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Construct the confidence interval for $\mu_{1}-\mu_{2}$ for the level of confidence and the data from independent samples given. a. $99.5 \%$ confidence, b. $95 \%$ confidence, $$ \mathrm{w}_{1}-68, \overline{\mathrm{n}}_{1}-115.5, \cdot \mathrm{.}_{1}-1 $$ $$ \mathrm{w}_{\mathrm{g}}-\mathrm{s} 4, \overline{\mathrm{x}}_{\mathrm{g}}-187 . $$

For the given exercise we want to let ffx equal 1- Co sign of two x -2. Okay. And that's going to be divided by um X -1 Squared. Mhm. That's going to be our function after Becks family monograph F to estimate the limit. And we're going to evaluate its values near one to support the conjecture. So we're going to look at 0.9 And we're going to get very close and it appears to be approaching two and then we're gonna look at about 1.01 And getting closer again, we appear to be approaching the value of two. So sure enough, if you look right here, We see that as we approach one it seems like we're getting to the value of two. So that's the mere final answer.

Hello. So today we're going to determine the ability of some major say, by inspection. So let's remind ourselves of the rank quality here. Um, the rank ability serum is such that the rank, that's the major, say the Milady or the dimension of the null space of a equals the number of college. So to solve affordability, we just have to rearrange this equation a little bit such that the dimension of the all space of a equals the number of columns minus the rank today. So for the number of columns, we just have to count 123 That gives us three tones for the rank. Let's look at our matrix. So we have a leading one in the second column. So we're gonna count that, and then we have another one right below that, however, uh, in each successive, bro, we actually have to have that step form with Rochel on form that red. So the second rock actually being cleared out by the first row using ah row operations. So we're not gonna help that. But we come and look at the third row and see that we again have a leading one that is to the right of the first lead one so we can count that. And for the same reasons as the second round, the fourth row is not counted. So we have to one to pinpoints, or, uh which gives us the rank of a so three months to equals banality, which is Thank you. Nullity of this matrix A equals one.

The following is a solution to number four and we're asked to find the 99.5% confidence interval for the difference of two population means given the following data here, and we're gonna use a two sample T interval. The reason why we have to use a T intervals because the sample size isn't greater than 30. So if at least one of these is not Uh at least one of these sample sizes less than 30, then we've got to use the tea interval instead of the z interval. So I'm gonna use a tt for here. If you got a stat, There are over two tests and it's gonna be option zero. So that's all the way down here where it says to samp t ent press center and just make sure data, I'm sorry, stats is highlighted and you can just punch in your data there. The confidence level is 00.995 stands for 99 99.5%. And then under pooled, you're just gonna want to keep that is Yes. The reason being, is that the two standard deviations are the same. Whenever that happens, then we have to have a pulled experiment And then we calculate in this top band here is our confidence intervals, so 10 4-9 and 14571. Okay, so we're 99.5% confident that the true population mean difference is between 10 4-9 and 14 .571. Okay. Mhm. And part B. It's very similar. We just want to be a little more confidence in 99.9% confident for the difference to population means again we're gonna use the tea interval because this 25 that's not greater than 30. So if we go to again stat and then tests and then the zero option. Now I actually have changed these because it's different data obviously. So 215 is the X one, bar seven is the standard deviation and 25 is that sample size? And then 1 85 is that I mean 12, is that standard deviation? And then 35 was the second sample size and we want to be .999. So 99.9% confident pull this still yes. And then we calculate and we get between 20.713 and 39.287. Okay, So let's go and write that down. 27 13 all the way up to 39 0.28 7. 99.9% confident that the true difference of the population means is between these two numbers. Mhm

Decent case chosen from the Kate sub interval of a regular partition of the interval. 0222 into in sub intervals of link Delta X Express limit as it goes to infinity. Summation cake was one to end C sub case square Delta X as a definite integral. Okay, let's unpack all of this and figure out what this means. Okay, so here's what they're saying. We have the interval on the X axes from 0 to 2. We cut it up into in sub rectangles, and in one of them, one random one was called the Cave Rectangle. There's an enlargement of it. We pick some value, which we called C sub k. And then, um the I forgot to say this one thing the length of these intervals is delta eggs. So we pick cease okay in that interval, and we plugged it into a function to figure out what the height of the Kate Rectangle is. Okay, so here's the wit. Here's the height. Then we're gonna add up all those rectangles, all right? And then we're going to take the limit is in, goes to infinity, which will make them infinitely staff in so that they take up exactly the space. So we want to know what integral does this represent? Well, I always think of these two things as being the integral sign, and this is the Delta X at the end is the DX, and they told it it. It was from 0 to 2. So here's the width of the rectangle. Here it says at all the rectangles, up from 0 to 2. All we need to know is, what's the height of the rectangle? Well, we took C sub K, and we squared it. So that means the function here must be X squared because it's C K Square. So all you have to do is look to see what's happening in the C K and then you'll know what the function is, and then you can write dinner.