Instructor: Hello students, this is Problem 3.
And of course, we are gonna start with Problem 3a.
Here we are going to go
over an example calculating probability
from a normal distribution using continuous data.
So first what I'm gonna do is read the problem.
Problem 3.
Total cholesterol in children aged 10 to 15 years of age
is assumed to follow a normal distribution
with a mean of 191 and a standard deviation of 22.4.
So part A is asking us what proportion
of children 10 to 15 years of age have a total cholesterol
between 180 and 190?
So first thing I would suggest
that you write down the information
that has been provided to us
that is mean equal to 191 and sigma equal to 22.4.
And then, next I recommend that you draw the normal curve.
So here, this is the normal curve
trying to draw it as best as I can.
And here is 191
and this is X.
So as per the question, now we will draw some lines
so we can understand what the question is asking us.
And basically, the question is asking us
what is the area under the curve?
So first, let's draw the lines
and I will be using different colors
to make this easier for us to see.
Okay, so this is 190
and this 180.
Okay?
And basically we are
looking for this area, okay?
And even though, we don't know the area under the curve
for this distribution,
we do know it for a normal distribution.
So what we need to do here
is transform our X values to our Z values.
So after that, we can use the appendix from the back
of our book to identify these probabilities.
So let's start the process.
I'm gonna go back to my blacking.
So here's the Z, and again, we know that the Z distribution
has a mean of zero.
So basically now what we have to do is figure out
what are the Z values for 180 and 190?
And to figure that out,
we have to be using the formula Z equal
to X minus mu divided by sigma.
So let's write down our formula here.
Okay, so now this was the probability we were looking for
180.
Okay?
So not as we try to convert this from X to Z,
we have to replace 180 and 190.
Okay? So now let's do the algebra and see what we get here.
We get negative 0.49,
negative 0.04.
Okay, so now what we need to do is go to the back
of our textbook,
page 342
and page 343.
So first, the way we look down the Z values
is we have to go down the left hand side
and find negative 0.4.
Then, we go across till we find 0.09.
And the way we find this probability
is the same way we'll find the probability
for negative 0.04.
So these probabilities, the way our table is set up
is giving us the value for less than the number, okay?
And again, this can vary from Z-table to Z-table.
And I know a lot of students when they're taking this class,
they like to go to Z-tables on the internet.
I have looked at some of them, yes, the Z-tables are all...
At least the ones I've seen, they're all correct
but it might be set up differently.
And if you use a Z-table that is set up differently
the way we are progressing
with solving the problem will change.
So for the purpose of our class,
please, please use the Z-table from the back of our book.
So again, we are looking for the area,
the green-shaded area.
And the way we are going to find out
is of course, we have to take the difference
between the two values.
So now we have to do what?
We have to look up the value and then we have to look at...
Subtract and get the final answer we are looking for.
So once we go to the back of our book, we see that the value
for this is 0.3121
and the value for this is 0.484.
Again, if you are having a tough time
figuring out these values from the Z-table,
please, please let me know.
I will be more than happy to have a Zoom meet with you
go over this one-on-one.
So now, the probability here is going to be...
And I know we are kind of short of space here
so I'm squeezing everything
0.484 minus 0.3121.
So when we do this,
and again I like to put this parentheses around.
So then, P is equal to 0.1719.
Okay, so basically, we can say that based
on the values provided,
there is a 17.19% probability
that the child will have a cholesterol between 180 and 190,
Okay?
And that is based on the values provided.
Again, just to go over the key points here,
I want to tell you that this part
was given to us from the Z-table and that is 0.3121.
And again, this part that was given to us
from the Z-table,
all the way here is 0.484.
And we are looking for the difference
and that's why we had to do the subtraction.
Now, a couple of things to keep in mind.
The area under the curve,
the total area under the curve always adds up to one.
And another thing to keep in mind is whenever you are
in doubt, which one should I subtract from which one?
Always keep in mind that the area under the curve
is always a positive at, okay?
So whichever number is bigger then the subtraction
should be from that number.
So here 0.484 is the bigger number than 0.3121.
So therefore, we are going to subtract 0.3121 from 0.484
because an area cannot be a negative value.
So again, based on the values that were provided here,
we can say that there is a 17.19% probability
that a child's cholesterol given the values
will be between 180 and 190.