- 00:01
SPEAKER 1: Hello.Welcome to this lesson of Mastering Statistics.Here we're finally going to startto put together a lot of the information and the skillsthat you've learned in the last several sectionsand use them for something that relates to statisticsand actually being able to solve problems.I could have really just compressed a lot of that stuffand kind of just crammed it down.

- 00:22
SPEAKER 1 [continued]: But, you know, I think sometimes whenyou do that, the details of what you're doing get lost.So what I instead attempted to do was show you all the piecesthat we're going to use.You know, we introduced a normal distribution.Then we introduced the standard normal distributionbecause we want to use these tables to look things up.Then we introduced z-scores.And then we did some conversions to find the z values.

- 00:43
SPEAKER 1 [continued]: And then we showed you how to use the tableto find the area under these normal curves.And all the time, I kept telling you, guys, we'regoing to use this to solve real problems.You're going to need to find these areas.And you just kind of had to trust me,but you really didn't have any real problems to motivate thatup until now.So what we're going to do is solve problems nowwhere you're going to be using those skills over

- 01:04
SPEAKER 1 [continued]: and over again.So you, really, if you've practicedwhat we've done up until this point,you should be very comfortable with the normal distributionin theory.Now, we're going to apply it.So the problem that we're going to solve is on the board.We've got a part a and part b.And as we go through these sections here,we'll do more and more different types of statistical problems.So the problem says body temperatures are normally

- 01:27
SPEAKER 1 [continued]: distributed with a mean of 98.6 degrees Fahrenheit.That's a normal body temperature when you're not sick.And the standard deviation of 0.73 degrees Fahrenheit.Don't even read the problem yet.Let's read what's in black first because that might benews to you, you know, anyway.A lot of times we learn, hey, normal body temp

- 01:48
SPEAKER 1 [continued]: is 98.6 degrees Fahrenheit.But until you start studying statistics,you don't really think too much about that.Obviously, if there's hundreds of millions or billionsof people on this planet, we're not alwaysgoing to have 98.6 degrees body temperature all the time.Different people are built differently.We're all pretty close to the same,but sometimes your body temperature

- 02:09
SPEAKER 1 [continued]: might be a little bit above 98.6, even when you're healthy.And sometimes your body temp might be a little bitbelow 98.6, even when you're healthy.When we say that people have a body temp of 98.6,it's an average value.That's why we say the temps are normally distributedwith a mean of 98.6.And that means that people do have body temps lower

- 02:31
SPEAKER 1 [continued]: and higher than 98.6, and they follow a normal distribution.That's what this means by normally distributed.So when you see some data and it tells youit's normally distributed, you should immediatelythink of that bell shape, that normal distribution shape,where the mean value is right under that peak.That's where most people are going to have their body temp.You can go lower and higher, but the probability

- 02:52
SPEAKER 1 [continued]: of getting people with those temperatures on either extremefalls off dramatically because of the shapeof the distribution.It also gives us a standard deviation of 0.73.So that's less than a degree, so 0.73 degrees.That means that most of the data, about 68%of everybody that you survey or you take as a random sampling,

- 03:13
SPEAKER 1 [continued]: are going to fall within plus or minus onestandard deviation around the mean.That's what a standard deviation tells you.So just by looking at these two numbers,you should understand a lot about human physiology.You all learned that the mean value of their body temperatureis 98.6.And you learned that the vast majority, about 68%of everybody out there, are goingto be within 0.73 degrees of that mean value.

- 03:38
SPEAKER 1 [continued]: Now, you can have people with normal body temperatureseven farther away, but, of course,the probability of that happeningis-- gets lower and lower because of the shapeof that bell distribution.So the question we want to find outis given that the population of people in the worldhave this characteristics, with the mean and standard deviationof their body temperatures, what is

- 03:58
SPEAKER 1 [continued]: the probability of having a normal body temperatureless than 96.9?Let me ask you a question.How many people have you ever metthat had a normal body temperatureless than 96.9 degrees?Pretty rare, right?Normally we say 98.6 is average.You expect some people have lower and higher.But this guy, this is greater than one standard deviation

- 04:21
SPEAKER 1 [continued]: away from the mean.So we expect there to be not a very high probabilityof finding someone with a temperature lower than that.So what you need to have is your information there.So what we want to do is you want to write the mean down.That's how we symbolize the mean, which is 98.6.And the standard deviation is 0.73.

- 04:41
SPEAKER 1 [continued]: And these are both in degrees.Now, if you wanted to draw a picture of whatyou were seeing here, we'll draw a quick picture.Sometimes pictures can be very helpful.So we're not going to make it a very big picture,but we'll draw a picture of a normal distribution, somethinglike this.And as I've said many times, it'snot going to be perfectly symmetrical.Actually, that doesn't look too bad.

- 05:02
SPEAKER 1 [continued]: So what we're going to have here is the mean value here.Right in the middle is 98.6.This is the temperature.All right?So most people at the peak of that curvehave a body temperature like that.And when you look one standard deviation, plus or minusthe mean, the standard deviation is 0.73 degrees.

- 05:22
SPEAKER 1 [continued]: So right around this inflection point and rightaround this inflection point, nowthis should be perfectly symmetrical here.What you get here, this is 99.33 degrees.And this one is 97.87 degrees because thisis the mean plus one standard deviation, plus 0.73.

- 05:44
SPEAKER 1 [continued]: This is the mean minus one standard deviation.So basically you add 0.73 and you subtract 0.73.That's the standard deviation.Now, what are we asked to do?We want to find the probability that the bodytemp is less than 96.9.So 96.9 is actually less than a standard deviation.So let's put a red dot right here.And this is 96.9 degrees.

- 06:08
SPEAKER 1 [continued]: And we want to find the probability that the bodytemperature is less than that.So if we draw this guy up to the curve, the probabilityof finding a person with a temperature lessthan that, meaning 96.9 or anything lower,is the shaded area in this tail down here,from here on to negative infinity.Now doesn't this ring a little bit of a bell to you?

- 06:29
SPEAKER 1 [continued]: Because now we're reducing this problemto finding the probability, which means areaunder the normal distribution from this temperatureall the way to the left.That's exactly what our table shows usin the back of the book.We just need to convert it to a z-scoreso that we can use the table to find the answer.Now, you should know from our previous lessons that youcannot just look 96.9 degrees up in the table in the back

- 06:53
SPEAKER 1 [continued]: of your book and find the answer.You can't do that.This distribution is created for this problem with this data.And so we only have one table in the back of the book.And that's for a standard normal distribution.So in order to find this area, the value of this area here,we need to convert this temperature to a z-score, whichwould be negative z because this is a z of 0 right here,

- 07:14
SPEAKER 1 [continued]: right under the mean there.And so what we want to do is figure out what that is.So we want to convert this to z.Remember, the little equation is x minus the meanover the standard deviation.So what we have is the value we careabout is 96.9 minus the mean, whichis 98.6, divided by the standard deviation, which is 0.73.

- 07:37
SPEAKER 1 [continued]: So when we do the subtraction, and we divide by 0.73,we get negative 2.33, negative 2.33.So if we were to redraw this, which you don't have to do.I'm kind of doing some of this stuffto just help you visualize.But if we were to redraw this distribution, justto kind of make sure we're all on the same page,it's going to have exactly the same shape as what

- 07:59
SPEAKER 1 [continued]: we have here.Not a perfect picture, right?This is 0.That's what the standard distribution is.Negative 2.33 is going to be out here, negative 2.33as a value of z.This surface area here, all the way to the left,is exactly what we're trying to solve,which is the surface area here.Now, the graphs are little squished.They're a little different, but you've

- 08:19
SPEAKER 1 [continued]: got to use your imagination here and realizeI'm trying to draw by hand.So what we're going to do, since you'retrying to find the area to the left of z,that's exactly what the table gives you.You don't have to do any manipulation.You just read it straight out of the table.And so what we're going to write down is our final answer.The way you write it is the probability, so we say p of z

- 08:40
SPEAKER 1 [continued]: being less than negative 2.33 because that'sexactly what the table gives us is the area under the curvethere.When you look up negative 2.33 in the table, what you get backis 0.0099.And that is the probability, 0.0099.Now remember, probabilities go from 0 to 1, 0 being-- meaning

- 09:04
SPEAKER 1 [continued]: it will never happen, and 1 meaning itwill definitely happen 100% of the time.This is a very small number, close to 0.So it leads you to believe that the probability of havinga normal body temperature less than 96.9is exceedingly rare, exceedingly rare.All right?That's basically how we use statistics to draw conclusions.

- 09:26
SPEAKER 1 [continued]: We know some information about our population.We know the mean and standard deviation,and we know that it's normally distributed.Then we can start asking questions.What's the probability of a temperature beingbelow a certain value?Also, we can ask questions like this.What is the probability of havinga normal body temperature greater than 100 degrees?Now, we all know that when we have a fever,

- 09:47
SPEAKER 1 [continued]: the temperature of our body goes up.That's not what we're talking about.We're saying the resting body temperature, when you're notsick at all, usually is 98.6 with some wiggleroom, standard deviation.You're going to be a little bit above, a little bit below,right?But what's the odds or probabilityof having a temperature greater than 100 degrees, evenwhen you're not sick, just somebody sitting on the couch,

- 10:07
SPEAKER 1 [continued]: watching TV.Their temperature is 100 degrees or greater.What's that probability?And that's what we're going to set out to solve.So if we were to draw something like this, a lot of timesa picture is good, especially in the beginning whenyou're starting statistics.Here we have the mean, which is 98.6.

- 10:28
SPEAKER 1 [continued]: What we're asking is here's 100 degrees out here.That's greater than the standard deviation.We want to know what is the area out here?See, this area is quite small compared to the rest of it.So we know that it's going to be very low probabilityof this happening.But we want to calculate the actual value of it.So what we do is we convert 100 degrees to a z-score.So we say x minus the mean over the standard deviation.

- 10:52
SPEAKER 1 [continued]: So 100 is the value we care about,minus the mean, which is 98.6, over the standard deviation,which was 0.73.When we do the subtraction, divide by 0.73, we get 1.92.So if we had to redraw this, just to show youwhat we're actually doing, this is

- 11:13
SPEAKER 1 [continued]: our standard normal distribution, like this.What we're actually after is a z-score of 1.92.And we want the area to the rightbecause it says a body temperature isgreater than this.So we're shading everything to the right.So you see, a lot of times in statistics,you're trying to find the probability that somethingis less than or greater than.

- 11:34
SPEAKER 1 [continued]: In a minute, we'll find out the probabilitythat you lie between a certain range of temperatures.So it all boils down to marking on that normal distributionthe values you care about and looking at the area.In this case, the area is greater than 100because we want to know what the bodytemperature, the probability of having a bodytemperature greater than 100.That means 100 degrees, 101, 102, 103, 104,

- 11:56
SPEAKER 1 [continued]: anything-- everything in between.That's why we want the area because it catch everythingafter.So what we're really after is the probabilitythat z is now greater than 1.92.Probability of z is greater than 1.92.But see, this is not what the table in the back of the book

- 12:17
SPEAKER 1 [continued]: gives you, right?We did this in a couple of sectionsago when we were just learning about z-scores.So whenever you're trying to find the area to the right,you need to flip it around and say the probability of zbeing less than negative 1.92.And we explained why you do this in a previous section.So go back and look.Basically, negative 1.92 is over here.

- 12:38
SPEAKER 1 [continued]: So we find the area to the left, which is the same as this one.So when you look in your table, youwill find that when you look negative 1.92 up in your tableand you get an answer, you'll get 0.0274.This is the probability.Again, probabilities can go from 0 to 1.

- 12:58
SPEAKER 1 [continued]: Right?0 meaning very unlikely or never going to happen,1 being certainly going to happen.Here, again, we have a small number.But notice, we compare this one to this one over here.The probability of having a body temperature less than 96.9is actually less than the probabilityof having a body temperature greater than 100 degrees.

- 13:20
SPEAKER 1 [continued]: And so I hope that you can understandfrom doing these things that-- a couple things.Number one, we're putting everything together.We're using the normal distribution.We're using the z-scores.We're using the table.We're using the concept that area under these curvesyield probabilities.Those are all separate concepts that if you justthrow them together quickly, can hit you real hard.But we've introduced them slowly.

- 13:41
SPEAKER 1 [continued]: And so now, when you read a problem,you need to think about it in termsof what's it going to look like on the curve?Where is the shaded area going to be?That's what you really need to do.And then the rest of it just comesfrom reading tables and subtractionand things like that.So follow me on to the next section.We'll continue working problems of this nature.We're going to do a lot of them to give youlots and lots of really great practice in statistics.

### Video Info

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

**Episode:** 12

**Publisher:** Math Tutor DVD

**Publication Year:** 2013

**Video Type:**Tutorial

**Methods:** Normal distribution, Probability

**Keywords:** body temperature; mathematical concepts; mathematics; probability learning

### Segment Info

**Segment Num.:** 1

**Persons Discussed:**

**Events Discussed:**

**Keywords:**

## Abstract

Instructor Jason Gibson brings together information from past sessions to start calculating statistics. He demonstrates how to calculate high and low probabilities.