Hello, Mr. Tarrou again.
Now in AP Statistics I have talking about Normal distributions I finally defined what that means using the Empirical Rule
and showed you how to use the Normal Probaliltty plot to check for Normality.
So, I have talking about z scores or Standard Normal Distributions
Now I have got these bell curves all over the place and I have defined what Normality is,
let's actually tie the two together and see how the z score calculation which is
estimate minus parameter over the standard deviation of the estimate actually works.
Now this is very early in the book so there just kind of spoon feeding us all the information.
So this formula is going to look like this,
x for the estimate minus the population mean that they tell us and the standard deviation that they tell us.
We don't know how to find any of this stuff yet.
I want to just try and run through in the next 15 minutes as many z score calculations as I possibly can.
We are going to be ending with a p value which is the probability of observing a certain event or one more extreme.
Which means all these problems are going to be inequalities.
Don't forget our density curves have an area of 1.
The reason why is because we can take that area that we get out of the bell curve
and easily convert into a percent by just moving the decimal over two places
and that will be the probability of observing whatever event that our word problem might be talking about.
Well, I'm gonna give you some generic data .
We have a Normal Distribution that has a mean of 200 and a standard deviation of 30.
Bell curve, center at 200.
Z score calculation, every single problem that gives you or says that a distribution is Normal
and then asks you for the probability of an event or what proportion of data lies within some interval,
all questions like that will be worked with the z score calculation.
We will learn other variables such as the discrete random variable that may not be using that z score calculation.
But if you hear Normal and then proportion or probability that will be a z score calculation.
And again these are on univariate unimodal quantitative data, that data that is along the x axis.
We are not talking about scatter plots, linear regression lines, or Chi Square that will talk about at the end of the year.
Just one nice set of data that is quantitative and continuous along the x axis.
So, lets answer some questions.
What is the probability in this distribution with a mean of 200 and a standard deviation of 30
that x is less than 155?
I have 155 here and I want to find the area to the left of that value.
So, I am going to do z equals
the statistic which is what the question is about minus the parameter over the standard deviation
of the statistic which is 30.
That comes out to negative 1.5 and I forgot my book
with your table in the front of your book, I know you cannot see this on the camera but hopefully you can see enough to follow along,
my z score is -1.5 standard deviations. It is negative because my value is to the left of the mean.
z scores are negative to the left of the mean.
-1.5 in the front page of my book where the negative z's are I am going to go down to negative 1.5
and I don't have a hundredths place so I am going to use the first column.
-1.5 has a p value of .0668
So my p value is from this z score .0668, that means that roughly 6.7% of my data is less that 155.
I would have 6.7% probability of observing a piece of data or observing an observation of less than 155.
Don't forget the unit of measure for z scores is standard deviations.
This is 1.5 standard deviations to the left of the mean.
This answer came out to be .0668
I will need to erase these examples as I go because I did not give myself very much room to do these examples.
What is the probability that x is less than or equal to 155?
If this is a continuous random variable, the probability of getting exactly 155
the probability that x equals 155 is one specific outcome
and if this is continuous there is an infinite number of possibilities.
Well 1 over infinity is 0.
When you are talking about bell curve calculations
the probability of observing 1 specific outcome is always zero.
So, the probability of observing a value less that 155 is the same as probability of observing 155 or less.
You have to use the proper notation for the wording of your question but it will not change your answer.
BAM!!!
Alright, moving on...
Let's take a look at another value.
Let's say that I want the probability that x is less than 228.
Well 228 is over here somewhere. This is just a sketch.
I want to find the area to the left of 228, so I am looking for this area all the way on the left hand side of this density curve.
Alright, Normal, Probability, that means z score.
Notice these patterns.
I am giving you the x's, we calculate the z score, and then find the p value.
We are going to do the reverse in a minute.
So, z equals 228...the question...minus the parameter which is 200, divided by the standard deviation of 30
and that gives me a z score of .93.
On your z score chart, on the part that has the positive z scores, you go down to .9 on the z column and go over to the hundredths column of 3 and you find your p value.
The area to the left of of this z score of .93, the area to the left of 228 is...
equal to .8238
What if you want the area of an interval.
What if I want this area just between 228 and 155?
If you want the area of an interval, it is the area to the left of the large number
that is how is how you take it out of the chart in your book
take the area to the left of the small number and you subtract them.
I want to take out this area that I had in the green, all this area is .8238, I want to cut out that tail and only have the interval left.
so I am just going to do subtraction. I am just going to cut it out like a pair of scissors or a minus sign.
The probability that x is between 155 and 228 is .8238-.0668
or that is . 757
So the probability of observing a value between 155 and 228 is 75.7%
AWESOME!!!
A couple of z scores and how to find the area of an interval...area to the left of the large minus the area to the left of the small.
Now, what if we want this area?
What if I want the area to the right of 228?
What if I want the probability of x being greater than 228?
Let's just throw and equal sign in there for good measure, because it is not going to change our answer anyway. You just have to match the wording of your question.
To the right of 228 if this is a density curve and I'm focused on this green area which is .8238
how much is on the other side? Well if it is a density curve the total area must equal 1.
Well if the left side is .8238 the right side must be 1-.8238.
I will be doing a video to show you how to use your calculator to get these numbers as well.
I am doing all these calculations as if we were reading a chart.
Do I have time?
Oh I got plenty of time. I thought I was running out but I have 5 minutes still.
What if I ask you what is...
Let me get my distribution back up here.
We have a Normal distribution again with a mean of 200 and a standard deviation of 30.
By the way if I was doing a quick sketch of this Normal distribution because of the 68-95-99.7 rule.
I could put tick marks where my standard deviations are and that would be 200, 230, 260, and 290. 170, 140, and 110.
You have to draw it this way if you put the tick marks every standard deviation because within 3 standard deviations left and right you have 99.7% of your data.
Empirical Rule
All this stuff is going to make my drawing look a little bit complicated.
I want to know within this distribution, what is the third quartile. What is the 75% percentile?
Where is Q3?
Now I am going back to this. You are either give a statistics, then find the z score, and then the p value or the area under the curve.
Or they simply work in reverse.
If I am asking for Q3, I am telling you what I want shaded in the bell curve and I want the statistic...I want the x
Q3, third quartile, 75% percentile...that means, and percentile is what?
What percent of data is below the observation you want to look at or discuss.
Somewhere there is a number whose area to the left is .75, the third quartile.
I could be asking you for the 90th percentile, the 10th percentile, the longest 25% of the values.
...if this were some question about pregnancy or something like that.
But where is Q3?
Enough talking, I am going to run out of time.
So Q3. If my p value is .75, I am going to take my z score chart and look inside the body of the chart, that is where to areas under the bell curve are, and find as close as I can to .75
If I do that, my z score is going to be .67
So this statistic we are about to find is .67 standard deviations to the right of the mean.
Now if I know what the z score is, guess what formula I am going to use.
Area...NO Perimeter...NO I am going to do the z score formula!!!
Which is z equals x minus mu over the standard deviation of the statistic.
Well my z is .67, my mu is 300, and my standard deviation is 30.
If I multiply both sides by 30 I get 20.1 equals x-200.
If I add both sides by 200, I might finish on time, and my statistic...my 75%...my Q3...is 220.1
BAM!