AP Statistics DESMOS Inference Review

Added: 2026-05-09

[00:00:00]Hey, welcome back. This is Mr. Kelly.

[00:00:02]Let's hop on over to Desmos and do a little review with our AP stats inference. What do you say about that?

[00:00:07]Make sure you're going to this version of Desmos so that you get the college board version that is in your blue book.

[00:00:13]Okay, so here we are in Desmos. The first thing I want to do, let's uh let's see if it'll do a normal curve for us, shall we? So, let's go up to plus and we'll do inference. And if we go to probability distribution, normal distribution is one of the choices. All right, so let's pretend like the mean is 50 and the standard deviation is 10. And I'm going to create that distribution. I have a nice-looking distribution here.

[00:00:35]And we can look at the cumulative probabilities. If I want to find the area between, you know, any two values, that's that's automatic right here. We say area between, we'll say uh I don't know. What do you want to do?

[00:00:45]30 and 70?

[00:00:48]Right? And it adjusts as we're typing it. And you want to look at the picture and look at what you have over here, 0.95. That seems about right. Right? If What if we want to work backwards? Maybe we want to find like the 80th percentile. Well, then I would go to bounds. That's like inverse norm. The 80th percentile means the bottom 80% is below this score, right? So, I'd go left and then X has to be less than or equal to. And if you go to bounds, you can just type in right here 0.80. And if you look, it tells you up here X has to be less than or equal to 58.42. Now, remember that the mean is 50. So, 80th percentile, you're going to be above 50.

[00:01:23]That's That seems about right. So, we're good to go with that. Now, suppose we're doing a sampling distribution and we pull samples of size I don't know. Let's say four. Then we could do the same thing.

[00:01:34]Let's go ahead and we'll do another probability distribution.

[00:01:38]And we'll make it normal, and the mean is What did we say the mean was up there? 50, right? But then the standard deviation will be standard error cuz we're doing sampling distribution, right? So, it'd be 10 divided by and then we'll say square root of four, which we know is two, right? But we have that distribution, and you can see that it is centered a little more, right? And it's a little higher in the middle, and that's because if we're taking samples and finding sample means, remember the central limit theorem says they're all going to be kind of like getting on top of each other here, getting narrowing in on the mean value there of 50, and then we can find the probability that that sample will be you know, we'll say again, what is the probability that the sample will be you know, from 60 down to infinity. So, that would be left, and I put a put a 60 here. So, it would be 0.977. That would be the probability that your sample of size four would have a mean of 60 or lower. So, that's how you can do normal distribution sampling distributions using normal curves. Let's see what else we have here. Let's clear this out, shall we? Whoops.

[00:02:44]Clear it out. Delete all. Great. Let's go into our inference here, and let's look at the different choices that we have. We have Z test, T test, Z test for proportions, independence, and goodness of fit chi square tests. So, here's where you're going to focus your attention. You're going to focus on T tests for means, right? And then one proportion Z test, two proportion Z test, and then our chi square test.

[00:03:08]That's where most of our focus is going to be. So, let's look at what happens when we have a T test. If we want to put in the data for a T test, maybe we have a matched pairs T test with a bunch of differences. So, we can put those in with brackets, and maybe our differences are mostly positive, but sometimes negative, and we just have a bunch of positive values mostly, right? And I want to do a test Look, I don't have to do anything else other than put those numbers in here, our data points right there. So, n equals what? 1 2 3 4 5 6 7 8 9. Create the test. And when we create our tests, we have the choice of confidence interval or significance test. So, let's look at confidence interval, and you notice that we have the interval right here. Our point estimate is going to be 2.111. That would be the mean, right, of our sample, 2.111.

[00:04:01]And we're trying to estimate it with 95% confidence. But, you know what? That's not good enough for me. I want 99% confidence. But, notice that my interval got a little wider, which is what happens when we have a larger confidence level, right? But, maybe we want to do a test with that. And we want to see Oh, you know what? I had a typo. That was an 18. And we'll see how that affects all of the math. You can just go up there and type 18.

[00:04:27]And it'll update automatically.

[00:04:30]Uh what do we have here? Let's do one-sided test, and we have a P value of 0.09. Still not quite significant enough for us, right? But, we have a null hypothesis mu equals zero.

[00:04:40]Um we can do left, which wouldn't make sense here, right? But, we have right.

[00:04:44]But, remember if it your alternative hypothesis is not equal to, we would do both sides. But, here's how you can do a T test in Desmos. Now, what if you have two samples? Let's go look and see what that looks like. We would go back to T test.

[00:04:58]And we have two samples. Let's go and do stats this time where they actually give you the sample size.

[00:05:04]We'll say 25, the mean is 50, and the standard deviation is 10.

[00:05:09]And then your next sample, if you have two samples, we'll say we have a sample of 30.

[00:05:16]The mean is 57, and the standard deviation is five. You create a test.

[00:05:22]And when you do that, we're going to look down at it. We again have a confidence interval. And remember, this is looking for the difference in the two samples. Oh, it looks negative there.

[00:05:31]So, that gives us some evidence that one sample is smaller than the other, right?

[00:05:36]And here's our significance test, where we have T is equal to -3.184.

[00:05:42]I actually, if I were going to do this again, I'd be a little more mindful, cuz I I keeping everything positive. So, maybe that would make my second sample my first sample and my first sample my second sample, but it doesn't matter.

[00:05:53]We're going to look at this right here.

[00:05:55]Our P value is.002, so we would reject the null. If we want to take this down, it'll tell you the degrees of freedom, the P value, the T score, all of that is fantastic. Here's our curve that goes with it. So, that is how you can do a T test inside Desmos. So, let's clear this up again and then we will look at doing a chi-square test now. The first one we'll do is the test for independence.

[00:06:19]Wow, look at that. You get a little table here. So, maybe we get three, four, and then five. Remember, all your counts have to be greater than five. I'm thinking about that. This side is like 34. Maybe put 22 and I will put 12 and we'll see when we do our test. Notice it doesn't do the checks for us. So, we're going to look at our expected values here. Notice we have a couple expected values that are less than five, so that's not good. That would be a problem right here, but you can click this box and look for contributions. That means which one of these cells is contributing most to the chi-square statistic. And if you want to see the chi-square statistic, it's down here. Notice our P value is really, really high here. It's not that significant. And one of the reasons why is look at that. Those are so small. Our expected values are so tiny. But that is a chi-square test uh for independence.

[00:07:07]Let's move now on to our proportions. We have a one proportion Z test and a two proportion Z test. In Desmos, you have to make sure that you have whole numbers here. So, maybe you go out and collect a sample and 12 out of 50 uh you know, whatever you're getting a sample of happens to work.

[00:07:24]You collect your sample of 50 and 12 of them happen.

[00:07:27]And you look at your confidence uh interval right here and you have a point estimate of.24, which is 12 out of 50. And you notice your confidence interval is.122 to.358. Your confidence level is.95, which means if we're going to interpret this that we're 95% confident that the true proportion of whatever we're measuring right here lies between.12 and.35, which means if we were to collect a lot of samples and make or make confidence intervals, 95% of those intervals would contain the true parameter, whatever we're looking for there. But maybe we do a test, right?

[00:08:05]And so we're doing tests here. We have Z equal to -3.677, so obviously we're working on the left there. Our P value is really small. Your null hypothesis, if we're testing one, is.5, but maybe for some reason you're checking.4.

[00:08:21]And when we update that, you notice that your Z score will update.

[00:08:26]And then we get -2.309.

[00:08:29]All right, so our P value is still kind of low, though, right? But that's how you do a one proportion Z test. If you're going to do a two proportion Z test, it's very similar with inference.

[00:08:40]We're just going to go here and you type in your first sample, and then you type in your second sample, and it'll do the test for you. But I think that's everything we need to know right now for your test. This is how you can do tests with Desmos, you know, for the AP stat exam. I hope you're all studying tonight and doing your best. This is Mr. Keller, remember.

[00:08:59]Good luck to you all. It's nice to be important, but it's more important to be nice. See you.