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The data below record the number of hours a team of workers takes to assemble a custom-built motorcycle: The data are recorded for 10 different teams each assemblin...

Question

The data below record the number of hours a team of workers takes to assemble a custom-built motorcycle: The data are recorded for 10 different teams each assembling a motorcycle: 89 78 48 85 67 45 60 62 62 56Create & QQplot in R and comment on the assumption that the population of times to assemble & motorcycle is well-approximated by a normal distribution.Construct a 90% confidence interval by hand for the mean time it takes & team of workers to assemble custom-built motorcycle_Con

The data below record the number of hours a team of workers takes to assemble a custom-built motorcycle: The data are recorded for 10 different teams each assembling a motorcycle: 89 78 48 85 67 45 60 62 62 56 Create & QQplot in R and comment on the assumption that the population of times to assemble & motorcycle is well-approximated by a normal distribution. Construct a 90% confidence interval by hand for the mean time it takes & team of workers to assemble custom-built motorcycle_ Construct the same interval above using R's t.test() command. By how much does the confidence interval width of a 95% interval differ from that of a 9% interval for this data carl complete in R or by hand)? Suppose instead; the manager had performed hypothesis test at the 10% level of the null that 60 hours vs the alternative that p 60. Compute the pvalue for his hypothesis test and summarize the conclusion that he would have drawn from the sample that he observed_



Answers

A researcher for the FAA wants to estimate the average flight time (in minutes) from Albuquerque, New Mexico, to Dallas, Texas, for flights with American Airlines. He randomly selects nine flights between the two cities and obtains the data shown. Assume that $\sigma=8$ minutes. (TABLE CAN'T COPY) (a) Use the data to compute a point estimate for the population mean flight time between Albuquerque and Dallas on an American Airlines flight. (b) Because the sample size is small, we must verify that flight time is approximately normally distributed and that the sample does not contain any outliers. The normal probability plot and boxplot are shown next. Are the conditions for constructing a $Z$ -interval satisfied? (FIGURE CAN'T COPY) (c) Construct a $95 \%$ confidence interval for the flight time. Interpret this interval. (d) Construct a $90 \%$ confidence interval for the flight time. Interpret this interval. (e) What happened to the width of the interval when the level of confidence decreased? Explain why this result is reasonable.

We're doing a hypothesis test to see if students are a good judge of how long a minute is. And so we're going to assume that the mean of the students when they guess the time is going to be 60 seconds and alternately it's not 60 seconds. So we're not we're doing a two sided test and I put my data into the calculator into my list and I found that the mean of those 15 numbers was 62.6 repeating, that the standard deviation of those numbers was 19.481 and that that sample size was 15. And so we're assuming that 60 is the main for those students. Now, obviously we're getting a number that's higher, 62.6 repeating. And we want to test and see if this is something that's unusual. But since the two tail test, our p value will not only be this end, but it will be this end that is symmetrically located on the opposite side. So let's find the probability, uh if the mean is 60 of having a sample of size 15 and getting a mean that is greater than or equal to 62.6, repeating and really or less than or equal to and this is 2.2 and two thirds. So if we subtract two, that's 58 subtract two thirds, that would be 57 a third, so that's where this other number is. And so how likely is it to get either this or this happening if the mean is actually 60? So we're just going to find one tail and double that to find our p value. So let's get our test statistic. Our test statistic is a T value with 14 degrees of freedom and we know we take what we got minus the mean, which is 60 divided by the standard deviation over the square root of an and I've already put that into my calculator to get that value. And we're getting a test statistic of and that comes out to be 0.53 So this is, you know, about half of a standard deviation above the main. And so we can use a table to look up r. P value, we can't get it super accurate that way. I used my uh my T C. D. F button on my calculator and plugged in that 0.53 as the left number put in like 1000 for the right number and 14 degrees of freedom. And then I remember to double it because we know that both these added together are my P value. And when I did that I got 0.604 So again, here's my p value, here is my test statistic and now I can make my decision. I'm supposed to use a 5% significance level, and we definitely fail to reject the null. Yeah. We have no evidence to show that this uh that the students are different from 60 minutes, so 60 minutes does seem to be the main make that I am same to be the mean for students. Yeah. Mhm. Yeah.

In this problem were given a word problem and as to construct a confidence interval and find the margin of error. Now, just to recap these steps in doing this would first to be fine would be to first find a T value which we can use to find emerging of error and finally, which you can use to compose a confidence interval. Okay, now, in this problem we're giving were given a word problem which can often be intimidating. And the first thing we need to do is to decompose this word problem to see what information were given. So just to read the problem on a random sample of eight people, the main commute eso first in a random sample of eight people, um, in the first sentence were given. The end is equal to eight. The sample size is eight and the next line the mean commute timeto work was 35.5 minutes and so that would be our X bar are mean 35.5 minutes. And finally we were told that the standard deviation is 7.2 minutes. So the sample standard deviation is 7.2 minutes. And if you look in the instructions were being asked to construct a 95% confidence interval. So this is a confidence level of 0.95 Now, First, we need to find our T Steve value. And to do so, we need to know the confidence level, which we know is your 0.95 And we need to find the degrees of freedom, which is equal to end minus one. So the degrees of freedom is equal to the sample size minus one, which is eight minus one, which is equal to seven. Now, look at the table in appendix B in table five and look for the rows starting with, uh and we need to find the T value based off of the row for the degrees of freedom of seven. And looking at this table, we see that the TV T value that we would like is 2.365 From here, we can calculate our margin of error, and we know that the margin of error is equal to tee times s over the square root of n and plugging. In our values, we have 2.365 times, 7.2 over square root of eight, and this is equal to approximately 6.104 now. Finally, the confidence interval is theatric, plus or minus the margin of error, and we know x minus. The margin of air is equal 2 35.5, minus 6.104 and we can also which is equal to 29.396 And we can also find 35.5 plus six point 104 which is equal to 41.604 And there you go. We have our confidence interval, which is 29.396 to 41.64 and we also have the margin of error, which is equal to 6.104

In this problem, we're going to be considering sample data and independent. Simple random samples were picked each or two samples each off Science 200. In the first sample, 112 people had the same attributes. In second sample, 88 people had the same attributes. So the a Z we can see here that were given anyone is 200 on X two. X one is 112 and then enter is 200 next to is 88. From thes values, we can obtain three other values, like P one hut, which is 112 divided by 200. So P one hunt equals 0.5 six. And we can obtain que one hot by subtracting 0.1 a 0.56 from one. That would be 0.44 from the second sample. We have p too hot, obtained from 88 divided by 200. On that is 0.4 form and cue. Too hot. He's one man, a 0.4 form, which is your 0.56 You can get PBA from adding the X one and x two, and that will give us 200 divide by the sum of anyone and to which is 400. And that P bob becomes your 4000.5 and Cuba will also be 0.5 because they needed to add up to one. Now, in the first part of the question, we're going to be getting the 95% confidence interval estimate for the difference. P one minus B two So and then we'll tell whether the result suggests what the results suggest about equality off P one and p two. So let's move on to find the 95% confidence interval on the next 5% confidence interval off P one when I speak to is obtained from adding and subtracting the margin of error, so you want to walk out the margin of error E, which is given by the formula, was P upon hot shoes. One hunt over n one speak to hut que to hunt over and to. So what we need to do to get imagine of area is to substitute the values that we have obtained on into the formula, and when we do that correctly, we find the value off. E is your point 0973 Next we add and subtract. Imagine of error from Deb. A P one had means be too hot. In this case, p one heart minus p too hot. He is 0.56 men and 0.44 which is 0.12 So the confidence interval limits are as your 0.1 to minus 2.973 which becomes 0.0 227 So that's the lower limit on for the upper one. We have 0.12 plus 0.973 which he calls zero point 21 73 So therefore the 95% confidence interval estimate will be your brains. You 227 It's less than p one in the spey too, just less than your point 217 We round it off to three significant digits so we can see that zero is not contained in this confidence interval. And for that reason, it appears that P one is equal to P. Two can be rejected, So it tells us that, uh, the equality off the one and P two can be rejected because zero is not contained in the interval. Next we walk out the 95% confidence interval estimate for each off the proportion. So we need to get the margin of error, um, for each proportion, and then get the confidence interval for P one on. Then we get the confidence interval for P two. So let's proceed. We get e the margin of error by working out the following. That is for the first for the first sample. And when we substitute the palace, it is going to be 1.96 Scratch it off P one hunt, which is the 0.56 times the value of Cuban 100 0.44 divided by end one, which is 200. And when we walked this out, we get the value off E zero point as usual. 688 So, for the first, uh, proportion, Imagina vera is 0.66688 and this valley will still be the margin of error for the second sample because we have the same numbers. Um, for P one, have you noticed that the value off P 100 Cuban heart that the pairs are the same 0.560 point 440.440 point 56 and also, the sample size is, uh, the same, and one is 200 extra 200. So the margin of error is identical for both, uh, the fast sample on the second sample. So then, to get the confidence interval for P one who would need to add and subtract the margin of error to the value off people? So the value of P one heart had you obtained it a zero point 56 And so the confidence interval for P one will be 0.56 minus zero point zero, 688 and uh, 0.56 plus 0.688 And when we walked this out to you, obtain zero point for nine want to this less than P one? It's just less than zero point 6 to 88 When you run off to three significant digits, it's going to be 0.491 It's less than P one, which is less than 0.6 to 9. So that's the 95% confidence interval for the first proportion p one now for the second proportion P two had we have 0.4 from, So we're going to address attract 0.688 eso. It's going to be there for the lower interview. It's going to be 0.44 my next year from zero 688 less than P two, just less than 0.44 class zero friends zero 688 And when you walk this out, it's going to be easier point 3712 less than P two, just less than 0.5088 And when you simplify, it's going to be 0.371 is less than P two. That's less than zero point 509 to 3 significant digits. So, according to this 95% confidence interval estimate for each proportion, we noticed that there is an overlap between the confidence interval levels. Interpol um confidence intervals. So, in other words, that the values the P to the confidence interval for P two is contained in the confidence interval for P one. Yes, and that suggests that there's an A P one equals p two cannot be rejected. You know, we could assume that P one and P two are equal according to this comparison off the individual proportions. Next read, go and conduct hypothesis test I used between these two proportions. In other words, we test the non hypotheses. H not p one equals p two on the alternative hypotheses. H one p one is not equal to Peter. So we need to proceed and walk out the test statistic. Zed, which is obtained from a p one hunt minus p to heart all that minus zero divide by people are sorry, this square it off, people Cuba off n one plus people cube off end too. So we had all these figures and we just need to substitute them into the formula. And when we do that correctly, the value off the test statistic is 2.4. And since this is going to be done with the 0.5 level of significance, the critical values that he is plus or minus 1.96 So now we can compare their value of that on the critical value. So in this case, this is a critical region shaded so one negative 1.96 and positive 1.96 And in our case our 2.4 is within the critical region and for that reason would have to make the conclusion to reject the non hypothesis. So I'm by rejecting ANALITICO. This is we we mean that a sufficient evidence to reject the equality off the off the proportions that is rejecting that p one is equal to P two Now, finally, we're going to I used this results to make a conclusion about equality off P one and p two. So the conclusion here is to reject, um, they reject p one he calls Peter. That is, according to them, confidence interval test. How the confidence next five percent confidence interval estimates of P one minute speak to on also from the hypothesis test that he led, uh, the critical the test statistic of 2.4. So meaning that we would have to reject the null hypothesis. And we also supposed to tell which of the three methods is least effective in testing the quality off p one and p two and we can see that the method off the overlapping the conflict individual confidence interval is the least effective method because according to this method, we note that the P two overlaps the confidence interval of P to overlap with the confidence interval of P one, we suggest that the two proportions could be equal, so that's the least effective method in testing the difference between the confidence between the two proportions.

So for this question I'm gonna have my difference be the eight a.m. 10th minus the 12 a.m. Temp. And I have my differences is negative 2.4 negative 1.4 negative one negative 10.8, negative 1.5 negative 1.5 and negative 1.8. And so I have a total of seven pieces of data. I have that the mean of those differences is equal to negative 1.4857 And I have the standard deviation of those differences equal 2.5 to 4 to. And so we will be assuming that the main difference is actually equal to zero and alternately that is different mean differences different from zero. So we're doing a two tailed tests and we want to find in the picture. We are again assuming that that mean difference is at zero but we're getting a value of negative 1.4 857 And since we're doing a two tailed tests were going to that shaded area and I shaded in red, that shaded area is going to be my p value. So let's continue on with our test. So we want to find what's the likelihood of sampling and getting a test statistic that is. And uh let's see, well let me write it up first A. D. D. Bar that is less than or equal to negative 1.4857 So that's A. D. Bar. We want to find what this likelihood is and double it. Now let's change it to our test statistic and that is going to have degrees of freedom of six. Yeah. And we need to convert it. So negative 1.4857 minus the mean of zero. That's what we're assuming it is. And then we take that standard deviation over the square root of and make sure you don't use n minus one there. And when we get that, our test statistic comes out to be negative seven point 499 And then we find that p value and that p value comes out to be about 0.3 So at a 5% significance level we have definitely have sufficient evidence to what make what decisions to reject the now. So we definitely think that that difference is not equal, that the difference is not equal to zero, that there does seem to be a difference in those temperatures. Now, if we want to go back and in in blue, if we want to find the confidence interval are significance level was five for a two tailed test, so we would find a 95% confidence interval and we would take that value that we got which was the negative 1.4857 plus remind us. And then we have to find the T. Star value that is going to correspond with six degrees of freedom for that 95%. And that corresponds with 2.447 And then we would take that 0.5 to 4 to divided by the square root of seven. And let me quick do my calculations here. And we get that those two values are negative 1.972 negative 1.1 And notice that this would cause us to reject No because zero is not contained in this. We would think that the differences different from zero. So the two results will always agree.


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