The following is a solution video to number 25 and this looks at a very small data set. In fact, there are only four data values of the amount that it costs for repairs for chevy cavaliers. And the first part of this asked for a point estimate for the mean and the point estimate for me, and it's just gonna be the sample means. So X. Bar. And we're also assuming that the sigma, the population standard deviation we know it is $220. So um the way you find X bar is just, you know, finding the average the arithmetic mean. So you add them together and divided by four Or you can use technology and I'm gonna use technology. So if you go to staten edit their your data values. So one of them was $225, 462 7 29 and 7 53. And then if you go to stat and then couch and it's one of our stats and we're gonna make that L. One And we calculate that. And we get this this x bar here. 542.25. Okay. So the point estimate is 542 points $25. Okay. And then in part B were given a box plot and a normal probability plot. And we're basically just checking to make sure that the distribution of the population is approximately normal. And looking at those two, it looks it appears to be approximately normal. So the short answer yes, we can use the Z interval here, even if the sample size is so small, because um the box plot appears there are no outliers and the normal probability plot looks like it's about linear. So, so in short, we're just going to say yes, it's approximately normal. The box plot says that there are no outliers. And since the normal probability plot is approximately linear, so the normal probability plot is within range will say, Which means that none of those data values are kind of outside those lines. And then the next couple of parts, we're gonna go ahead and use our inference procedure and we're gonna find the 95% confidence interval. And I'm gonna go back to technology and do that. Now. You can use the Formula film, but technology is much faster, so I'm going to do that. So if you get a stat test and we're gonna use the Z interval since we know what the population standard deviation is, So it's the seventh option. And usually we leave it on stats because we're giving summary stats, but this time we're actually giving a data set, so I want to leave it on data. Now. The sigma Is was given to you in the problem is 220 for your list. Go and change that to L. one unless you put it in different column and then put it as L. Two or L. Three or whatever. But I put in L. One and then the frequency is one and then the sea level, the confidence level is 95% of 950.95 Then we can calculate. And then this front first line here that gives us the 95% confidence interval. So 326.65 and 757.85. It's between those two. So it's gonna write that down before I forget it. So 326 0.65 2757 0.85 And then we also need to interpret that's the way we interpret that. We'll say we can be 95% confident that the mean repair cost mm for all chevy cavaliers is between mhm $326.65 cents and $757.85. So between those those two, Okay then the third part or four fires should say the fourth part is a 90% confidence interval. We're actually going to compare these two. Now I'm not gonna write everything down. Um Now we're gonna do the same thing. So we go to stat and then test and we're still using that Z. Interval. So go to the seventh option there. Z interval not the T. And everything stays the same except the sea level. We're gonna change 2.9. And we calculate that we get these numbers. Let's go and write these numbers down and we'll just compare it real quick. So 90% confidence interval between 361 0.32 Let's draw a little line here so we see it. Okay? And then 7 23 point 18 and then whenever we interpret this, I'm not gonna write everything down because it's basically the same thing. So we can be 90% confident everything else stays the same and then, you know, confident that the mean repair costs for all chevy cavaliers is between between $361.32 And $723.18. Okay, So everything else, you know, basically stays the same. So let's look at these two confidence intervals, the 92 the 95. So how do they compare? Well, the 90% confidence interval is actually a little bit narrower, so the lower bound is higher than the 95% and the upper bound is lower than the 95%. So it's a smaller confidence interval, so the width decreased as the confidence level decreased And the next part of that is is this reasonable? Yes, it is. If you just look at those critical values, so if you look at the margin of air, the critical value for the 95% confidence interval Is 1.96, whereas for the 90% confidence interval that Z star is 1.645. So we're multiplying by a smaller number whenever it's 90% confident. But also, you know, if you just think about logically the more confident you you have to be, the wider you need to make that interval. Otherwise we would just want to make it, you know, 99.9% confident every time. And so that that's the trade off. If you want to be less confident, well then you can narrow down that confidence interval and you can give you better information. But if you want to be more confident, then you do need to widen that confidence interval up just a little bit. So yes, it is reasonable. Um, as the confidence level decreases, you do, you will have a narrower confidence interval, which is a good thing.