The following is a solution to number 48 9. And this just looks at three different data sets that are very, very similar. Just the sample sizes are a little bit different and this is taken from population where the mean is 100 the population standard deviations 15. So we're gonna do a few Z intervals. But the first part of this is to find the sample means for each data set. And I did technology use technology for this, I use a. T. I. T. Four because it, you know, really cuts down especially as the days get bigger, but you can use any sort of technology, Excel works great mini tab. Or you can do the formula, it might take a little longer but that's fine too. So stat and edit. And you see I have these three datasets here and I'm just basically gonna do the data set X. Bar for one, L two and L three. These are my three different data sets of stat. And then air over to cal can it's the one of our stats and I'll just go and change that to L one and this is my first X bar. So X one bar 99.125 So X1 bar is 99.125. Okay, so I do the same thing, go to stat, cowpoke one of our stats, but this time it's going to be L two. So the medium data set and I get about 99.1 99.1 for X bar. And then very last go to stat, cowpoke One of our stats and then I'm going to change this to L three And that's about 90.03, repeating. So we'll say 90, I'm Sorry, So, you see these sample means they're roughly the same. There may be slightly off by, you know, a couple hundreds, but it's not huge. Right? So that you can essentially think, okay, these are about the same um sample means. So now we're gonna do the 95% confidence interval for each one of these and then we're gonna compare. So let's take a look if we go to stat and then tests now, since we know the population standard deviation, we can use the Z interval and we have data Um since we're given a list of data, the sigma is 15 and I'm just gonna go in order. So my list is a one first, the frequency is always one and then the sea level, the confidence levels, 95% of .95. And then I go in and calculating this top one here, That's my confidence interval. So it's a pretty big confidence interval. And that's 20 Ish, you know, maybe a little more than 20 Units apart. So let's go and write that down. 88.731 88.731 All the way up to 109.52. Okay, so that's whenever the data set was quite small, it was about eight data values I think. So we're gonna do the same thing if we go to stat and then tests and it's that seventh option again. But this time we're going to go to L two. So the medium data set size and let's see what happens here. So 92.5-6 all the way up to 105.67. So you see that's already getting a little bit smaller, it's about 13 units apart as opposed to 20. Alright, so 92 .5-6 All the way up to 105.67. Yeah. And then finally the last data set So that's whenever an equals 20 so stat tests and it's still the seventh option. But this time it's going to be the third column and let's see what happens here. So this is where the sample size was 30 93.666 to 1 oh 4.4. So you see that's even smaller, it's only about 11 10 10.8 I guess units apart. So let's go and write that one down. So 93 .666 to one oh 4.4. So these are three data sets and were asked to compare or you know, look at as the sample size increases. You know what happens? And as we've already talked about it as in increases that margin of air is going to decrease. Which makes sense because you're dividing by a bigger number. If you look back at that format sigma over squared of in your divided by a larger number. So therefore that's a horrible. Therefore I guess it's not really shorthand whenever it's that bad. So therefore that confidence interval width decreases. Okay, so as in increases, the confidence interval will get narrower and you can see that as we increase the sample size, It gets narrow, narrow goes from 22, 13 To like 10 and change. Okay, so then these last two questions kind of deal with the same thing. So let's say you you miss um put something, answer that first. I think it's the first data value was 106. Well let's say you put that in a 016 so let's see what happens with the new confidence intervals. So let's go ahead and make that error. So stat edit. So yeah, it's that first one, so 016 which is 16, so I'm just gonna change it to 16, 16 and 16, So let's see what happens here. So stat tests And it is the easy and also seven, and then let's just kind of keep the same order. I could do L three first, I guess, but let's just keep it to the same order and see what happens. So 77.481 to 98.269 77.481 to 98 269. And this is kind of the last part, but you see that this actually already doesn't it's been uh Affected so much by that outlier of 16, that this has pulled it down so much that it doesn't actually contain the population mean. So you can see the smaller that sample size, the more it's affected. So let's go and take a look at the second one. See what happens here, stat tests seventh option and list is L. two this time. So it is significantly bigger, 88-101, So 88 points 0 to 6 All the way up to 101.17. So that one barely contains The mean 100 but it's definitely been shifted down. So I'm also comparing two up here, you know, this first one's been shifted down like 11 units right from 88 to 77. Well this one's only been shifted down about 4.5 units. So you can see it's not nearly as affected by that outlier. And then the very last one stat Tests 7th option. And then this time we're gonna change this to L. three and 90.66-1 on 1.4 9666 To 1014. So you can see this one also contains the population mean? And it's been shifted down looks like three units instead of four, you know? So it's even less affected because the sample size was 30 instead of 20 and that's kind of what part E talks about Which ones are still good um indicators or which one is still capture the population mean? Are are good estimates and the last two seem to still be good estimates. So data set golden scroll down. So if you look back this dataset one that's the only one that doesn't contain 100. So data sets two and 3. Data Sets two and 3 still capture New Equals 100. And so what does this show? This shows. Okay that as in increases the more accurate the confidence interval becomes because outliers become less influential. Okay, so the greater the sample size, the less influential those outliers become.