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On December $25,2009,$ an airline passenger was subdued while attempting to blow up a Northwest Airlines flight headed for Detroit, Michigan. The passenger had smuggled explosives hidden in his underwear past a metal detector at an airport screening facility. As a result, the Transportation Security Administration (TSA) proposed installing full-body scanners to replace the metal detectors at the nation's largest airports. This proposal resulted in strong objections from privacy advocates, who considered the scanners an invasion of privacy. On January $5-6,2010,$ USA Today conducted a poll of 542 adults to learn what proportion of airline travelers approved of using full-body scanners $(U S A$ Today, January $11,2010$ ). The poll results showed that 455 of the respondents felt that full-body scanners would improve airline security and 423 indicated that they approved of using the devices.a. Conduct a hypothesis test to determine if the results of the poll justify concluding that over 80$\%$ of airline travelers feel that the use of full-body scanners will improve airline security. Use $\alpha=.05$b. Suppose the TSA will go forward with the installation and mandatory use of full-body scanners if over 75$\%$ of airline travelers approve of using the devices. You have been told to conduct a statistical analysis using the poll results to determine if the TSA should go forward with mandatory use of the full-body scanners. Because this is viewed as a very sensitive decision, use $\alpha=.01 .$ What is your recommendation?
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a. There is sufficient evidence to support the claim that over 80$\%$ of airlinetravels feel that the use of full-body scanners will improve airline security.b. There is not sufficient evidence to support the claim that over 75$\%$ ofairline travelers approve of using the devices.
Intro Stats / AP Statistics
Hypothesis Testing with One Sample
Hypothesis Testing with Two Samples
Oregon State University
The first nuclear bomb was…
this again is a very basic question based on hypothesis testing for proportions. So what is happening over here is that U S A today has conducted a poll off 542 adults to learn what proportion off airline travelers approved off using full body scanners. Okay, so one very important piece of information here is that end is equal to fight 42 right? This is our sample size. Now. The whole reserves showed that 455 off the respondents, right? 4 55 respondents felt that the body scanners would improve security. They said that this would improve security. They said that this would improve security. Okay. And 4, 23 of them. 4. 23 of them indicated that they approved off using these devices. They approved off using these devices. Okay, So let us move on to the questions. What is part easy. But he says that we have to conduct the hypothesis has to determine if the results of the poll justify concluding that over 80% of the airline travelers feel that the use off full body scanners will improve will improve the airline security. Okay, So what is going to be the null and the alternative hypothesis in this case, Donal and the alternative hypothesis. Okay, so the null hypothesis will be that the proportion is less than are equal to 0.8 get. And the alternative will be that r p is greater than zero point. What are they saying? They want to test to determine if the results of the poll justify concluding that over 80% off airline travelers feel over 80%. So, yes, this is going to be an alternative hypothesis. Be greater than 0.8. Okay. Now, in order to conduct the hypothesis test, we're going to use the statistic. And what is the formula for the C statistic formula for the statistic is P bar, which is our sample proportion minus peanut, which happens to be a hypothesized proportion are 0.8 is what we're going to take in this case upon route over 0.8 in 20.2 upon in what is the value of N n is 5. 42 or let me is right, the formula way. This is the formula, right? This is what we are going to use so this is going to be upon peanut in tow, one minus peanut upon in. This is the formula that we haven't used. Okay, so what is PBA fever is going to be 4. 55 by 5. 42 p. Bar is going to be 4 55 by 5. 42. Right. Necessarily before 55 5. 42. And if I use a calculator for this, what if I made about 42 is 420.8394 is 0.8394 83948 Or let me illustrate. This has 8395 Okay, let's put in the values for the set. Sadistic. This is going to be 0.8395 minus 0.8 upon peanut, which happens to be 0.8 multiplied by one minus. Peanut, which is 0.2 upon in what was in five 40 Do. Right. So this thing actually becomes 0.395 upon route over this thing. So if I use a calculator for this, this is a rood over 0.820 point two, divided by by 42 which happens to be 0.1718 0.1718 Okay, so this is 0.395 develop a 0.1718 2.299 So let me just write this as 2.3. Right. This becomes 2.3. So my is that statistic is 2.3. All right, So now, in order to find the p value, we can use any statistical software that we lie or an online tool Or maybe just a manual zet table. So what I'm using over here is that table. My the statistic is 2.3. So we have a very useful visualization here. Let me just set my Z value to two point three. If I said this 2.3, this is 2.32 but we want to 0.3. Okay, so this is 2.3 Nice. So as we can see that this is a bell shaped curve and it's symmetric around zero. So which means that the area under the curve after zero is going to be 0.5. So what is going to be the P value. The P value is going to be This is my This actually is my cough. Right? This is 2.32 point three. Over here I have zero. So I want this reason in the tail. So what is that going to be? That is going to be 0.5 minus 0.4893 So this will become my people. We will become 0.5 minus 0.4893 So this is going to be 0.1 07 If I am not mistaken, let's just see this Little white by minus 0.48 19 Yes. 010107 This is my P value. What is the Alfa that we're using? The Alfa is 0.5 Alfa is 0.0 fight in this case. Okay, So I can clearly see that my P value is less than Alfa right. Which means that I can reject minor hypothesis. I will reject. I will reject h not now. What exactly does this mean? My it's not Was that P is less than 0.8 less than or equal to. And my alternative hypothesis wars that the proportion is greater than zero pointed. So I can say that I have enough statistical evidence to suggest I love statistical evidence to suggest that my P is greater than 0.8. Now, if I put this in words, what exactly does this mean? That is the question. Yeah, So hence we can say that, uh, over 80% of airline travelers feel that the use off full body scanners will improve the airline security. Okay, we are done with part A. Let us move on to part B. What is part deep now? The party says that suppose that TSA will go forward with the installation and manage the use of the full body scanners over. If 75% of airline travelers approved off these devices now 0.75 So we can say that are null. Hypothesis for part B is going to be P is less than or equal to 0.75 and that he is greater than 0.75 Right. If this happens, if alternative hypothesis, uh, is suggested by our test, they're going to go ahead with this. Okay, So what? We have been told is to conduct a statistical analysis using the poll results to determine if he s a should go forward with the mandatory user. Full body scanners. Okay. And this time we are using Alfa as 0.1 Again, we're going to use the same formula to find the C statistic, which is this one? What is P Bar? In this case, P Bar is going to be 4. 23 I think. Yes, for 23 by 5. 42. So this is going to be 4. 23 upon 5. 42. So if I use it calculated, this is 4 23 food ready trick 4 23. Divided by 5. 42 0 point 7804 So this happens to be 0.7804 All right, let's put this in the formula. This is 0.7804 minus 0.75 upon route over 0.75 in 20.25 upon what was in 5 42. Right, So this actually turns out to be 0.3 04 upon now. I have a calculator for this new doors you don't mind. Sometime in you tore a 0.75 into zero. Going to fight. Divided by 5. 42. This is 0.185 0.185 Okay. 0.304 Divided by 0.185 1.6432 zero point. I'm sorry. This is 1.6432 This is my zed statistic. Now, just looking at this, I can say that this will not reject an alive offices. Let's just set this over here. Zero point 64 Okay, so let's just get this 20164 this is 201640.4495 So what is my P value? My P value is the tail right. The area under the tail that is to the right off 1.64 So 0.5 minus my P value is going to be 0.5 minus 0.44 What was this? 44954495 So this actually turns out to be 0.505 So what was our Alfa R Alfa was 0.1 We can see that RPI value is greater than Alfa. Hence we fail to reject Arnold hypothesis we fail to reject. It's not right. Hence I can say that I do not have enough statistical evidence to suggest that the authority should move ahead with the installation off these new devices. Right. So the authorities, yeah, should not move ahead. Should not move ahead. Why? Because we do not have enough statistical evidence to say that the proportion is the proportion of people who approve of these devices is more than 0.75 Okay, they should not move ahead. And this is going to be our answer, Yeah.
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