WEBVTT mathematics/statistics/son
00:00:00.000 --> 00:00:02.600
Hi and welcome back to www.educator.com.
00:00:02.600 --> 00:00:06.500
Today we are going to be starting to talk about normal distributions.
00:00:06.500 --> 00:00:12.300
When will talk about normal distributions in the next couple of lessons and it will come up again and again in the future.
00:00:12.300 --> 00:00:15.900
It is a pretty important one.
00:00:15.900 --> 00:00:19.300
First let us talk about what a normal distribution actually is.
00:00:19.300 --> 00:00:22.600
I want to distinguish it to what we have been talking about before.
00:00:22.600 --> 00:00:29.700
When a distribution looks normally shaped, a normal distribution is an entire different thing.
00:00:29.700 --> 00:00:37.100
We are also going to talk about a lot of normal distribution yet they all have the same shape and what that means.
00:00:37.100 --> 00:00:46.100
Then we are going to talk about some normal distribution problems but only problems that use the what we call empirical rules.
00:00:46.100 --> 00:00:48.100
What is a normal distribution?
00:00:48.100 --> 00:00:49.900
It is also called a normal curve.
00:00:49.900 --> 00:00:54.000
A normal distribution is a theoretical model.
00:00:54.000 --> 00:00:59.500
It is not based on data necessarily but it is a projection.
00:00:59.500 --> 00:01:05.700
So far we have looked at frequency distributions actual data points that have a normal shape.
00:01:05.700 --> 00:01:24.100
By normal shape what we really meant was symmetrical, unimodal, point of inflection, those kind of features.
00:01:24.100 --> 00:01:29.000
They do not just have a normal shape, they are not lead normal distribution.
00:01:29.000 --> 00:01:35.900
The normal distribution, the actual model is actually a probability density function.
00:01:35.900 --> 00:01:44.800
When I draw a normal distribution, you might say that it just looks like those normally distributed shapes that we have looked at before.
00:01:44.800 --> 00:01:55.100
The difference is that her, notice this curve is actually represented by a function that we are going to talk about in a later lesson.
00:01:55.100 --> 00:02:03.800
What is underneath the curve, the area underneath the curve is the probability with which some value will occur.
00:02:03.800 --> 00:02:10.900
On the x axis we have some values, presumably values of some variable.
00:02:10.900 --> 00:02:12.600
For instances, height.
00:02:12.600 --> 00:02:18.700
The values might be like 60 inches, 61 inches.
00:02:18.700 --> 00:02:33.900
Here what we see is the mean, mode, median, but here in this area that is the probability of
00:02:33.900 --> 00:02:47.400
which a value that is clocked out of the road with far in between x1 and x2.
00:02:47.400 --> 00:02:53.800
We are interested in the relationship between the area and values.
00:02:53.800 --> 00:03:08.200
That is why a normal distribution is really important because there is this nice, regular, relationship between probabilities and values.
00:03:08.200 --> 00:03:16.400
Let us talk about the range of the probabilities that are possible underneath the normal distribution or normal curve.
00:03:16.400 --> 00:03:20.000
The area underneath the curve represents probability.
00:03:20.000 --> 00:03:29.500
The entire area underneath the curve, if I colored everything in, that probability is equal to 1.
00:03:29.500 --> 00:03:35.000
All individual probabilities, this one and this one.
00:03:35.000 --> 00:03:51.300
All of these add up to, from negative infinity to infinity, all the way equals 1.
00:03:51.300 --> 00:03:55.700
A probability of 0 means that , that value never occurs.
00:03:55.700 --> 00:03:59.500
By never, I mean never.
00:03:59.500 --> 00:04:06.900
What we are going to introduce is this new notation.
00:04:06.900 --> 00:04:13.700
It might not be new to you if you have probability before but this just means that this is a probability function.
00:04:13.700 --> 00:04:20.400
Remember that we plot a probability – density function, it is a probability function we are given some value x
00:04:20.400 --> 00:04:24.100
you will get as the output of the probability.
00:04:24.100 --> 00:04:36.600
P of an exact square, a very particular square, the probability of height being 60m of dot is 0 in a normal distribution.
00:04:36.600 --> 00:04:41.000
Because it is a density distribution, it is not about frequencies.
00:04:41.000 --> 00:04:43.000
It is about probability density.
00:04:43.000 --> 00:04:49.600
What we are interested in is the probability of a square following a certain range.
00:04:49.600 --> 00:04:59.900
That probability is not 0 and it actually does not matter how far you will get, that probability is not actually 0.
00:04:59.900 --> 00:05:11.500
It might be like .000000000001 but is not 0.
00:05:11.500 --> 00:05:14.800
The normal distribution, we could summarize it a couple of ways.
00:05:14.800 --> 00:05:19.800
We could summarize it using the mean and the standard deviation.
00:05:19.800 --> 00:05:27.000
Here we see that the mean is represented by the μ sign and the standard deviation is represented by the sigma (∑).
00:05:27.000 --> 00:05:34.400
If you recall back to population versus samples, this is not much more like a population than it is a sample.
00:05:34.400 --> 00:05:42.300
That is because the normal distribution is a theoretical model, it is not actual data.
00:05:42.300 --> 00:05:47.100
It is a theoretical model, a projection very similar to a population.
00:05:47.100 --> 00:05:50.700
It is where we think the samples are coming from.
00:05:50.700 --> 00:06:03.100
Some properties are just like the normally distributed shapes that we have looked at before, symmetrical, unimodal, asymptotic.
00:06:03.100 --> 00:06:13.200
This is just talking about this fact but it actually approaches but never reaches 0 probability.
00:06:13.200 --> 00:06:21.100
It asymptotes with the x axis and it comes close but does not actually touched it.
00:06:21.100 --> 00:06:24.800
The final things is that it is continuous.
00:06:24.800 --> 00:06:29.500
The normal distribution actually goes from negative infinity to infinity.
00:06:29.500 --> 00:06:38.200
Most values and most measurable things in the world do not actually have values for negative infinity to infinity.
00:06:38.200 --> 00:06:41.000
For instance, height.
00:06:41.000 --> 00:06:49.700
We do not really have anybody who is infinite height or negative height for that matter.
00:06:49.700 --> 00:06:55.500
But the normal distribution stretches on forever and ever.
00:06:55.500 --> 00:07:00.200
There are going to be what seems like a lot of different normal distributions.
00:07:00.200 --> 00:07:03.900
For instance, you might see normal distributions that look like this.
00:07:03.900 --> 00:07:07.100
You might see normal distributions that look like that.
00:07:07.100 --> 00:07:09.600
You might see normal distributions that look like that.
00:07:09.600 --> 00:07:13.800
You might see them as having slightly different shapes.
00:07:13.800 --> 00:07:19.300
Like this one is sort of fat, this one is skinny, this one is even skinnier.
00:07:19.300 --> 00:07:24.800
But you do see that each of that are symmetrical, unimodal, asymptotic, and continuous.
00:07:24.800 --> 00:07:28.100
Actually there is more to the normal distributions than that.
00:07:28.100 --> 00:07:29.800
It is not that those for features.
00:07:29.800 --> 00:07:38.100
There are other properties that has an empirical rule.
00:07:38.100 --> 00:07:47.700
We talked about how can they be fat or skinny, but not all shapes that look like this is a normal distribution.
00:07:47.700 --> 00:07:49.700
Let me show you why.
00:07:49.700 --> 00:08:00.900
In a normal distribution, it follows some strict set of relationships between the probabilities and values down here.
00:08:00.900 --> 00:08:02.600
That is going to be important.
00:08:02.600 --> 00:08:04.300
Here is what I mean.
00:08:04.300 --> 00:08:24.700
When you go 1 stdev out form the mean or the line of symmetry, when you go 1 stdev out, that area right here should be about 34% of the curve.
00:08:24.700 --> 00:08:28.200
That is always the case in a normal distribution.
00:08:28.200 --> 00:08:44.200
Let us say we have this shape and it looks like a normal distribution, but we go 1 stdev out.
00:08:44.200 --> 00:09:05.900
Let us say we go 1 stdev out from the normal distribution, does this looks like just 34%?
00:09:05.900 --> 00:09:08.200
Does it look like more than 34%?
00:09:08.200 --> 00:09:12.000
That means that this is not a normal distribution.
00:09:12.000 --> 00:09:15.700
Even though it looks like one superficially.
00:09:15.700 --> 00:09:26.000
On the other hand, let us say we have something that looks like too skinny to be a normal distribution
00:09:26.000 --> 00:09:37.700
but if in this case going 1 stdev out, looks like that.
00:09:37.700 --> 00:09:43.300
But this is sort of 34% of the total curve, right?
00:09:43.300 --> 00:09:49.500
Because of that, even though it looks skinny it is a normal distribution.
00:09:49.500 --> 00:09:58.000
In this one, it looks like a perfect in a very typical normal distribution but it is not because the area does not fit.
00:09:58.000 --> 00:10:01.600
That is one thing you need to know, that 34%.
00:10:01.600 --> 00:10:18.700
If you flip around that side because it is a symmetrical distribution, if you flip it around the area between 0 and stdev 1, this is also 34%.
00:10:18.700 --> 00:10:23.200
When you add those together, it is about 68%.
00:10:23.200 --> 00:10:36.000
Just by going 1 stdev out on each side from the line of symmetry this should get 68% of the area covered more than halves.
00:10:36.000 --> 00:10:55.800
If you go out another stdev and you look at the area between 2 stdev out and 1 stdev out, that area is equal to about 14% or 13 ½.
00:10:55.800 --> 00:11:12.600
Let us write this t our new notation where we use P and I will just use this as my values for now.
00:11:12.600 --> 00:11:47.000
Where x is in between 0 and 1 = .3413 and the area in between 1 and 2 = .1359, this is how we would write this in algebraic notation.
00:11:47.000 --> 00:11:54.400
Finally let us cover the small area right here where now we are going 3 stdev out.
00:11:54.400 --> 00:11:56.500
I’m just going to write my notation down here.
00:11:56.500 --> 00:12:10.600
Here I’m going to write P where in between 2 and 3, that area is about 2%.
00:12:10.600 --> 00:12:13.900
This is what we call the empirical rule.
00:12:13.900 --> 00:12:20.100
I advice that you memorize these 3 numbers, 34, 14, and 2.
00:12:20.100 --> 00:12:33.800
Note that by knowing this you could know at least approximately how much of the area is covered by going 1 stdev out and 2 stdev out, 3 stdev out.
00:12:33.800 --> 00:12:48.900
Because we are rounding if you add all of these numbers out 34, 34, 14, 14, 2, 2 = 100% but in a normal distribution it is asymptotic.
00:12:48.900 --> 00:12:50.600
It is continuous.
00:12:50.600 --> 00:12:57.100
It is a little bit less than 100%, it is 99. Something percent.
00:12:57.100 --> 00:13:05.300
There is a tiny little bit area that goes out and out forever.
00:13:05.300 --> 00:13:18.700
It is handy to know that if you 1 stdev out then you will cover 68% of that total area.
00:13:18.700 --> 00:13:29.500
If you go 2 stdev out, that is about 95% or 95 ½ %.
00:13:29.500 --> 00:13:41.000
When you go out all the way to 3 stdev you will get .9974%.
00:13:41.000 --> 00:13:46.700
It is really close to 100% but not 100%.
00:13:46.700 --> 00:13:50.900
That is important for a normal distribution.
00:13:50.900 --> 00:13:53.300
Let us talk about the different kind of problems that you might be able to solve just by knowing the empirical rule.
00:13:53.300 --> 00:14:04.200
Usually in a normal distribution problem, you have to first look for rather the distribution that we give you is normally distributed.
00:14:04.200 --> 00:14:10.200
If it do not say that it is normal, then you cannot use the empirical rule.
00:14:10.200 --> 00:14:14.700
Let us read the prompt and let us get with the different kinds of problems there might be.
00:14:14.700 --> 00:14:26.100
The distribution of SAT scores for incoming students in a university is approximately normal with a mean of 550 and a stdev of 100.
00:14:26.100 --> 00:14:31.600
They told us it was approximately normal so we could use our empirical rule.
00:14:31.600 --> 00:14:40.100
Usually in a normal distribution problems where you need to use the empirical rule they will probably give you the mean and stdev.
00:14:40.100 --> 00:14:48.200
In other problems with a normal distribution that we will look at later, we would not give you the mean and you might be able to figure it out.
00:14:48.200 --> 00:14:53.500
They might ask you what percentage of scores with 450 or below?
00:14:53.500 --> 00:15:06.800
Here they give you the score and you are supposed to find the probability of getting that score or below.
00:15:06.800 --> 00:15:11.500
Where x is less than 450.
00:15:11.500 --> 00:15:16.200
Note here that what is missing is probability.
00:15:16.200 --> 00:15:25.500
Another question that they could ask you is that the same prompt above, they could ask you what math scores separates the lowest 2% from the rest?
00:15:25.500 --> 00:15:28.800
Here they give you the probability.
00:15:28.800 --> 00:15:33.000
What they actually want you to find out is, what are the scores?
00:15:33.000 --> 00:15:40.100
What is this blank right there, right?
00:15:40.100 --> 00:15:44.000
Here we are missing the score.
00:15:44.000 --> 00:15:51.900
Here I will write missing probability.
00:15:51.900 --> 00:16:01.000
Here we are missing the score or value.
00:16:01.000 --> 00:16:05.100
Now let us try to solve these 2 problems by using the empirical rule.
00:16:05.100 --> 00:16:10.400
What percentages of scores will 450 and below?
00:16:10.400 --> 00:16:17.900
It has to stretch out in a normal distribution first.
00:16:17.900 --> 00:16:24.600
Here is 550, that is the mean, point of symmetry.
00:16:24.600 --> 00:16:33.600
450 is just 1 stdev in a way.
00:16:33.600 --> 00:16:36.200
The distance is 100.
00:16:36.200 --> 00:16:42.100
You could think of stdev as new units of measurements.
00:16:42.100 --> 00:16:51.900
We knew about inches and meters but now we are interested in how far way things out in terms of stdev rather than inches or feet.
00:16:51.900 --> 00:16:55.400
Here the stdev is 100.
00:16:55.400 --> 00:17:05.500
We know that even though these are the scores or what we call the raw scores, we know that these in terms of stdev,
00:17:05.500 --> 00:17:11.000
this means that this is 1 stdev away on the negative side.
00:17:11.000 --> 00:17:21.600
We could just draw a little border here and sketch the part that we need to find and that is the area that we need to find.
00:17:21.600 --> 00:17:28.200
We know that this part from here to here is 34%.
00:17:28.200 --> 00:17:33.400
We know that this part is 14%.
00:17:33.400 --> 00:17:36.800
We know that this part is about 2%.
00:17:36.800 --> 00:17:51.400
We could just add 14 + 2 and get the probability where x is less than 450 = about 16%.
00:17:51.400 --> 00:17:54.800
That is the percentage of scores.
00:17:54.800 --> 00:18:00.100
What about when we are missing a score that we have the probability?
00:18:00.100 --> 00:18:05.600
Let us sketch this one as well.
00:18:05.600 --> 00:18:13.800
Here we have this 2% and we want to know what is this value right here.
00:18:13.800 --> 00:18:24.400
We know that although we do not know the raw score right now, we do know in terms of stdev
00:18:24.400 --> 00:18:31.800
that at about in between 2 or 3 stdev away, that is about the 2% mark.
00:18:31.800 --> 00:18:37.100
Just to show you I will draw the other one.
00:18:37.100 --> 00:18:47.400
In terms of stdev, here is 0, -1, -2, -3.
00:18:47.400 --> 00:18:53.200
This seems to correspond with about being 2 stdev away on this side.
00:18:53.200 --> 00:19:10.700
We know that this middle is 550, if we take stdev 100 jump then this would be 450 and this value will be 350.
00:19:10.700 --> 00:19:13.500
That is a little small but I will write it up here.
00:19:13.500 --> 00:19:37.900
We know that what we are looking for is 2% that is equal the probability where x is less than 350.
00:19:37.900 --> 00:19:50.100
So far it has been pretty easy just adding and subtracting, and memorizing the empirical rule so let us go ahead and do some more problems.
00:19:50.100 --> 00:19:55.600
Before some more problems we are going to just look at this in terms of a shape analogy.
00:19:55.600 --> 00:20:05.100
Just to sum it in, when we think about shapes like rectangles, we know that rectangles are defined by their length and width.
00:20:05.100 --> 00:20:08.800
If you know their length and width, you could draw that rectangle.
00:20:08.800 --> 00:20:14.600
In the same way for the normal distribution, all normal distributions follow the empirical rule.
00:20:14.600 --> 00:20:21.000
All you need to know is the mean and the stdev.
00:20:21.000 --> 00:20:35.600
You could draw that particular normal distribution and we know that it is unimodal, symmetric, asymptotic, and continuous.
00:20:35.600 --> 00:20:41.900
Rectangles can all look a little bit different like sometimes their length is longer than the width.
00:20:41.900 --> 00:20:43.900
Sometimes the width is longer than the length.
00:20:43.900 --> 00:20:47.800
Sometimes the length is equal the width in the special case of square.
00:20:47.800 --> 00:20:51.100
They could all look different but they are all rectangles.
00:20:51.100 --> 00:20:55.900
In the same way normal distributions can look slightly different from each other.
00:20:55.900 --> 00:21:01.300
Because the x axis can be stretched out or can be shrunk.
00:21:01.300 --> 00:21:08.700
But as long as that normal distributions follows the empirical rule where about 1 stdev out
00:21:08.700 --> 00:21:14.700
on either side is about 68% then you know that it is still a normal distribution.
00:21:14.700 --> 00:21:18.100
Even though it looks a little too skinny or too fat.
00:21:18.100 --> 00:21:22.200
What can you find out when you know rectangles?
00:21:22.200 --> 00:21:33.400
If you have the area and width, let us say you have this and this, you could find out the length because they have this relationship with each other.
00:21:33.400 --> 00:21:41.100
Same thing with perimeter, if you know the perimeter and the length then you could figure out the width.
00:21:41.100 --> 00:21:46.300
These are what we call constraints because they constrain the system.
00:21:46.300 --> 00:21:52.800
They are like these little boundaries and you can balance within the boundaries and figure out the things that are missing.
00:21:52.800 --> 00:22:01.400
In the same way, when you have normal distributions you could balance from probability to raw score because they have this relationship.
00:22:01.400 --> 00:22:06.700
And that relationship that you have for now is called the empirical rule.
00:22:06.700 --> 00:22:20.000
We have only covered knowing that relationship between probability and score and only wonder even intervals away.
00:22:20.000 --> 00:22:24.200
That 1 stdev, 2 stdev, 3 stdev.
00:22:24.200 --> 00:22:30.600
But we really do not know how to get the probabilities when it is like 1.5 stdev away.
00:22:30.600 --> 00:22:34.300
That is what we will cover in the next lesson.
00:22:34.300 --> 00:22:36.600
Let us go into some problems.
00:22:36.600 --> 00:22:38.600
Example 1.
00:22:38.600 --> 00:22:48.200
Using my empirical rule, what percentages of values in a standard normal distribution is used to solve below the SAT score of -1.
00:22:48.200 --> 00:22:53.600
What falls above of these scores of -1?
00:22:53.600 --> 00:22:57.500
Although we have not yet covered the standard normal distribution yet,
00:22:57.500 --> 00:23:01.400
let us assume that it is the standard normal distribution we have been talking about.
00:23:01.400 --> 00:23:04.800
We will define it in the next lesson.
00:23:04.800 --> 00:23:15.400
It is actually when we do not know any of the values and we just know the c scores or what we call the standard normal deviations.
00:23:15.400 --> 00:23:19.300
That is the c scores.
00:23:19.300 --> 00:23:27.100
We could easily do this problem just by knowing the empirical rule.
00:23:27.100 --> 00:23:39.000
In a standard normal distribution, we pretend that we do know the actual values, but we really do not know.
00:23:39.000 --> 00:23:52.600
We just know the standard deviation or what we know the c scores.
00:23:52.600 --> 00:24:01.100
What they want to know is what percentage of values fall below a c score of -1?
00:24:01.100 --> 00:24:13.900
Here we know that this area is about 14% and this area is about 2%, this area is pretty negligible.
00:24:13.900 --> 00:24:22.600
If we add these up this would be .16.
00:24:22.600 --> 00:24:35.900
To write it in an algebraic expression it is p where z is less than -1 = .6.
00:24:35.900 --> 00:24:41.900
Once you know this, it is now asking about what about above the z square of -1.
00:24:41.900 --> 00:24:45.100
Now it is talking about this area.
00:24:45.100 --> 00:24:59.100
There are two ways you could do this to figure out p where z is greater than -1, we know that the entire area is 1.
00:24:59.100 --> 00:25:07.800
We could just subtract out this red part, .16.
00:25:07.800 --> 00:25:14.900
We could just do that and we know that it will end up like this area which will then be .84.
00:25:14.900 --> 00:25:16.900
There is another way that we could this.
00:25:16.900 --> 00:25:28.300
What we could do is out at this part, this part over here which we know it is exactly half of the normal distribution .5.
00:25:28.300 --> 00:25:36.800
And then add 2 to it to this little part right here which we know is the 34%.
00:25:36.800 --> 00:25:47.800
When we add those together, we could do 34 and that also gives us 84% of the curve.
00:25:47.800 --> 00:25:55.800
Those are two different ways of doing it either way, whatever your preference.
00:25:55.800 --> 00:26:00.300
Here is another example.
00:26:00.300 --> 00:26:13.700
Using the empirical rule, what percentage of values in a standard normal distribution fall below the z squares or stdev of -1 and 2.
00:26:13.700 --> 00:26:21.000
It is nice to sketch it so that you will know where you are headed.
00:26:21.000 --> 00:26:32.700
Here is 0 and what we need is between the stdev of 1 and -2.
00:26:32.700 --> 00:26:36.600
Once again there are multiple ways that we could find this out.
00:26:36.600 --> 00:26:51.800
Probably a very simple and straightforward way is knowing p and we are trying to find p where x or z lies in between -1 and 2.
00:26:51.800 --> 00:26:56.200
What we could do is add up all this separate little probabilities.
00:26:56.200 --> 00:27:10.600
The probability between -1 and 0 , and add that with the probability between 0 and 1.
00:27:10.600 --> 00:27:17.500
Add that with the probability between 1 and 2.
00:27:17.500 --> 00:27:38.500
You could just add all those up and that would be 34, 34, 14.
00:27:38.500 --> 00:27:48.600
That would give us 68, 70, 82% of the curve.
00:27:48.600 --> 00:27:50.300
That is one way of doing it.
00:27:50.300 --> 00:28:06.000
Another way you could possibly do this is I just know that this is 2%, I could deduce that from this whole thing being 50%.
00:28:06.000 --> 00:28:09.800
I could just subtract that 2 and get 48.
00:28:09.800 --> 00:28:21.600
I could actually just do 34 + 48 that would give us the same answer of 82%.
00:28:21.600 --> 00:28:26.500
Just different ways of summing this up figuring out the distributions.
00:28:26.500 --> 00:28:37.100
Some of you may memorize the middle part between -1 and 1 is 68 then it would be like 68 + 13.
00:28:37.100 --> 00:28:43.500
Either way you want to do it but the point is I want to show you there are lots of different ways you could cut this out.
00:28:43.500 --> 00:28:50.600
You want to think of this little distributions like a chocolate piece or something that you could
00:28:50.600 --> 00:28:57.400
just break this off in lots of different ways and add them together again in lots of different ways.
00:28:57.400 --> 00:29:01.600
Example 3.
00:29:01.600 --> 00:29:03.900
What is this problem missing, sketch and find what is missing.
00:29:03.900 --> 00:29:08.600
We know that there are only two things that could be missing in the problems that we have introduced so far.
00:29:08.600 --> 00:29:16.000
One is probability that could be missing or the value, the boundary.
00:29:16.000 --> 00:29:26.300
It says, given a normal distribution with a mean of -25 and a stdev of 10, find out where the middle 95% of values would lie.
00:29:26.300 --> 00:29:38.600
Right now we have the probability here.
00:29:38.600 --> 00:29:44.100
What we do not have is the actual values.
00:29:44.100 --> 00:29:46.100
This is a values missing problem.
00:29:46.100 --> 00:29:49.000
That is what we are going to need to do.
00:29:49.000 --> 00:30:04.800
It helps to sketch out what we have done, my x axis is a little bit off.
00:30:04.800 --> 00:30:15.700
We know that the mean is -25, here my raw scores and here I’m going to write my z scores or standard deviations.
00:30:15.700 --> 00:30:28.200
Here is 0, -1, -2, -3, 1, 2, 3.
00:30:28.200 --> 00:30:31.800
We are trying to find the middle 95%.
00:30:31.800 --> 00:30:43.200
If you remember the empirical rule, we know that around here and here this is approximately 95%.
00:30:43.200 --> 00:30:50.600
If you want to check that you could add 34, 34, 13.5, 13.5.
00:30:50.600 --> 00:30:54.800
That is about 95% of the curve.
00:30:54.800 --> 00:31:00.800
Let us try to find what these values are right here.
00:31:00.800 --> 00:31:07.700
We know that the stdev is 10, each of these little jumps are worth 10.
00:31:07.700 --> 00:31:17.500
Let us go out 10 jumps from -25 and that would be -35, -45, in this side.
00:31:17.500 --> 00:31:26.800
If we go on the positive direction it would be -15, -5.
00:31:26.800 --> 00:31:43.300
To solve it we would say this probability .95 is the probability between -45 and -5.
00:31:43.300 --> 00:31:52.600
Many way of writing it without these values is to write it in terms of the standard deviation but it asks about values.
00:31:52.600 --> 00:31:54.800
I just wanted to show you this other way.
00:31:54.800 --> 00:32:03.900
The other way we could write it is also like this.
00:32:03.900 --> 00:32:12.600
What z scores does the middle 95% cover and that would between 2 and -2.
00:32:12.600 --> 00:32:24.800
Note that the 45 corresponds to -2 and the -5 corresponds to 2, just like here.
00:32:24.800 --> 00:32:26.600
That is the nice thing.
00:32:26.600 --> 00:32:30.200
There are these relationships between the raw scores and the z scores.
00:32:30.200 --> 00:32:34.600
We are going to get more into that in the next lesson.
00:32:34.600 --> 00:32:36.100
Here is example 4.
00:32:36.100 --> 00:32:37.800
What is this problem missing?
00:32:37.800 --> 00:32:39.800
Sketch and find what is missing.
00:32:39.800 --> 00:33:00.800
Given the normal distribution of the mean of 46.4 and a stdev of 6.1, find the score that comes at the largest 16% of values.
00:33:00.800 --> 00:33:06.800
This one gives us again the probability and we need to find the score or the values.
00:33:06.800 --> 00:33:11.000
We know that the missing thing is the missing score.
00:33:11.000 --> 00:33:15.900
Using our empirical rule, we could find that 16% is.
00:33:15.900 --> 00:33:22.800
Here are my raw scores and here are my z scores or standard deviations.
00:33:22.800 --> 00:33:34.000
Since I am looking at the top 16% that need to go in this side, 0, 1, 2, 3.
00:33:34.000 --> 00:33:40.500
We know that this about 14% and this is about 2%.
00:33:40.500 --> 00:33:49.800
Here is my elusive 16% and we need to find this cut off score.
00:33:49.800 --> 00:33:59.900
That cut off square is 1 stdev away and here my stdev is 6.1 so my little jump is 6.1.
00:33:59.900 --> 00:34:25.400
What I need to do is add 1 jump to 46.4 and that would be 52.5 or you could write is the probability where x is greater than 52.5 or 16%.
00:34:25.400 --> 00:34:31.700
That is it for using the empirical rule to find the answers for normal distributions problems.
00:34:31.700 --> 00:34:33.000
Thanks for using www.educator.com.