- 00:00
[Finding Z Values with the Normal Distribution, Part 1}

- 00:01
SPEAKER: Hello, welcome to this lesson of mastering statistics.Here we're going to learn how to find z-valueswith the normal curve, when we'regiven the area or the probability involved.And so what we're going to do in this sectionis kind of like what we did before.We're going to gain some practice with justhow to use the chart here.And then we'll get to some problems

- 00:21
SPEAKER [continued]: where you'll see where it's practical.So just like before, we first started outby learning how to use the chart,learning how to find the z-values, and go into the chartand get the probabilities, and then welearned how that was useful with actual problems.Same thing's going to happen here.So we're basically doing the reverse.You know, usually we're given a problem.

- 00:42
SPEAKER [continued]: We find the z-values.We calculate the probability.Usually in statistics, we want to knowthe probability of things.Here we're going backwards.I'm going to give you the area under the curve,and you're going to return to me the value of z thatmakes sense for that.And it's going to depend a little biton the type of problem as well.So for this particular one right here--what z-value-- and you have to read these very carefully

- 01:06
SPEAKER [continued]: to make sure you don't make a mistake-- what z-valuehas an area-- or you can think of it as probability,because that's also the same thing-- of 0.0038 to its left.All right.There's two pieces of information that's important.

- 01:26
SPEAKER [continued]: First one tells us is-- it has an area point 0.0038,and also it's to its left.Now if you remember, the chart in the back of whatever bookyou're using-- or what we're certainly using in this class--the chart of z-values-- gives youthe area of the probability to the left handside of the z-value.So this particular problem lines up exactly

- 01:48
SPEAKER [continued]: with what your book is going to give you in that table.It gives you the area of the probabilityto the left of the z-value.So really what you have to do hereis just look this value up in the chart,and just, kind of, go backwards and find out what value of zis there.So if you think about it, here's a probability distribution.What we're saying is that there is some value of z

- 02:09
SPEAKER [continued]: here-- I don't know exactly where it is-- but thereis an area out there that to its left of whatever z-valuethis is, the area is 0.0038.Basically, we're going backwards.So the area here, the shaded area here, is 0.0038.We want to find what value of z that is,

- 02:30
SPEAKER [continued]: and since it's all defined to be the left, toward the leftthere, we can just look it up.So go in your table and look for 0.0038--it might take a little while to find it,but eventually you're going to find an exact value for 0.0038.And then that-- when you find that guy,you can read off of the column and off of the top row there--

- 02:53
SPEAKER [continued]: you can read the value of z.So the corresponding value of z for this problemis negative 2.67.Now, you might say, why do we care about this?There are going to be types of problemswe're going to work very soon where knowing thisis going to help us solve the problem.So for now, we're just learning howto use the table to go look for a value

- 03:13
SPEAKER [continued]: and return the value of z.So as another problem-- what z-value hasan area of 0.0212 to its left?Again, it's to its left, which is exactly how this chart is

- 03:38
SPEAKER [continued]: situated.So, again, what it's really asking is-- is it's saying,here's a normal distribution, right?And somewhere here is a value of z--I don't know what it is right now--but the area to the left of that z-value is 0.0212.And so we have to work backwards.

- 03:60
SPEAKER [continued]: So go grab the chart.Go in there and see if you can find an area or a probability0.0212.Scan around.You'll see they're ascending and descending.Find that exact value, and you'll actuallyfind a value for 0.0212, and thenread the corresponding borders of the table to figure outwhat value of z that is.

- 04:20
SPEAKER [continued]: And again, in this particular case,since we have exact values, and the table gives us whatwe need, we get negative 2.03.That's the value of z that correspondsto this area or this probability.All right.So finally, let me show you somethinga little bit different.It's basically the same thing, but we're

- 04:41
SPEAKER [continued]: going to do a little bit more thinking.What z-value has an area of 0.9706 to its right.

- 05:02
SPEAKER [continued]: Now you notice immediately that there's a difference here,because it's right.Now you know that the table is not going to give you back--it's not set up to give areas to the right of anything.So you're going to have to do some thinking here.And that's why we're doing these different kinds of problemsto kind of show you how to kind of wrestle with it.The first thing to do is draw a picture,because that will instantly help you visualize

- 05:23
SPEAKER [continued]: what you're supposed to do.So if you have a normal distribution here-- first thingto do is realize what it is we're really trying to do.All right, so somewhere there is a z-value--I don't know where it is, but I'mgoing to draw it here for right now.Let's say z is equal to question mark.Such that the area to the right of itis a big number, almost 1.

- 05:43
SPEAKER [continued]: That's why I'm drawing it way over here,because I know that it's going to be a large area veryclose to 1.And so the area is 0.9706, right?So what I need to do is find this value of z,but because it's giving me the area to the right,I cannot just look this up in the chart.If I look 0.9706 in the chart, and get the answer,

- 06:07
SPEAKER [continued]: and circle it, it's wrong, because thatwould be giving me-- the chart is set upto give you areas to the left.And so since this is an area to the right,it's not going to work right.But what you need to realize though isthat since the entire area under this curveis 1, if I take 1 minus this area, 0.9706,

- 06:28
SPEAKER [continued]: I will get an area of 0.2-- I'm sorry-- 0.0294.So what does this physically represent?If this shaded area is 0.9706, then 1 minusthat means that this shaded area-- here I'lluse a different color-- this area here that we're

- 06:49
SPEAKER [continued]: interested in is this value here.So 0.0294.So see all we care about is finding out what the value of zis here.That's all I care.If I can find that, whatever that value of z is,I've done the problem.No matter how I can get there, I get there.So I take 1 minus this guy.That gives me this shaded area.

- 07:10
SPEAKER [continued]: Now I can go look this number up in the chart,because the chart is set up to give me the areato the left of the value of z.So if I go to the chart, and look that up, 0.0294, that I'mgoing to get an exact value.I can find 0.0294 in the chart, and I'mgoing to find that that correspondsto z-value of negative 1.89.

- 07:32
SPEAKER [continued]: And that is the answer.So you see you kind of have to go backwards a little bithere, because since it's the area to the right, I shade it.That clues me in that I can find this by taking 1 minus that,and then this problem becomes exactly what we've done here,where we're just looking in the table, finding that value of z,so the area to the left of it is this guy right there.

- 07:53
SPEAKER [continued]: Now, again, you have to trust me a little bit.These types of things are going to be usefulhere in just a minute whenever I show youhow to solve some problems that involve needing to beable to go backwards like this.The other thing I'll mention to you is in all threeof these problems-- see here we had to look up a value of 0.294in the chart, in the area of the chart, probability,and we found an exact match.

- 08:14
SPEAKER [continued]: Here we had to look this number up, whichwas also an exact match.Here we had to look 0.0038.So these are all carefully chosen hereto be exact matches in your table.You should be able to find a number that matches thison the board in your table.99% of the time your table is going to match mine.

- 08:34
SPEAKER [continued]: You're going to be able to find this.But eventually you'll hit a problemwhere it won't be-- if you look at your chart carefully,you'll see that you can't find every single number.What if I change this to 0.0213 or 0.0219?Or what if I change it to 0.211?Some of those values may not be in the chart.So the way that we're going to do it-- you

- 08:55
SPEAKER [continued]: have to read your own book and your own professor-- all right.But the way we're going to do it hereis, if it's not exact, if you can'tfind an exact match for what you need,you just pick the closest value.All right?And if the exact value you need is exactlyin the middle of two values, then youcan average the z-scores on either side to get the value.So basically, you kind of do a little bit of interpolation.

- 09:16
SPEAKER [continued]: If the value you need-- in other words, just to kind of clarifyhere-- let's say we need this value--I don't care what it is-- I need this value.But it's not in the chart.But there is a number to the left and a number to the right.All right.Then what I'm going to do, since this one is not in the chart,is I'm just going to choose the one that's closer to the one,

- 09:39
SPEAKER [continued]: and I'm going to use the z-score associated with that.All right.And if this one that I need is exactlyin between two adjacent ones, then I'lljust average the z-scores to get an exact value of the z-score.That's the easy way to do it.Your other professor may have a little more complicated wayto find the z-score, if he wants to be more exact.But choosing the one that's closest gets you pretty close

- 09:60
SPEAKER [continued]: most of the time.So follow me on to the next section where we'llcontinue building these skills.And you'll find out that it's not hard at all.

### Video Info

**Series Name:** Mastering Statistics, Vol 2

**Episode:** 17

**Publisher:** Math Tutor DVD

**Publication Year:** 2013

**Video Type:**Tutorial

**Methods:** Normal distribution, Standard deviations, Probability

**Keywords:** mathematical concepts; mathematics

### Segment Info

**Segment Num.:** 1

**Persons Discussed:**

**Events Discussed:**

**Keywords:**

## Abstract

Instructor Jason Gibson discusses Z-value calculation and uses tips, shortcuts and technique to guide students through difficulties. His demonstration tackles challenge areas and shows when shortcuts can help approximate answers.