts Hin! A To test the effectivenes; Of ( new doug designed to relieve pain, patients were randomly selected and divided into two equal groups, Groups and Group inchuded 80 patients who were given @ pill coritaining thie drug: of these 62 felt pain relief. Group included 80 patients who were given @ piV cuntaining placebo; of tnese, 54 fel pain relief The researcher wishes detunin f there i5 sufficient cvidence that the true proportion experiencing pdin relief from the drug mote than the: proportion experiencing pain relief from the placebo The test results in 3 P-value 0f 0.078 Conclusion Usne level of significance 0.05,state the concluslon afthe test incontcxt I [*1 Progaaph Fuitora)



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Please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) What sampling distribution will you use? Do you think the sample size is sufficiently large? Explain. Compute the value of the sample test statistic and corresponding $z$ value. (c) Find the $P$ -value of the test statistic. Sketch the sampling distribution and show the area corresponding to the $P$ -value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level $\alpha ?$ (e) Interpret your conclusion in the context of the application. Diltiazem is a commonly prescribed drug for hypertension (see source in Problem 19). However, diltiazem causes headaches in about 12\% of patients using the drug. It is hypothesized that regular exercise might help reduce the headaches. If a random sample of 209 patients using diltiazem exercised regularly and only 16 had headaches, would this indicate a reduction in the population proportion of patients having headaches? Use a $1 \%$ level of significance.

So in discussion, we know that the null hypothesis age not is that P is equal to 0.23 On the alternative hypothesis H A, there's a P is not equal. Does your zero three? As for the rejection region, this is the two tailed tests for Alfa is equal to 0.5 on the critical values of Z. Yeah, critical r Z is equal to negative 1.96 z to is equal 296. Therefore, we know they have to reject checked our no hypothesis H not. Yes, G because last time negative 1.96 or greater. Don't welcome, friend six. Okay, Now for the test statistic, we know that Z is equal to P virus. He over the square root off P times one minus p. That s p over. Okay, so, Playgirl, all our values and we get 0.258 Okay. Minus June 23. Over. Discouraged. What was your point? 03 times one minus your appointment. Your three over. But was 924. She's our end, which will give us a Z value of negative one. No. So as P value is zero point. You're five second for which is greater than did you appoint your five. Therefore you can fill to reject the no happens is h not.

We're told that a company is testing a new type of pain relief, new headache medicine called pain free. And they're running a test to see whether this works or not. Okay. And so were given this matrix appear to begin with a focus on just this matrix for now. So we can see that it says when people took pain free, we can see who actually got pain relief and who did not. And then when people took the placebo meaning nothing Riel, how many of them were pain free and how many of them were not okay? And the first question it asks is of those who took the placebo. So right off the bat, we know we're on. Lee focused on this bottom row. So of the people that took the placebo, how many got relief? Meaning, yes, there was pain relief. Well, if we look at the intersection of the placebo row and the yes column, the answer to part A would be that we have eight people that were pain free, right? Right here. That's where we'd be looking at. So the answer for part A is eight for Part B. It says of those who took the new medication. I mean, now we're focused on the people from the pain free call row of those who took the new medication. How Maney got no relief. So the no call and we can see that the intersection between the pain free row and the No column is three. So our answer for Part B is three for part C. It says the test was repeated on three Mawr groups of 50 patients each, which is all of this data that we have down here. All three of these matrices. It says it would like us to find the total results for all 200 patients. Not be careful here because recognized. It said that each of these three groups down here represented 50 groups are 50 people 50 patients each when it asked for all 200. It is also including our original matrix. So make sure you repeat that down here as well. 22 3 8 17. You should have all four matrices you've been given down here now. Besides that, it is simply asking for the total results, which is another way of saying they want us toe. Add all of these together. Remember, when you are adding matrices, you simply combine the corresponding entries meaning I'm going to start by taking the top left entry top left element of every matrix and add those together. So if you take 21 plus 19 plus 23 plus 22 you would get 85. Then I'd combine all the top right elements. Top right entries. So four plus six plus two plus three. If you take four plus six plus two plus three, you would get 15. Then we're gonna take the bottom left elements and add those together. If you take six plus 10 plus three plus eight, you would get 27. And finally we will combine all the bottom right elements. And if you take 19 plus 15 plus 22 plus 17 you would get 73. That is your matrix for part C. Final part Part D is just simply a question of analyzing the data. And it asked, Does it appear that pain free is effective? Remember, the first row represented pain free. The bottom row represented placebo. And this was the people that got relief. This was the people that did not. Well, if we look at our first row are placebo or, I mean, are pain free row. Clearly, the majority of people did experience pain relief. Yes, they got pain relief. If you compare that to our bottom row are placebo row? More people did not receive pain relief or did not report pain relief as those that did so. It looks pretty clear to us that yes, pain free does work because more people on pain free had pain relief than those that didn't.

Problem Number 20 The H node is the drunk. That the drunk is not effective is not effective. And the alternative is that that drunk is effective. Eso by creating the Alfa is opening one and the chi square is equal to 10.643 and the degrees of freedom is equal to one. So the P value is smaller than 0.5 which is Oh, point or one. So we re inject. The non hypothesis is there is enough evidence to support that The claim that the drug effect

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.


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