 Good morning, good day, and good afternoon. This is Monica Wahee, your library college lecturer, here moving you through chapters 7.2 and 7.3 z scores and probabilities. I decided to mush these two chapters together because I thought they actually kind of belong together. I didn't really understand why they were separated. So at the end of this lecture, you should be able to explain how to convert an x to a z score, show how to look up a z score in a z table, explain how to find the probability of an x falling between two values on a normal distribution, describe how to use the z table to look up a z corresponding to a percentage, and describe how to use the formula to calculate x from a z score. Well, that sounds like a lot, but you'll understand it at the end of this lecture. First, I'm going to go over what a z score is and what the standard normal distribution is. Then I'm going to talk about z score probabilities and what those are. I'm going to show you how to use the z table to answer some harder questions besides the ones I talk about during the z score probabilities section. Then I'm going to show you how to use a slightly different formula to calculate x from z. Finally, I'm going to just remind you some tips and tricks about using z scores and probabilities correctly. So all this talk about z scores. So what is a z score? And what is the standard normal distribution? Well, let's take a look at this very pretty thing I made. You may recognize it from the last lecture. It was my little empirical rule diagram. So remember the empirical rule. Remember how it required a normal distribution. Well, that worked well for the cut points available, right? Like mu, mu plus or minus one standard deviation, mu plus or minus two standard deviations. If we ask questions that were right on those cut points, we had good answers. But what about in between those cut points? So I wanted you to notice in this empirical rule diagram, these numbers at the bottom, like I just circled them like negative three, negative two, negative one, and then mu doesn't have a number. So pretend there's a zero there. And then there's one, two and three. Okay. That is the standard normal distribution. And that is also called Z. So these things on the right, those are Z scores. So see the green area, zero is the Z score that's on the lower limit of that. And one is the Z score at the upper limit of the green area. So you can see that this whole curve, the standard normal distribution on the right, whole the me with the whole curve is zero in the standard deviation of the whole curve is one. And that is what Z score is. So I just want you to notice the concept of standard. I'm in the US and in the US, we use, you know, the US dollar. But one of the things I've noticed is that a lot of countries see it as a standard. So they'll map their currency to the US dollar. So maybe the Euro will map its currency to the US dollar. Maybe the Egyptian pound will also map its currency to the US dollar. And once it does that, it's a lot easier to compare them, right? And so that's the main reason for the standard normal distribution is it helps you compare X's from different distributions, different normal distributions that have different mutes and different standard deviations from each other. It helps you map them to this normal standard normal distribution here that's standard. So you can compare them. So let's talk about Z scores. Every value on a normal distribution, so every X can be converted to a Z score. Just like I was saying how you can convert any currency to dollars, there's some formula for that. So you can convert every X on a normal distribution to a Z score. But you have to know how to use the formula, right? And what goes into that formula? Well, first you need the X that you want to convert to a Z score. So you need to pick one. Then you need to know the mu of your distribution, your normal distribution, and the standard deviation of your distribution. And here are the two formulas that are used. The one I was just talking about is on the left, is the formula for calculating the Z score. And we'll go over the one on the right later in this lecture. So remember in the last lecture I was talking about a class that had 100 people in it and that all took a really hard test. It was so hard nobody got 100%. And it was a 100 point test. So nobody got 100. The top score was in the 90s. So and remember in the upper right, there was, there's the mu. The mu was 65.5, which is pretty bad score on a 100 point test. And the standard deviation was 14.5. So I'm going to give you an example of calculating a Z score on that particular distribution. So let's say you got a friend, you have a smart friend. And that smart friend got a 90 in the face of all this. Well, let's calculate the Z score for 90 on this particular distribution. Okay, so here's what we're going to do is first we're going to remind ourselves, you don't have to do this in real life when you're doing it, but I'm just doing this for demonstration purposes, is what our empirical rule stuff look like. Remember at mu plus one standard deviation was 80. And mu plus two standard deviations was 94.5. So already, you know, whatever your answer is going to be for 90 is it's going to be between one and two, right? But we just don't know exactly what it's going to be. So I'm just showing you this for demonstration purposes to relate it to the last lecture, but you don't have to do this in real life when you calculate it. Okay, so we know that the Z we're going to calculate is going to be somewhere between one and two. And as you'll see on the slide here, I labeled over on the Z curve, I labeled where Z equals zero, which is the mu that 65.5. So we're going to anticipate we're going to get a z score that somewhere between one and two. And you'll see in blue, I listed the ingredients, right? So we have the smart friend score 90. We have the mu 65.5. And we have standard deviation 14.5. And then we have our Z formula. So let's do it. Okay, so x minus mu is going to be 90, which is our x minus 65.5. You do that out first, and then you divide it by 14.5. And look, our Z score is 1.69. And that's exactly where we thought it would be, it would be somewhere between one and two. And so as you can see, you can take any x and convert it to Z. Here, we'll do another example, only this friend is not so smart. This friend actually got a score that was kind of low, it was so low, it was below the mu of 65.5, this poor friend only got a 50. So let's try it again, let's do a Z score for 50. So again, you know, this is just for demonstration purposes. But remember, in empirical rule land, 51 was at mu minus one standard deviation. So we're going to expect that between again, negative one and negative two, Z is where our 50 x is going to land if we calculate the Z score. And so here we are, we calculate the Z score, we have 50 minus 65.5 divided by 14.5 and we get negative 1.07. And the reason why it's negative is, as you can see, it's on the left of the mu, so then the Z score is going to be negative. And so as you can see, it's exactly where we thought it would be, it would be a little bit to the left of negative one. So now we're going to get into something that's a little bit harder, which is the Z score probabilities, you're feeling pretty good about the Z score. But now let's talk about the probabilities. Okay, so remember the probability from the empirical rule, this is just old empirical rule stuff. So remember, I gave you a question at the end of that lecture, I said, what is the probability I will select a student with a score between 36.5 and 51. And remember, the answer was like this orange area, which is 13.5%. But what if you have Z scores like 1.69, the smart friend, and negative 1.07, which are the not so smart friend, you know, in other words, you have X's of 90 and 50, which are not on the empirical rule. How do you figure out the percent or the probability? That's the next step with your Z scores. Okay, so now let's ask this question. Let's say what is the probability that students scored above the smart friend? Now we could also ask for below, but I'm just choosing to ask for above this time. So in other words, what is the area under the curve from Z equals 1.69 all the way up. So see, like a little ways through that blue wedge, we wish we knew the area for everything up from 1.69 Z through the purple area through the little black thing at the top, we wish we knew that area. We only know from the empirical rule what's on the cut points of like one and two, but we don't know this in in between things. So how do we figure that out? Well, this is another problem here. What is the probability that students scored below the not so smart friend, right? And in that case, see the diagram, we'd have to figure out what is the part of the orange that that friend gets plus the red and plus a little black part at the bottom. What is the percent or the proportion of the curve that represents that. So that's what we're getting into now. And that's what we do is we look these up in a Z table. So what the Z table is is basically they figured out every single Z score you could have between negative 3.49 and I'll go into why negative 3.49 between negative 3.49 and positive 3.49. And they went like every 100th. So they figured out for every single one of those Z scores, what the probability is, and they actually fit that all on a table. And so now what I'm going to show you how to do is how to use that table to look up the probabilities. And by the way, if you look up a probability that happens to be on one of those empirical rule cut points, you'll get what the empirical rule says. It's just that the empirical rule is nice because you don't have to pull out the table. But if you have something that's not on the empirical rule cut points, get out your Z table. So how do you use the Z table? Well, the first thing is you want to figure out what area you want, right? So we're going to start and do the not so smart friend, because that's a little bit easier actually to demonstrate. Okay, so what is the probability that students scored below the not so smart friend? So which is a secret way of saying, what is the area under the curve that makes up most of that orange part, all of the red and the little black part at the bottom? What is that proportion? And so for areas left of specified Z value, you're supposed to use the table directly. So I'm going to show you how to use that table to look up negative 1.07. And then I'm going to come back and tell you what they mean by use it directly. Hi there. So here we are at the Z table. And if you have the book, you can look it up in the appendix in on page eight. But there's also a lot of Z tables on the internet. Sometimes they're arranged a little differently. So I'm using this one because it's from the book. So remember the Z that we're looking up, we're looking up the Z of negative 1.07. So remember, I said they had to somehow calculate all the different probabilities for every single Z between negative 3.49 through positive 3.49, every hundredth, they had to come up with that. Well, how did they fit it all on their table? Well, this is what they did. See, this is the beginning of the table. Remember, I said negative 3.49. Well, this is negative 3.4. And then to find the Z and negative 3.49, you have to imagine that the nine is here, but it's going to be the last one here. So see this nine here, this is what it would be. So just for pretend, if we had a Z score of negative 2.58, I go 2.5 and then I have to go over to the eight right here. Okay, or if I had one that was negative 2.10, right, or negative just plain 2.1, right, then I'd go over just one to this zero line. And see, these these little tiny things in here, those are all probabilities. In fact, let's go look up our probability, which is negative 1.07. So we're going to go down here, negative, here we are at negative 1.0. And then we have to go over to the seven column, right? So what's the cell, here's a cell and three from the left, I guess I could guess that. So we have negative 1.0987. So this is 0.1423. Otherwise known as 14.23%. So that's actually what you get out of the Z table. That's the probability that's the percent you're looking for. And just in case you're wondering, these aren't all negative, the first page is negative. The second page is positive is all the positive C scores all the way up to 3.49. But what I want you to hold in your head is what we just looked up, which was negative 1.07, which is 0.1423. Okay, hold that thought. Okay, here we are back at our slides. And so look at that green part where it says for areas to the left of a specified Z value, which we're doing with the not so smart friend, use the table entry directly. So here was our table entry, it was 0.1423. So we're just going to use that number that we found. And we're going to say the probability then is 14.23%. And that kind of makes logical sense knowing the empirical rule. Now I'm going to show you an example of what why I was saying use it directly. In this next example, we're going to look at the smart friends probability. In fact, we're going to ask what is the probability that the students scored above the smart friend and the smart friends at z equals 1.69. So I'm going to demonstrate now for areas to the right of a specified Z value, you either look them up in the table, then subtract result from one, or you use the opposite Z, which is in this case would be negative 1.69. And you'll get the same answer, whether you do it the first way or the second way, but I'm going to demonstrate both. Okay. So first, I'm going to demonstrate what happens when you look up the probability in the table for that Z, and then you subtract that probability from one. So let's go look up Z equals 1.69. All right, here we are back at our Z table, only this time we're looking up a positive Z. So we don't want this first one, we want the second one. So remember, we're looking up Z equals 1.69. So we're looking under here for 1.6. And that's right here. And now we have to go over to the nine column. So that's going to be 0.9545. So hold that thought 0.9545. Okay, we're back with our probability that we looked up in the Z table. Now remember, we were supposed to look it up in the table and subtract the result from one. So that's what we're going to do now. So we found 0.9545 in the table, we're going to take one minus 0.9545 and we get 0.0455 or 4.55%. This is a little tiny piece, which kind of makes sense, because it's right at the top of the distribution just a little piece of the blue in the purple and then a little black at the top. All right. And so what you want to imagine is that 0.9545, which is like 95.45%. That's the whole piece below Z equals 1.69. That's most of the blue, the green, the yellow, the orange, the red, and the little black at the bottom. That's all in the 0.9545. Okay, so again, we were looking up in the area to the right of the specified Z value, and I showed you the first way of doing it. There's another way of doing it. And that's where you just use the opposite Z from the get go. So we're going to now use the opposite Z, we're going to look up negative 1.69. All right, here we are back at the Z table, only this time we're looking up negative 1.69. So negative 1.6 is the first thing we need to find in this column. So here we are negative 1.6. And then we know nine is the last column, I'm learning that. So we'll go over here. And so that that looks familiar, right? 0.0455. Okay, hold that five. All right, we'll back. And so as you know, if you look it up in the table directly, like the 1.69 directly, and you take that probability, and you subtract it from one, which is what we did last, we got the same answer we got now, right? 0.0455 or 4.55%. So it is kind of more efficient to just use the opposite Z. If you're looking for areas to the right of the specified Z value. But I always say, when you're done looking it up, compare it to the picture. And I always say draw a picture too. You know, I don't mind if you have normal curves, drawn over all of your homework, or all over the wall, I guess, or maybe a whiteboard, that's probably more efficient. But it's best to draw it out, label on there where your Z and your X are, and then just look at it. Because we know that the little piece above Z equals 1.69 is not 95% of that curve, it's just not it that's over 50%. And we can tell that little tiny piece is under 50%. So if you accidentally do the first way and forget to subtract from one, you know, maybe if you check it against your normal curve drawing, you'll realize, Oh, I made a mistake. So even though there's two different ways to find the probability, if it's to the right of the Z value, just try to make sure no matter how which ways you use, that you finally do a reality check against the drawing you make, just to make sure you got the right piece, because there's only two pieces, there's a big piece and a little piece of the skirt. And we got 4.55%. We know that's a little piece. And we know from our drawing that we were looking for the little piece. So that's how you do your reality check. Okay, you thought that there weren't any harder questions. Well, here are some harder questions. So this is a little bit more on probabilities in the Z table. So here's another question we haven't handled yet. What if you were looking at a probability between two scores, such as the probability the students will score between 50 and 90. So it's somewhere in the middle. Okay. Note that in that case, when you have a between one, you actually have two axes, and we'll label them x one and x two. So the not so smart friend is going to be x one, and the smarter friends going to be x two, just to keep these axes straight. Okay. So the next step is you're going to calculate z one and z two. And I'm kind of cheating because we already did these we already knew the z one for the not so smart friend was negative 1.07. And we already knew the z two for the smarter friend was 1.69. So I just put them on the diagram. Okay, and then I'll here's this beginning of the strategy, and I'll just explain the strategy, and then I'll do the strategy. So for z one, you find the probability to the left of the Z. So you find the little piece to the left. And remember, you can take the direct probability from the Z table. So that's what direct means is you just get to copy it directly out of the Z table. Then for z two, you find the probability to the right or above. So you find the little piece there. And you use one of those two methods I showed you, which we did together. And then finally, imagine like the whole curve, you're subtracting the piece at the bottom, the z one probability, and you're subtracting the piece at the top. So you're trimming with those two pieces to get the between probability. So that's the strategy is basically you find out the the size, the probability of each of the little pieces on the sides, you should track both of those from one, and that traps whatever's left in the middle. So I'll demonstrate this. So remember for z one, the probability to the left of z one was point 1423, we did that together. And then we use both of those methods. And they got the same answer to find the probability to the right of z two, which was point 0455. Okay, so that's a little piece at the top, and then we got the little piece at the bottom. And now we'll take one minus the piece at the bottom, minus the piece at the top and the total is 0.8122, or 81.22%, which kind of makes sense. That's a big piece in the middle. So it wouldn't be surprising if it was about 80% of the curve. So this is how you do a between one. Here's another question I haven't really handled. What if you were looking at a probability more than 50%. So such as the probability that students will score greater than 50, right, like like the big side. Okay, well, actually, you just do what you normally would do you say for areas to the right of the specified z value, either look up in the table and subtract the result from one, or use the opposite z, which in this case would be 1.07. So if we did method one, we end up going one minus 0.1423, which we already looked at. And we get 0.8577. We use method two, we'd take the z of 1.07, not negative 1.07, but 1.07. And we could go look it up in the z table, and we get 0.8577 again 85.77%. So it's this isn't actually a harder question. I just wanted to show you how it works when you're getting like a bigger piece, a bigger than 50% piece of the distribution. And here's another sort of similar example where we're looking at the probability that students will score less than 90. Okay. So that's easy, right for areas to the left of the specified z value, just use the table directly. So when we went and looked up z equals 1.69, we got 0.9545. So that's the answer. It's 95.45% of the curve is below z equals 1.69 or below x equals 90. So as I mentioned before, but I'll just mention again, you're supposed to treat all probabilities to the left of z equals negative 3.49 as P equals zero. So I showed you what negative 3.49 looks like in the z table, it's like 0.02. Well, there's not much smaller than that. So just if you actually calculate a z and you get like negative four, just say the P is zero. Okay, then the second thing is treat all areas or probabilities to the right of z equals 3.49 as P equals one or 100%. So as you couldn't imagine, you know, 3.49, that's at the top of the curve. So if you calculate a z and you got like a five, you can just assume that's 100% right or one. Okay, so we've gone through how to calculate z. And we've talked about looking up probabilities in the z table, and we've even talked about manipulating those probabilities to get certain probabilities. But we haven't talked about calculating x when C is given. So sometimes you're actually given a z, and you are have to calculate the x back from the z. In fact, sometimes it's even harder. Sometimes you're given a probability. And the probability is not a z, but you can use the probability. Remember, that's those little percents in the middle of the table. You can go find it in the middle of the table and look up the z that keys to it, and then put it into this equation. And so I'm going to just give you examples of some real life questions that you might see, like on a homework or on a test, probably not in real real life, that where you need to calculate x and you need to use that formula in the red circle. So let's say I was just bored, and I was wondering, what is the score, the test score on the distribution that is at z equals 1.5? Okay, so see where z equals 1.5? We never asked that question before. So let's say I just out of curiosity wanted to know what would the test score be of a student who was at z equals 1.5? So what I would do is I would take 1.5 times 14.5, because that's what the formula says, it's z times the standard deviation. And then I do that first because of order of operation. And then after doing that, I'd add the mu, which is 65.5. And I get 87.3. So the x, the student who got 87.3, that student got a score that's at z equals 1.5. Now, as you probably imagine, people don't go around asking so much about, well, I wonder what that person score is at z equals negative 2.3 or whatever, they don't usually phrase it like that. Usually, you see more like a question like this, which is what is the score that marks the top 7% of scores. Now that's a secret way of saying, we are looking for the z at p equals 0.7 Oh, so it's like we turn that 7% backwards into probability, and we say, we're actually looking for the z at p equals 0.7 Oh, so how do you do that? Well, I'm going to show you. Okay, so we're on the hunt for probability 0.0700. Okay, so let's start at the top of the table here, you'll see we're digging around the middle of the table, right? And you'll see like 0.00. That's nowhere near the ballpark because we're looking for 0.7 Oh, so let's scroll up here, or scroll down actually. So now we're more and we're in the 0.04 neighborhood. Here's 0.06. Okay, we're getting close. Well, here we have a 0.0708. And that's 0.0008 more than we want it to be. Well, here next door, we have 0.0694. And that's only 0.0006 less than we want it to be right because if it had 0.006 more, it would be 0.0700. So this is technically closer than this one because this is 0.008 off. And this is only off by 0.0006. So we're going to choose 0.0694 as the probability of record for this for the top 7%. Only, we're not going to just choose this, we're going to figure out what is Z at that score. So what are we going to do? We're going to map back here, negative 1.4. And then we got to go all the way up, which we can guess is 8. So it's negative 1.48. So hold that thought. Okay, we started out looking for the Z at P equals 0.0700. And but the closest we got was 0.0694. And that map to Z equals negative 1.48. Now, what I want you to notice is negative 1.48 is actually on the left side of mu. Okay, so that is the Z score at the bottom 7% of the scores. So we're going to use the positive version of that Z since we want the top 7%. So we're going to use 1.48. So the opposite Z. And now we're going to plug it into the equation. So 1.48 times 14.5, which is the standard deviation, plus 65.5 equals 87. So now 87 is the score that marks the top 7% of the scores. I'm going to do another exercise for you that does the this time the bottom 3% of the scores, because this is often kind of challenging for students. So I'll just give you a second demonstration. So as you can imagine, we're going on the hunt now for Z at P equals 0.0300. So let's go over to the Z table. Alright, now we're getting a little good at this, right? So we're digging around in the middle and we're looking for 0.0300. Okay, and starting at the top, we're in the zero, zero department. Oh, here's 0.01 something, zero two. Okay, we're getting close to 0.0300. So Oh, here, point, 0.0301. Could you ask for anything closer? Totally perfect. Okay, so that's what we're going to use for Z is the Z at 0.0301. So let's look up that Z. So that Z is negative 1.8. And then we look up eight. So it's negative 1.88. Hold that thought. All right, well, we were on the hunt for P equals 0.0300. And we didn't find that. But we did find P equals 0.0301 in the table. And that mapped back to Z equals negative 1.88. Right. And now we go back to the question, we see that we want the bottom 3%. So we keep the negative. Now, if I'd asked about the top 3%, we'd lose the negative, we use 1.88 in the equation. But since we want the bottom 3%, we're going to keep the negative. Okay, so now let's do the equation. So x equals and then in the parentheses, negative 1.88 times 14.5, which is our standard deviation, then plus our mu, which is 65.5. And the score we get is 38.2. So 38.2 is the score that marks the bottom 3% of scores. And just be happy, your score is not in there. Okay, now here's another challenging, hard question. What if the question on the test or probably not in real life, but on a test says what scores mark the middle 20% of the data? And so I put a little arrows on there just to point out, well, when they say middle, they mean it's hugging the mu. It's actually assuming that there's gonna be 10% on the right side of the mu and 10% on the left side of the mu. And so how you start to do this is you figure out the Z score for one minus point two, which is the 20% divided by two, which equals four, right? So then after that, you know, because one minus point two is point eight, and point eight divided by two is point four. So we get this point four. So we go find the Z score at point four, which you get at using the Z table now. So so I'm, you know, looked around and I found point 4013 in that digging around in the middle of the Z table. And that map back to negative Z equals negative point 25, right? And so that is then what I would put on for the lower limit on that one. And then Z equals point 25, the positive version goes on the other side. So once you figured out both of the Zs, the Z on the left and the Z on the right, you just have to put them through the equation. So for the left side, we use the negative Z. And for the right side, we use the positive Z. And that's how we get our limits. So what scores mark the middle 20% of the data, 61.9 and 69.1. It's not weird how that worked out. But anyway, 61.9 and 69.1 mark the middle 20% of the data. I didn't totally didn't do that on purpose. It just worked out that way. All right, I can't believe you made it through all this, I'll bet your brain is ready to explode. So now is a good time to talk about just a little review just help you come down a little bit from this whole really intense lecture. Okay. So first, I'm going to do a little Z score quiz game show style stuff here, right? So if you ever get the question when you're on the test, and you're like, Oh, my gosh, where is X? Where is X? Well, if you can't find X, it's usually in the question. So usually the way these questions go is somebody like maybe me will put a meal in a standard deviation at the top of the question. And then there'll be like maybe five questions about that pertain to that meal in that standard deviation, but they ask about different axis. And when I would teach this class in person, you know, people will come running up to me in the middle of the test, which you probably shouldn't do. And they would say, Where's the axe? Where's the axe? You gave me, you know, these pieces of the equation, but I can't find the accent, I'd be like, walk in the question, look in the question, you know, because I don't want to give it away. And then they'd all run back to their seats and find it. So that's so if you're wondering, you're panicking, where is X? Look in the question, it's usually in the question. Okay, so let's say you find an X. And what do you do with an X? Okay, and you're stuck with an X, what do you will usually what you have to do is calculate a z score. So remember, if you've got an X, you probably have a mu in a standard deviation, you can calculate a z score on that. So if you're panicking on a test, and you have an X, a mu standard deviation, just for fun, calculate a z score and see if it gets you anywhere. Okay, well, let's say you have a z score, what do you do with the z score? Well, you always look it up, right? I mean, if you're if you're going this direction, if you're getting if you started with an X, and you get a z, you got to go to the z table with it. Okay, so that's your next step. So if you're doing all this work, calculate a z score, and then you're done, you're like, Oh, my gosh, what's my next step? Go look at the z table. Well, what if the question asks for an x, right? Well, remember, we have a whole formula for that. So use the x formula. So if there's no x anywhere, and it's asking for an x, then use the other formula, use the x formula. And what if the question gives you a p, or I just said p for probability, but it could be a percentage like remember the top 7% and the bottom 3%. Well, if they give you a percent, just start digging around in the middle of the z table, just start digging around looking for that percent. Because once you start digging around, you realize it maps back to a z. And then you can get into the groove of using the x formula and you'll probably get yourself out of this panic. So here are some final tips and tricks for getting z scores and probabilities right. And I've said this one before, draw a picture. And what do I mean by that graph out the question, draw the curve, draw the line for me, which goes in the middle, and where the x goes above or below the me, just start with that, it doesn't have to be to scale. But mainly, you want to get those elements in there. There's one x, shade the part of the curve wanted either above the x, or below the x, you know, just color it in so that you get an idea of do you want the big part, the one that's greater than 50%, or the little part, the one that's less than 50%. If there are two x's, then shade in the area wanted, which is usually in between them. If it's a calculate the x question, put where the z or the P is. So if it was like the top 7%, you could shade in the top little part of the curve. If it was the bottom 3%, you could shade in the bottom little part of the curve. So make this picture and do it at the beginning. Okay, then note that x is usually in the question, if you can't find x, and you're trying to do the z formula, and you're saying, okay, I'm trying to make a z score, that's what it asks for, I'm trying to find a probability, that's what it asks for. I'm looking the question and you'll probably find the accent there. A big problem that I see is people mistake little z's for P's. Now obviously, if you've got a z, that's like negative, you know, a P can't be negative, a probability can't be negative. So you won't make that mistake, even if it's like negative 0.25, right? You won't make that mistake. And if the z is bigger than one, you won't make that mistake. So if you see a z equals 2.5, you're like, obviously, that's not a probability. But when you have a little baby z score, that's between zero and one, like 0.023. It looks a lot like a P, but it's still a z. So a lot of times people get a little lazy, like they hate using the z table, and then they calculate the z score and it's really little so they don't look it up. Don't be fooled. You still have to look it up. So if you're calculating z, you need a little baby z like that, it's still a z. Still go look it up. Okay. Then finally, remember how step one was draw a picture and I went on and on about that. Step 99 or the last step before you're done with the question is check your logic against that picture. So if you shaded a big part of your picture, your probability should be bigger than 0.5 or 50%. If you shaded a little tiny part of your picture and you're getting like 0.95 something, you know that that's wrong. So please check your logic against the picture before you say that you're done with your question. Okay. So you made it through this long lecture about z and about probabilities. So I gave you an introduction to the standard normal curve into those two z score formulas. I showed you how to calculate z scores and how to look up probabilities. And I also showed you at the end how to calculate x if given a z score or a probability. Okay, and all I want to say is, unfortunately, those students, those pretend students on that distribution, they were another them got 100%. Okay, that's not the case in our class. A lot of times people get 100% on the quizzes. That's why I can't use your grades as examples. Okay. So good luck on the quiz.