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Welcome back, everyone. So in the last couple of videos, we've seen how to find probabilities from z scores and vice versa using our z table. We just look up the value that we're given to find the missing one. Now obviously, this means that we have to have our table with us, and this gets kind of tedious and a little bit more complicated when you start getting harder problems like finding areas between two z scores. So what I want to do in this video is show you that if you have the option of using a calculator on a homework or test or quiz or something like that, how to find probabilities from z scores using a TI-eighty four.

Alright? We're going to see that using our calculators, we're going to find probabilities much, much quicker. I'm going to show you the two functions we'll need to do that. Alright? Let's get out our calculators and we'll do some examples together.

Alright? Let's get started. So let's just go ahead and jump right into our first problem and I'll show you the steps. Basically, we're going to sketch a graph in order to represent each problem and then use a calculator to find the probability. Now we've seen how to calculate probabilities of these types of problems before, where z in this case, the probability that z is less than some value, which is negative 0.81.

We would just go to our table, and we would look up that value. And spoiler alert, it would be this one right over here, 0.0209. But we want to do this on a calculator, right? So first, let's go ahead and just sketch out the problem. And again, we've seen how to do this before.

I'm going to go ahead and draw a little graph like this. Remember, this is going to be our zero point down the middle, like this right over here. So a z score that's negative and slightly less than one is probably going to be around over here somewhere. Right? So this would be negative 0.81.

So we're going to just draw the line like this, and basically, this should just be the area to the left of that value. So it's going to be something like this. All right? So we've got these. This is going to be our highlighted area.

Actually, let me go ahead and just do that in green. So then how would we do this on a graphing calculator? All right? So what we're going to do is we're going to go over to our calculator. And actually, before we get started, I want you to go ahead and access the window settings because these are going to be the window settings that you should use.

Otherwise, if you do all these steps here, you may not actually get to see the graph because it might be out of focus. So I'm just going to go ahead and assume that you paused the video and plugged in these window settings, and we'll see why in just a second. So the first thing we're going to do is we're going to go ahead to our calculator and we're going to access the distribution function. So you're going to hit second and then a little button that says variables or vars. The thing on top of it should say distr, so distribution.

If you're going to go and do that, it's going to open up a menu with a couple of functions here. These are functions that will basically just spit out numbers on the main screen of the calculator. One of them is called normalcdf, which basically will just give you a number only. But if you go to the right tab, it's the draw tab, and it's actually going to use the calculator's graphing function and graphing tools to do this. I like using this because it gives you the graph, and it's much less likely that you'll make a mistake.

So go ahead and hit that shade normal, and also the plug the inputs are going to be exactly the same for both of the functions. So I'm going to use shade normal. Okay? So now we're just going to move on to step two, which is we're going to plug in the lower and upper bounds. Essentially, what this is doing is telling the graphing calculator where on the x axis, where on this axis over here to use for its calculation of area.

Alright? So, what happens is if you're using a if you're finding a probability that z is less than some number, like our problem here, then what you're going to do is your lower bound is going to be essentially negative infinity. Right? Your calculator can actually hit negative infinity, but it can hit some really, really high numbers, which is negative one times 10 to the ninety ninth power. This is what it could be the number that's there by default, so you just don't change it.

Then for the upper bound, you're just going to plug in whatever z score you're trying to find here. Right? So in this case, this is going to be our negative 0.81. Right? So let's go over to our calculator and actually do that.

So this is going to be negative oops. This is negative one. If you needed to plug this in again, you should do that. It's negative 1 e 99. And this is going to be negative 0.81.

Now remember, for normal standard normal distribution, mu is zero and sigma is one, so you just leave those alone. And then you just go ahead and hit the draw function. Essentially, what we get is we just get a nice pretty little graph right over here that shows exactly what our sketch graph looks like. It's pretty close. Right?

So this is essentially the graph that it's showing us. Right? So, again, if you plug this in and you kind of just compare your two graphs, it's going to make it a lot less likely that you'll make a mistake. Okay? The number that we're looking for is right over here.

The area button, the area number, which in this case is 0.2. And then we said this was going to be nine zero, but this is actually, five numbers, but it actually rounds to basically 0.209, which is exactly what we saw from our table. So if you compare those two numbers, you'll find that they're exactly the same. Okay? So that's it.

That's how to calculate the probabilities. All right? Now, what you're going to do here is when you're done and actually, after you've done the three steps over here, what you're going to do is you're going to go over to your calculator and you'll hit the Clear button to go back to the main screen. And then what you'll do is you'll have to go back into the menu, so second program, and then you can hit the Clear draw. This basically just wipes the graph so you can just start on the next problem.

So I always recommend that you do that. So alright. So let's move on to now our second problem over here because it's just going to work the exact same way. So here we've got a probability of the z score being between negative one and positive one. So let's go ahead and draw that on a graph.

What would this actually look like? Well, so now we're going to go over down here, and we're going to look at our z scores. So again, this is sort of like the middle over here. This would be like our dotted line for the mean or the center. And negative one would be something like over here, and then positive one would be something over here.

Now again, if we had to do this using a Z table, we'd have to figure out two Z scores, we'd have to subtract them and things like that, but I'm going to show you this is much easier by using our calculator. So we've got our two values over here, which basically means that the shaded region is going to be this one right over here. It's everything that is between these two numbers. So how do we do this? So let's go back to the steps.

You're going to go and go to your graphing calculator. Hit the second vars button. Then again, just go to your normal sorry. Go to the draw, shade normal. Now what we're going to do here is because we're finding a value or probability between two values, our inputs are going to be slightly different.

Instead of going from negative infinity all the way up to the value that we're looking for, basically what you're going to do here is you're going to plug your lower value as just obviously the one that's to the left and the upper value as the number that is to the right. Right? This is essentially saying to the calculator, only calculate the area between these two Z scores. Okay? So that's what we're going to do here.

So we're going to change our lower and upper bounds, and this is going to be just negative one, and you're going to set the upper one to be positive one. Again, your mu and sigma don't change them, and you're going to hit the draw. See how it basically just gives you the area and it leaves off everything to the left, which is really cool because it doesn't do the same thing that your z table does. Alright? So over here, we can see here that the area that it's telling us is going to be, is going to be this number right over here, which is going to be 0.682.

And we can go ahead and round that if we wanted to keep it up to four decimals, 0.6827. That is correct. So 0.6827. And, we've seen this number before, actually, when we've done this. So that's correct.

So now that we see how to use this function for different types of problems, let's go ahead and take a look at some practice. Thanks for watching.

Welcome back, everyone. So in the last couple of videos, we saw how to find a probability from a given z-score using a TI-84 graphing calculator. But just as we saw when we did this stuff by hand, oftentimes in problems, you'll be given the reverse. You'll be given a probability and asked to find a z-score. That's what I want to show you how to do in this video, and it basically comes down to one function in your calculator called the inverse norm function.

I'm going to walk you through the steps, and I'm going to assume that you have a TI-84 Plus CE calculator, the slightly more advanced version. I'll talk about how to do this with a regular TI-84. Most of the steps are the same, but there's one difference, and I'll talk about that later. Let's jump right into our problem here. So get out your calculators and follow along with me.

So we're going to sketch a graph to represent each problem, then we're going to use a calculator to find a z-score. So in the first part here, we have a probability that z is less than some unknown z-score is equal to 0.0853. Remember, that's an area, that's a probability that is not a z-score. So first things first, I like to just draw a little graph before I start plugging numbers into my calculator just to make sense of this, right?

So I've got my normal distribution. And what I've got here is I've got an area that is actually pretty small, 0.08. So what that tells me is if I'm coming in from the left, which is what this area represents, then basically, this is going to represent a small sliver of the normal distribution. So in other words, it's going to be an area something like this, right?

So I've got a z-score that's all the way out here, which means that this is going to have to be a negative number. So whatever I get out of my calculator, it should be a negative number because this number here represents the small little area that's to the left of that line. Alright? So let's go ahead and do that. How do we do that?

The first step is you're going to open up your calculator. You're going to hit the second vars button, and then you go down into the third option, which is called inverse normal. Alright? So it's going to bring up a menu that kind of looks like this, where you have to input some numbers, and that's the second step. You're going to enter the area or the probability that's just given to you straight out of the problem.

That's all there is to it. So in other words, we're just going to go ahead and plug in 0.0853. Now you're going to plug in your mu and sigma, which are just going to be zero and one for a standard normal, and you just move on. Right? So then the last thing that you have to do is choose the tail of, basically, the sort of function.

And all that really means here is the area that you just entered in step two, is that an area to the left, to the center between two values, or to the right? Your calculator actually can calculate all of these things depending on the areas that you give it, because sometimes you'll be given areas to the left, sometimes you'll be given areas to the right. So all you have to do is just choose the appropriate tail, and the calculator does the rest for you. Now, this is an area that we were told coming in from the left, so we just pick the left over here, and we're done. So then all you have to do is just go ahead and go down to the last little button, which is we're just going to go into Paste.

All right? You're going to hit the Paste button. It's going to spit out this thing, and you hit Enter, and it just gives you a z-score. So what we just found here is that the z-score that corresponds to this area is negative 1.37. If you look this up on a Z table, you would actually and look for this probability, you would get negative 1.37.

This is pretty consistent with what we found in our drawing. We saw that we were going to get a negative number, and that's exactly what we got. Alright? So, one thing I want to mention here, by the way, is if you have a regular TI-84 Plus calculator, what happens in this third step is that the TI-84 Plus, not the CE version, will always assume that the area that you're giving it is from the left.

So it always assumes here, that this area is going to be the number that you plug in is going to be an area from the left. Alright? Okay. So let's go ahead and move on to our second example, which is basically now where we have a probability that in error that something is greater than some number. We have p that z is bigger than some number, some unknown number we're trying to find, and it's 0.3409.

So let's go ahead and set up our little graph over here and, get started with that second problem. So I've got the probability, and this is the probability that this is an area to the right problem, right, because we have z that's bigger than some number, is going to be 0.3409. This is a number that's kind of close to, like, 0.5, which would be in the halfway point, but it's going to be slightly smaller. So in other words, it's going to be something that looks like this, and we're going to have to shade the area that goes all the way to the right. Right?

If I was all the way at the midpoint, this would be if I shaded all of this area, this would be exactly 0.5. So I have to kinda cut a little bit off so that this covers about 30% of the normal distribution. Alright? So my z-score is going to be a positive number, and it should be kind of small. That's kind of what I expect here.

Let's go to the steps again. We're just going to go ahead and clear. We could just go to second vars. We just go down to that third option.

And now you just plug in again, if you have a CE calculator, you just plug in the area that's given to you in the problem. You don't have to do any extra work here. Alright? So you're just going to plug in the point 0.3409, and you're going to go ahead and keep your mu and sigma as zero and one. Now for the tail, because, again, we're looking at an area that's to the right of some number, we're not going to do a left tail or a center tail.

You're going to have to change that option to the right. This basically says, hey, this area that I was given in this problem is to the right of whatever z-score I'm trying to find, so go ahead and calculate that for me. Alright? And then the calculator does everything else. So you're going to hit the paste button, and you're going to get a number which is 0.41, which is exactly what we would expect, a small positive number.

And, again, if you go ahead and look this up in a table, actually, if you look this up in a table, which you're not going to get is, you're not going to get 0.41 if you look at this number in the table. Remember that when you use a Z table, the number that is here when you look this up at the Z table is always assuming that you have areas from the left. And so this is what I want to talk about here. If you have a regular TI-84 Plus that's not a CE, what's going to happen is you won't get this third option in your calculator. And so what happens is it's always going to assume that the areas are coming from the left.

So basically what you have to do is one extra step, which is you have to convert this probability to a left probability, which is you're going to have to do 1 minus whatever this number is, 1.3409, and you would get something like 0.6591. This is the number that you would plug into the calculator, which basically represents the area that's all the way over here. This would be that area. And you'd have to figure out that z-score. Alright?

So it's a little bit complicated. It's a little bit different for a regular TI-84 Plus. But if you have a CE, you can just go ahead and always plug in whatever numbers are given to you right off the bat, and the calculator does everything. Alright? So that's it for this one, folks.

Let me know if you have any questions. Thanks for watching, and let's get some practice.