Tom randomly selects flights and the times (minutes) required to taxi out for takeoff were measured: 37,13,14,15,31, 15. Create a normal quantile plot:b) Does the s...

The taxi and takeoff time for commercial jets is a random variable $x$ with a mean of 8.5 minutes and a standard deviation of 2.5 minutes. Assume that the distribution of taxi and takeoff times is approximately normal. You may assume that the jets are lined up on a runway so that one taxies and takes off immediately after another, and that they take off one at a time on a given runway. What is the probability that for 36 jets on a given runway, total taxi and takeoff time will be (a) less than 320 minutes? (b) more than 275 minutes? (c) between 275 and 320 minutes? Hint: See Problem 21

So we're gonna be looking at taxi times on the runway and we'll go to set 15 and appendix speed, we'll find the range variance and standard deviation of these things. So start us off, we can find the range quite easily. All we have to do is subtract our largest value from our smallest value or high and low. So that's gonna be 49 and 12. Take the difference between that to get a range and that's 37. And since we're looking at minutes as our units were an attack on that minutes Now, if the variance, uh there's a couple of data values here, we have a sample size of 48. So quite a few numbers to go through. What I would recommend doing is in Excel you write down all of the 48 values and trim down, You know, one through 48 in this column is called that A. And then in column B. I want you to find what the average is. Actually, that's not just gonna be that's gonna be a single cell. So all you have to do is say highlight a single cell in there and in there you're gonna write equals uh mean, actually no go for average. So average. Maybe it's a biggie or biggie average. So you'll just open the parentheses, highlight all these cells here. Mhm. And that's what you'll highlight and then I'll give you your average. Then what you can do is copy that number that you get into a second column. Let's call that column seat and then what you'll do is say that this equals open through parentheses. We highlight the first cell and then subjected from the mean, which is in b we'll call that be one, make sure to put a cash sign in between these two. So cash sign, cash sign, that'll lock it into that one cell, so it won't change the average and then go ahead and drag it down. Um when you drag it down, just make sure you find that little cross cross thingy in the bottom right corner. So you can drag it through everything and then that's going to get you your difference. And then in a different column, we'll call that column deep. What you're gonna want to do is simply square everything from this. So it's gonna equal open parentheses. E highlight a cell, I like the top sal drag it down, square it and then everything here will be squared. And then once you get that, all you have to do is take the sum of everything in here. So let's call that E one. What you could probably do is just use an auto some on the top right corner and just take the son of deep. Mm And then you'll just eat one and a different cell. You can just say that he won Equals E one divided by Your sample size of 48 -1 souls 47. And what will end up getting for our variance is going to be 85.51. Yeah. Yeah. And that's gonna be minutes squared for our variants to get the standard deviation. All we have to do is square root. That variation. And that is going to give us 9.24 minutes. Yeah. Mm. And I'm just gonna take away that square root. Yeah.

So I know we have some negative numbers but that's not going to affect how we do our process. The mean is still going to be the sum of all the numbers. And this is going to come out to be negative 114. And we have Uh eight numbers to look at. So that's going to meet our average. And when you divide that that's gonna be negative 14.25 as our mean. Now for the media and all we have to do is order up our numbers from at least the greatest in other words in ascending order. So we're going to find the values at the 4th and 5th place. Since those will be our middle to values, We're looking at even numbers. So we have to average out that's gonna be -9 and -15. So we'll just add those two up and divide by two. Okay, And get our median of -12. Now the mode is the number that appears the most often. But seeing as we have no duplicates, there is going to be no mode. And finally the mid range all we have to do is at the smallest value which is 1932 to our largest value which is going to be 11. Divide that by two. And that is going to be a mid range of negative 10.5. Mhm. And since we're looking at airline wait times, having a bunch of negative numbers means that they are arriving early, which means that american airlines has very good performance

So we're gonna be looking at a few different measurements here, the variants the standard deviation and the range. Uh for a problem, we're gonna be looking at a couple of different airline flights, so we're going to first calculate the range, which is going to be our high value minus are low value, and that will be uh most 11 minutes minus are lower value of negative 32 meaning that the flight came half an hour early And that's gonna give us a range of 43 and this is gonna be in minutes. So there we go. And then we can calculate the variance since we found out the standard are the mean from last section. So the way we find the mean is by adding up each one of our values divided by the number of observations we have. So we have eight different observations. And if we add up everything and divide that by eight, we should end up getting a negative 14 and a quarter of 14.25, this will also be in minutes. So what we can do is plug this negative 14.25 back up into this variance equation right here. And what we'll get is after subtracting the observations from the mean and squaring that difference and then doing that to every single value we have, will divide that by one, less than our observation. So that's gonna be 8 -1 on the bottom, And on top, I'll do the first one, that's gonna be negative 15 -3 -14.25. Mhm And then we'll end up squaring all of this. Remember there's a parentheses outside here, and then we're just gonna add up all the other numbers, all seven of them. And what we'll end up with is 231 0.36 and says, since this is a variance will square that square those units. All right. And to make this into a standard deviation, all we have to do is take the square root of our variants, so it's going to be the square roots of 231 3 6 minutes squared. Okay, And what this comes out to be is 15.21 minutes for our standard deviation.

Following is a solution video to number 26 and this looks at the flight times from Albuquerque to I think like Denver or something. The average flight times and minutes. And the first part is to ask for the point estimate. The point estimate to find the mean is the X bar the sample mean. And you can do that just using the formula or if you want to you can use technology and I'm gonna use technology. If you go to Staten edit this on a T. I 84 Here are the data values 117 minutes, 95 minutes, 109 minutes etc. And if you go to stat and then air over to Calcutta and then it's one of our stats List is L one and then we calculate and then this X bar is about one of 3.44 So that's what we're going to put here. So one oh 3.44 minutes is the point estimate. Then we're giving a box plot and a normal probability plot. And it says because the sample size is pretty small. If you look back at that, In fact, n equals nine. There are only nine native values. So in order to use the Z interval, um we need to see if the population is normally distributed and so we can look at a box plot in the normal probability plot to determine that. And by looking at those, the short answer here is yes, the data appears normal, so the box plot looks fairly symmetric so the data appears normal. And also the normal probability plot looks about linear, kind of has heavy tails, but that's fine. So it looks about linear. All those data values are kind of within range free from any skin this or outliers. So that's Looking at the box pot normal probability plot. We can go ahead and use those conditions for inference or we basically verify the conditions of inference. Yes, the sample size is small, but the original data appears to be approximately normal. So now we're going to find the 95% confidence interval. So again, you can use the formula if you so wish or you know, it's probably a little easier if you have a stat dishonesty I 84 tests and it's the seventh option here. The Z interval. Alright, so since we know the sigma, we know the population standard deviation, we're going to go to the Z interval and then make sure the data is highlighted. So it's not stats. We usually we have summary stats at this time, we actually have data sets of data and then sigma is eight. That was given to you in the problem. The list in my case was L one, so if you have a different column then there was changes to L. Two or three or whatever you have. And then the sea level is 95 95% of 950.95 And then calculate. And then this top line here that gives you the Confidence intervals about between 98.218 and 108.67. So the 95% confidence that it will is between 98 .218 And. 108 0.67 And then we also need to interpret that. So the interpretation here follows the same kind of pattern. We say we can be 95% confident that the main flight time mm between in Dallas and Alburquerque. I should say for all american airlines flights for all american airline flights, mm is between 98.2 and 108.7 minutes. Now, I've rounded there, you can be as accurate as you want, but Somewhere between those two numbers, we can be 95% confident notice it's for all American airline flights between Albuquerque and Dallas. So that's the 95% confidence interval. Now we're gonna do the same thing, but this time we're gonna change it to the 90% confidence interval. So we go to the Z. interval and all this stays the same except I'm gonna say .9 this time And it's between 99.058 and 107.83. So let's go and write that down. So the 90% confidence interval is 99 point 058 And 107.83. And we interpret it the same way really. The only difference here are the numbers. So we could say, you know, we can be, I'm not gonna write everything down Just, you know, the things that change. So we can be 90% confident that the mean flight time between Dallas and Albuquerque for all american airline flights is between, so is between so dot dot, dot is between 99 058 And one of 7.83 minutes. So it's between, you know, basically an hour and I don't know, maybe an hour, 40 minutes or so. Um, so let's look at these a little bit more closely. So the 95 compared to tonight. So everything stays the same except for the confidence interval or the confidence level, I should say. So, notice that the interval gets a little bit narrower. So here we've got uh, oh, maybe about a 7.5 minute difference, but here we've got almost actually more than a 10 minute difference between the low end and the high end. So whenever we get less confident we can widen up that, I'm sorry, narrow that interval down. So that's kind of the trade off. The more confident you are, the wider your interval has to be the less confident you are, then you can be a little bit narrower on that interval and that's kind of what what party is talking about. So in playing our words will save the width of the interval decreased when the confidence level decreased, which makes sense. The less confident you are, the narrower the interval can be. Yeah.