- 00:01
SPEAKER: Hello.And welcome to this lesson of Mastering Statistics.[Finding Probability using the Normal Distribution, Part 2]We want to continue solving problemsinvolving the normal distribution we'vebeen working with.Now, a lot of times you can get a lotof value out of one single concept of a problem,and so we're going to recycle what we've used before.The body temps are normally distributed with a mean of 98.6and a standard deviation 0.73 Fahrenheit.

- 00:23
SPEAKER [continued]: But now part c, we want to find, whatis the probability of having a normal body temperaturebetween 98 and 99 degrees?So you see, this is a little differentthan what we have gotten before.In the previous sections, we weretrying to find, what is the probability of havinga temperature lower than or greater than a number?Now we want to find, what's the probability

- 00:43
SPEAKER [continued]: of lying within a range?And you start to see why marking offyour areas on these charts, on these are normal distributions,are useful.So if we wanted to draw kind of a picture, whichI almost always recommend, we'll draw it over here, right?So here's our normal distribution like this, right,

- 01:06
SPEAKER [continued]: 98.6 here.And basically what we have is 99 is going to be,let's say, right here, and let's say98 is going to be right here.And so what we're trying to find is a range.And we're going to go up here, and wecan see that we're collecting a large area under this curve.

- 01:27
SPEAKER [continued]: So we expect that the answer that we getshould be pretty substantial.In the previous sections, we had really low probabilitiesbecause we were looking in the tailof these curves for real extreme cases of body temp.Now we're finding body temp in a very normal range.All right?So even when you're not sick, yeah, there'sa chance you could have anything in this range

- 01:47
SPEAKER [continued]: by an average typical person.So we're going to try to see that.Now, what we're going to do is we'regoing to convert both of these values of interest to z-scores.So let's convert 99 degrees into a z-score.So it's x minus the mean over the standard deviation, so 99minus the mean, which is 98.6, over the standard deviation.

- 02:11
SPEAKER [continued]: And whenever you do this, you get 0.55.That's a z-score for this guy here.All right?So if you wanted to draw it here,you could just say z is equal to 0.55.And then over here you could say x minus the meanover the standard deviation.The other one we have is 98 degrees minus 98.6 over 0.73.

- 02:37
SPEAKER [continued]: And when you do that math, you get negative 0.82.So this number here corresponds to a z-score of negative 0.82.So you could redraw this.But essentially it's the exact same thingwith the standard distribution has a mean of 0and the standard deviation of plus/minus 1,and you put your z-scores on there.But the area that we're trying to find

- 02:58
SPEAKER [continued]: is exactly what we have drawn here on the board.We're trying to find the area between these two z values.Now we know the table in the back of the book'sonly going to give us area to the left of our z-score.So if we find the area correspondingto this, or the probability correspondingto this z-score from here, we're goingto get the entire area off to the left.

- 03:19
SPEAKER [continued]: But if we take this one and put it in the table,we're going to only get this part.If we subtract the two, we're goingto be left with what's in the middle.So the way you write this is you saythe probability, the answer, is goingto be the probability of z being less than 0.55minus the probability of z being less than negative 0.82.

- 03:43
SPEAKER [continued]: This is how you write it down before youput any numbers in there.And so when you look at the table, whatis the probability of z being less than 0.55?You just look this up in the chart, and you get 0.7088.And you subtract from that what youget when you look in the chart, negative 0.82,and you get 0.2061.

- 04:06
SPEAKER [continued]: And when you subtract these guys,you're going to get 0.5027.Now look at your answer and see if it makes sense.Probabilities can go from 0 to 1.This is about 0.5.So a little bit more than 0.5, so a very significantprobability.That's like 50-50 odds.If you flip a coin, a perfect coin,we say it's 0.5 odds of heads, or probability of getting

- 04:29
SPEAKER [continued]: heads on any given flip, right?So what we're trying to say here is,if you're trying to find what is the probabilityof a random person that you pull outof a giant population of people having a bodytemperature in this range, and the answer'sgoing to be 50%, which is a large, large number.You could occasionally have people out hereand occasionally have people out here that

- 04:49
SPEAKER [continued]: wouldn't fall in that range.That would be the roughly other 50% because this is almost 50%here.So the basic idea of how do you do it,you find the z-scores of your boundaries.You put this in the chart, and it gives youthis area minus what you get from this one,and you're left with what you're interested in.That's why drawing these picturesis sometimes extremely useful to do

- 05:10
SPEAKER [continued]: because it helps you visualize the math that youneed to pull off.For instance, what if you have a question that said,what is the probability of a bodytemperature less than 97.6 degreesor greater than 99.6 degrees?All right?So this is an OR in there, and thisis when it becomes extremely useful to draw a quick picture.

- 05:33
SPEAKER [continued]: It just takes a second.So let's draw a distribution that looks like this, right,and we have 98.6 in there.And so what we're asked to do is-- notice that the rangehere, 97.6 to 99.6.So it's one degree on either side of this guy.

- 05:56
SPEAKER [continued]: So let's say this is a degree, so let's saythis is 99.6 degrees.And I would say this is degree 97.6 degrees.This is probably not exactly right,but we're going to do it for the purposes of our drawing.So effectively what we're trying to do iswe're trying to find-- we have these two boundaries here.Well, what is the probability?Is it going to be the shaded area between them?

- 06:18
SPEAKER [continued]: Well, no, it's not.Because if it were the shaded area between them,the problem would be phrased differently.It would say, what would the probabilitybe of having a temperature between these two numbers?That's not what it says.What is the probability of body temp being less than 97.6,meaning this guy, or greater than this one, whichis this guy?So effectively you're trying to find the area in the two tails.

- 06:42
SPEAKER [continued]: And we've done problems like this before when we firststarted learning about this.And also further note that because thisis one degree below the mean and thisis one degree above the mean, this area and this areashould be exactly the same, surfacearea in each of these tails.But we want to add them together to findthe probability of having a low body temp or a high body temp.

- 07:04
SPEAKER [continued]: So we want to calculate the z-scoresfor both of these boundaries.So we'll have x minus the mean over the standard deviation,so we'll have 99.6 minus 98.6 over 0.73.And when we do the subtraction and the division,

- 07:25
SPEAKER [continued]: then it will give us 1.37.And for the second number, we willhave x minus the mean over standard deviation.97.6 minus 98.6 over 0.73 gives you negative 1.37,and that goes for there.Notice that these z-scores are exact negatives of one another,

- 07:45
SPEAKER [continued]: and that's because of the symmetry of the factthat this is one degree below and one degree above.So in order to find the total probability,we need to find the shaded area here and the shaded area here.So the total answer, the probability,which will be the answer, is goingto be equal to the probability that z is greater than 1.37.

- 08:06
SPEAKER [continued]: That's this one, because if you think of this one,this is a z-score of 1.37.That corresponds to this value right here.This value is a z of negative 1.37 right there.So the probability of being greater than 1.37is that one plus the probability of zbeing less than negative 1.37.

- 08:29
SPEAKER [continued]: That gives us this area.So this gives us half of the area.This gives us the other half of the area.And we just have to use the chart to look this stuff up.So the probability of z being greater.Remember we already said that we can't get that directly outof the table.We have to convert this and say that zis less than negative 1.37.

- 08:51
SPEAKER [continued]: Basically, in order to find the answer to this,we have to change it.Because we're trying to find an area to the right,we have to change z going less than the negative.All right?And then we have over here the probability of zbeing less than negative 1.37 just left over from before.Notice these two are exactly the same-- basicallythe same answer.So when you find negative 1.37 in the table

- 09:15
SPEAKER [continued]: and you look it up, you're going to get an answer of 0.0853.And of course, this is the same thing, 0.0853.And so I'll switch colors, and I'llsay that the probability to solve this problem,you answer it, you get 0.1706.This is the probability of getting a temperature less

- 09:36
SPEAKER [continued]: than 97.6 or greater than 99.6.The answer is 0.1703.So the way we did this problem, the way I did it for you here,is to draw it on the board and to say, all right,we need to find the shaded area in here and the shaded areahere.Here's the z-score for this one.Here's the z-score for this one.

- 09:56
SPEAKER [continued]: And we find the probability of z being greaterthan that number, the probability of zbeing less than that number.But in order to get this answer, we have to switch it to a lessthan and make it z negative.We've been doing that a lot lately,so you should be familiar with that.This one just comes along for the ride.So when we look in the chart, we get an answer of 0.0853.We get another answer of 0.0853.

- 10:18
SPEAKER [continued]: Notice these are the same numbers.That's because of the symmetry.This surface area is exactly the same as this one, right?So the way we did this particular problem isto show everything explicitly.I wanted to show you to find this area,and I want to show you to find this area,and you add them together.But if you're pressed for time on a testand you know that it's symmetric like this,

- 10:38
SPEAKER [continued]: you know that the area here is exactly the same as the areahere because of the symmetry, thisis one degree less than and one degree greater than the mean,all you really need to do is find this z-score in the table.And you're going to get an answer,and you just multiply it by 2 because youknow that this area is exactly what this one was.So if you happen to notice that ahead of time,

- 10:59
SPEAKER [continued]: that the symmetry says I only have to find half of itand just multiply by 2, save yourself a little time.But if you don't realize that, then you just kind of gothrough it, and you'll see that, in the end, that's exactlywhat's happening anyway.So you can see why I spent so much time on z-scoresand getting area under the curve and all that.We did one thing at a time.And now we have the skills for you

- 11:20
SPEAKER [continued]: to go and do lots and lots of problems.You'll be given the characteristicsof the normal distribution.You'll be given the mean and the standard deviation,and you'll be asked questions about, find the probabilityof this happening.Draw a picture.That's my number one recommendation,especially in the beginning.Draw a picture so you can visualize what you're doing.Because a lot of times if you don't do that,you'll look up a z value in the chart,

- 11:40
SPEAKER [continued]: get a number that isn't really what you want.You may have to flip it around, as we'vebeen doing here sometimes.So draw a picture and then proceed.That will be very helpful for you.So follow me on to the next section.We will continue working these problems.We will use these skills that we've learnedto solve statistical problems.

### Video Info

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

**Episode:** 13

**Publisher:** Math Tutor DVD

**Publication Year:** 2013

**Video Type:**Tutorial

**Methods:** Normal distribution, Probability

**Keywords:** body temperature; mathematical concepts; mathematics; Study guides

### Segment Info

**Segment Num.:** 1

**Persons Discussed:**

**Events Discussed:**

**Keywords:**

## Abstract

In the second half of his presentation on finding probability using normal distribution, instructor Jason Gibson demonstrates how to calculate probabilities inside and outside of a given range.