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• 00:00

[Finding Probability using the Normal Distribution, Part 5]

• 00:01

JASON GIBSON: Hello.Welcome to this lesson on mastering statistics.We're going to wrap up this problem typewith using the normal distribution table,putting all of it together to solve these problems.And then we'll move on to the subsequent sections,where we'll work on some different concepts.So the problem here is in a city,the wages are normally distributedwith a mean of $700, and a standard deviation of$50.

• 00:26

JASON GIBSON [continued]: These problems are all made up, so when we say this,it's probably a weekly salary of $700, a standard deviationof$50.So that means most of the people in the cityare going to have an average value of $700 a week,but of course some may earn less than that per week,and some may earn greater than that.But most of the data will be within plus or minus$50 from

• 00:46

JASON GIBSON [continued]: that mean.In fact, about 68%, that's what a standard deviation is.About 68% will be about $50 plus or minus the mean.That's how you read that and interpret it.Then we want to know, what is the probability that a workermakes between$620 and $770?So, we know we're going to have to write down we know. • 01:08 JASON GIBSON [continued]: So the mean is$700, the standard deviation is $50.Then we want to go ahead and draw a picture.I'm always a very big advocate of that.So let's draw a picture when we can.And so what we will have is, this is our normal distributionhere, like this. • 01:29 JASON GIBSON [continued]: The average value is 700, and we're trying to find out,what is the probability that a worker makesbetween$620 and $770?So 620, right here, is going to be 80 away.Let's just call it right there, 620.And 770 is a little bit closer, it'sabout 70 the other direction. • 01:49 JASON GIBSON [continued]: So we'll call that 770.And basically, what we're trying to do is we're trying to find,because we're trying to find whatis the probability of having a wage between those values,we want to find that shaded area.So we know we're going to need to convertthis value to a Z-score, and this value to a Z-score.So Z is equal to x minus the mean • 02:10 JASON GIBSON [continued]: over the standard deviation.So 770 minus 700 over the standard deviation,which was 50, for this particular normal distribution.And when you get that, you get 1.40 for that Z-score.Now for the other Z-score-- again,as x minus mu over sigma, 620 minus 700, • 02:34 JASON GIBSON [continued]: that's the mean value, over the standard deviation of 50.And when you do that subtraction and do that division,you get negative 1.60.You can redraw this if you want, but ultimately,just know that this one corresponds to a Z-valueof 1.40, and this one corresponds to a Z-value • 02:55 JASON GIBSON [continued]: of negative 1.60.That helps you visualize what you're going to end up doing.So when you're trying to find the area between two items,you look up the chart for the first one, that gives youthis whole area, look up on the chart, the second one, thatgives you this whole area, subtract the two,and that's what you end up with.So the way you might see it written on the test is, • 03:16 JASON GIBSON [continued]: find the probability that the random variable x, meaninga random person we draw, has a salary less than 770,but also greater than 620.You read it from the inside out.Less than 770, greater than 620, which we have now convertedto our Z-score implementation.How are we going to find this? • 03:36 JASON GIBSON [continued]: We're going to find the probability that Z is less thanthis value here, and this is going to be 1.40.And we'll subtract from it the probability that Zis less than negative 1.6.So this gives me this giant area, • 03:56 JASON GIBSON [continued]: this gives me this little tail, and I subtract the two.So whenever I put this into place here,what I will get for this one, we look 1.40 up in the chart.And we get a probability of 0.9192.And then we put this one, the negative 1.6,and we get a value of 0.0548. • 04:22 JASON GIBSON [continued]: So when we subtract these guys, the probabilitythat we care about-- this one minus this one, 0.8644.So hopefully this is getting a little bit like clockwork.Draw your picture, find your Z-scores,figure out how to use the chart to get the area we care about.And our last problem of this type • 04:43 JASON GIBSON [continued]: says, what is the probability that a worker makesless than$620 or greater than \$780?So you can visualize that as beingin the tails of the distribution.But let's draw it, just for giggles,to make sure we understand everything.So here's a distribution here, like that.

• 05:03

JASON GIBSON [continued]: The mean value of this to 700.So we want to find out if it's less than 620.That's going to be 80 away, so this is 620.So from here, we want this tail of the distribution.And then 780 is the other side.That's also 80 away from the mean.So it's this guy right here.

• 05:25

JASON GIBSON [continued]: So what we're really trying to figure outis let's find the Z-score.We can find both of these Z-scores very easily.620 and 780.So what we're going to have to do-- and in fact,you can actually make it a little easier on yourself.You don't have to do this, but you can look at thisand say, well, look, this is very symmetrical.

• 05:47

JASON GIBSON [continued]: I'm trying to find this area plus this area.And this area should be exactly the same as this area becauseof the symmetry, because this value here is exactly80 away from the mean, and this valueis also 80 way from the mean.So basically, this shaded area is by definitionexactly the same as that one.So really, when you notice symmetry like that,

• 06:08

JASON GIBSON [continued]: you only have to look at half of it and just multiply it by 2.So let's find the Z-score just for this one.It's going to be 620 minus the mean, dividedby the standard deviation, which is 50.And when you take the subtraction, and you get that,you'll get negative 1.60.So really, what it boils down to is finding,

• 06:31

JASON GIBSON [continued]: what is the probability of Z being less than negative 1.6?Whatever this gives me, 1.60, whatever this gives me is goingto be this area.And then I'm going to multiply itby 2 to get the other area on the other side,because of the symmetry.So what I'm going to have is when

• 06:53

JASON GIBSON [continued]: I look at this up in the chart, negative 1.6.I'm going to get an answer of 0.0548.I'm multiplying by 2.So the probability, 0.1096, and that's the answer.Now of course, I could get a Z-score over hereand find this area.

• 07:13

JASON GIBSON [continued]: But what I'm going to find, it isgoing to equal the same thing.So I'll add it to itself.So when you notice symmetry like this,you can work with one half of the tail, find the answer,multiply it by 2.It can save you a little bit of time.So, make sure you understand all of these problems.All of these problems we've done with the normal distributionand calculating areas and probabilities in areasunder the curve-- make sure you know how to do them,

• 07:34

JASON GIBSON [continued]: work them all yourself.Well, in the next couple sectionswe'll tackle one of our final topicsin this volume of mastering statistics.Everything we're doing here's bedrock material.We're going to get into confidence intervalsand other things later, where this stuff is goingto be your basic, basic material that you'llhave to understand for everything that follows herein statistics.So make sure you understand how to do this,

• 07:56

JASON GIBSON [continued]: then follow me on to the next sectionwhere we'll learn a new skill in statistics.

### Video Info

Series Name: Mastering Statistics, Vol 2

Episode: 16

Publisher: Math Tutor DVD

Publication Year: 2013

Video Type:Tutorial

Methods: Normal distribution, Probability

Keywords:

### Segment Info

Segment Num.: 1

Persons Discussed:

Events Discussed:

Keywords:

## Abstract

Instructor Jason Gibson demonstrates how to calculate probability with z-values and a normal distribution. In this chapter, his example includes calculating wage ranges in a pair of problems.