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.