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.
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 a number of 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. 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 3rd 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 you'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 0 and 1 for a standard normal, and you just move on. Right? So the last thing that you have to do is choose the tail of, basically the of the sort of function. And all that really means here is the area that this that the that you just entered in step 2, 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, the last little button, which is, we're just going to go into paste. Alright?
So 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. Alright?
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 an error that something is greater than some number. We have \( p \) that \( z \) is bigger than some number, that's 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, 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, right, 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 the \( z \) score is going to be a positive number, and it should be kind of small. That's kinda what I expect here. Okay?
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 work and any extra work here. Alright? So you're just going to plug in the 0.3409, and you're going to go ahead and just keep your \( \mu \) and \( \sigma \) as 0 and 1. 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 under 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 that is here when you look this up the \( z \) table is always assuming that you have areas from the left. Alright? 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 3rd 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 to 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, 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.
