The following is a solution to number 11 and were given summary stats that the sample mean X bar is 18.4 and the sample standard deviation is 4.5. Now we only know the sample standard deviation, not the population standard deviation. So since we don't know the population standard deviation sigma, we have to use the tea interval instead of the z interval, so keep that in mind moving forward. So I'm gonna use technology for this problem here. We basically have three questions that are very, very similar, and then um one where we talked a little bit about conditions for inference, um now it doesn't say anything about the normality of this population, that's going to come up here in part D, But it doesn't matter because these sample sizes are big, they're greater than 30, so it doesn't matter how this population is distributed, that sample size is big enough. So we're asked to find the 95% confidence interval whenever end is 35, Then we find the 95% confidence interval will never end this 50, we're gonna compare that so the same confidence level but the sample size is a little bit bigger. Then we come down here and we increase that sample that confidence level of 99% confidence. And then we're back down to the n equals 35. So let's take a look and see what happens. So I'm gonna use the T. I. T. Four but you can use any sort of technology want or you can go and do the formula but if you go to stat and then arrow over two tests and it's this eighth option here, the T. Interval and um make sure that the summary stats is highlighted because we don't have data. We're just giving summary stats and X bar was 18.4. The s was 4.5. The end for this first one was 35 then that confidence level is 350.95 and that represents 95%. So whenever we calculate this top band here that's our confidence level. So about 16.9-19.9 is going to be our interval. So let's go and write that down. So 16 Point I'll go ahead and do what they did. So 8854 And then the upper bound is 19.946. Okay So that's our first confidence interval. Now let's see what happens when we increase that sample size from 35 to 50. So we go back to stat tests and it's this eight option here, the t interval And this time the only thing that changes is the end and that's 50 now. So we calculate And we go from about 17 to 19.7. Okay, so let's write that down. So 17 .1-1 All the way up to 19 six 79 So let's compare now. So it looks like the lower bound has actually increased by, I don't know maybe about point 2.5 4.3 and the upper bound has decreased. That means that interval is narrower. So what does that say about in? And the margin of air? Well, as in increases, which is what happened here? The margin of air decreases. Making the confidence interval more narrow. Okay, so they're inversely related. So as in increases or degrees of freedom, increase. That margin of error is going to decrease, making it less variable. And that's the reason why that interval is a little bit narrower. So now let's go back to the 35 sample size. But this time we're going to increase that confidence level to 99%. So go back to stat tests and the 8th option. Okay. And we're back down to 35 but we're going to increase that sea level of the .99. And then when we calculate, let's see what happens here. So we go from 16.3 to 20.5. Okay, so let's write that down 16 point 3 to 5, 20 point 475 So compare this to part A where we had that same sample size but different confidence levels. So the lower bound actually decreased in the upper bound increased. That means that interval is wider. Okay, So what does that mean as the confidence level increases? That margin of error is also going to increase because that that critical values actually greater. So this is larger whenever you increase that confidence level, um making the confidence interval wider. So as the confidence level increases, that confidence interval is going to increase or get wider as well. All right. And then part D. Now it says what conditions for inference are needed? If in equals 15? Well, if N equals 15, then the population from which the sample was taken must be normally distributed. Okay, Since And is less than 30. Okay. So notice in part A B and C and I already talked about this, but A B and C. Um those sample sizes are greater than equal to 30, so it doesn't matter how this population is distributed. It still works. We can still use that procedure But if N is less than 30 in this case 15, then we need that statement in there that, Hey, don't worry about the population, it's normally distributed.