Instructor: Hello, students.
Now we are in part C, which is slightly different.
So first I'm going to read the problem again.
So, it's problem three.
Total cholesterol in children age 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 in part C, the question is if a sample
of 20 children is selected, what is the probability
that the mean cholesterol level in the sample
will exceed 200?
So what is going on here is that
now instead of looking at the entire population,
we are looking at a sample of 20 children.
So we will draw the normal distribution,
just like we did before,
but now instead of the total population,
this will be our sampling distribution of means.
And as per the central limit theorem,
the mean of our sampling distribution
is going to be equal to our mu, our population mean,
and the standard error of that distribution
will be equal to the sigma,
or population standard deviation,
divided by the square root of our sample size,
which is here, 20.
So, without further delay I'm going to start
writing some of this information.
So here X-bar is equal to mu,
and SE, which is the abbreviation for standard error,
is equal to sigma divided by square root of 20,
which is our end.
So now I'm going to draw our normal distribution,
because nothing changes there.
We are going to draw that.
So here now is 191,
but this now will be an X-bar.
Okay?
Now, we will do exactly the same thing as we did before
because looking at the probability
that we are trying to find out the value greater than 200.
So just like before, we will have to transform it
to the standard normal distribution
so that we can use our table in the back of our book
to find the probability.
So now, again, I'm going to draw the area
we are looking for under the curve,
which is, again, that 200.
Kind of seen this before.
Okay, this is the area.
And now we have transformed this to
the distribution, and this is zero.
Okay, so we have our Z-axis here with a mean of zero,
and we need to now calculate the values.
Again, this time the formula
is going to be a little different.
So it's going to be Z equal to
X-bar minus mu
divided by sigma and square root of N.
So, again, the probability we are looking for is the
value above 200, and we have to go through
all the transformation as before.
So now it is Z greater than.
So we are going to use the formula here.
Square root of 20.
Alright, so once we do the algebra, what we find is
this will be 1.80.
Now, again, we have to go to the back of our text,
and what our text will provide us, as before,
is the value less than.
So this is what we are getting
from the back of our textbook.
So, again, going back to our normal distribution
and using a different color,
this is what we have right now.
Oops, color didn't work well.
Let's see, yes.
So this is the area for which we have the value,
and we need to find the value
for the area that's kind of highlighted in red.
So what do we do here, again?
Thankfully we know
that the entire area under the curve is one.
So basically we have to deduct it from one.
So let's do that without any further delay.
So P, probability
that the Z is greater than 1.80
is basically one minus the probability
that the Z less than 1.80.
So the what's gonna then be one minus point-
Oops, sorry.
It should be 0.9641,
and this will be
0.0359.
And that is basically
3.59%.
So what does that mean?
So, again, given the values,
if we take a sample of 20 children from this population,
then there is a 3.59% probability
that the mean cholesterol of those 20 children
will be greater than 200.