The following is a solution to number 19 and this is where X. Bar the sample manner. The point estimate is one. Oh wait. And then the standard deviation for the population. So we do actually know what sigma is, is 13 and we're asked to find the 96% confidence interval whenever and equals 25 96% confidence interval whenever and equals 10. So a smaller sample size and then an 88% confidence interval whenever and equals 25. So a lower confidence level. And I should note that they do actually say that the population from which these samples were drawn is approximately normal. That allows us to use these smaller sample size. If we didn't know what the population distribution was or if there was skin is to the population distribution then we would need to have a larger sample size greater than equal to 30. And we kind of talk about that a little bit down the road. All right, so let's go to our calculator. So I'm gonna go to my calculator and um you know you can use the form if you want but calculators much quicker and it's always accurate. So I'm gonna go to stat and then there are over two tests and I'm gonna go to the seventh option here the Z interval. So since I know that I I know what the population standard deviation is, that I was given that sigma that allows me to use that Z interval. And here you're given either data or stats. Now we're given the summary stats. I'm gonna go ahead and just keep it on stats And the Sigma is 13. The x bar is 108. And then the end at least for this first time is 25. And then that confidence levels .9696% confidence level. And then whenever I calculate I get 102.66 So 1 13.34. So I'm gonna write that down Before I forget it. So 102 0.66 All the way up to 113.34. That's my 96% confidence interval whenever and equals 25. So now I'm going to do a 96% confidence that or whatever and equals 10. So almost the same thing. I go to stat Tests and it's that 7th option, Everything else stays the same. But this time I want to change that end to 10 And then I calculate and I get 99.557 and 116.44 99.557 In 116.444. So if you compare these two real quick um the confidence interval for part B is much wider than Part A. And that's because it's a smaller sample size and that kind of tells you what that second part of the answer is as in decreases. The margin of error is going to increase. So the smaller number that you divide by means that it's going to give you a bigger number. So the margin of error increases and therefore the confidence interval is wider. So the smaller the sample size, the wider the confidence interval. Now let's look at part. See this time if we go to stat tests Z interval this time the ends back to 25 but we're going to do an 88% confidence interval And that's gonna give us one of 3.9 612.04 103 0.96 And then 1 12.04. So between one and 3.96 and 112.04. And it says compare part C to a well, this is narrower. They're over than part A. Okay. So the reason is as the confidence level decreases, which is what happened went from 96% confident to 88% confident, then the margin of air Emmy also decreases. Therefore the confidence interval is narrower. Okay, so if you decrease the confidence of all, that will make the confidence interval narrower because the margin of error is going to be narrower too. All right. And then partied says, could we compute the confidence interval if the data is not normal? So if the population from which the data was taken, if it's not normal, can we compute a confidence interval? And the short answer is no um reason is because and is not at least 30. So if n is Great and equal to 30, then it doesn't matter what the population is. If it's normal, if it's not normal, if it's skewed, whatever Central limit theorem applies and in equals 30 and it's created equal 30, then we're fine. But um With these sample sizes, this is 25, 10 and 25, these aren't big enough. So the the population has to be normal in order to conduct these confidence intervals, which such with such small sample sizes and then the last part part e If there are three outliers above the main, how would that, you know, affect everything? So if there our three outliers above the mean? Well, first off, we probably couldn't do a confidence interval because that would skew the data and our sample sizes aren't big enough, but let's just assume that, you know, we may be truncated or something. Um In theory, the confidence in the rule would b shifted up, or it would kind of increase. Everything would just shift up towards the outliers. Thank you. So that's what's going to happen if you apply a few outliers, at least on the high side. If they the outliers on the low side, the opposite would happen. Everything would just kind of shift down a little bit.