Sampling Distribution Example Problem

The average male drinks 2 liters of water when active
outdoors with the standard deviation of 0.7 liters.
You are planning a full day nature trip for 50 men and
will bring 110 liters of water.
What is the probability that you will run out of water?
So let's think about what's happening here.
So there's some distribution of how many liters an average
man needs when they're active outdoors.
And let me just draw an example.
It might look something like this.
So they're all going to need at least more than 0 liters,
so this would be 0 liters over here.
The average male, so the mean of the amount of water a man
needs when active outdoors is 2 liters.
So 2 liters would be right over here.
So the mean is equal to 2 liters.
It has a standard deviation of 0.7 liters or 0.7 liters.
So the standard deviation-- maybe I'll draw it this way.
So this distribution, once again, we don't know whether
it's a normal distribution or not.
It could just be some type of crazy distribution.
So maybe some people need almost close to-- well,
everyone needs a little bit of water, but maybe some people
need very, very little water.
Then you have a lot of people who need that, maybe some
people who need more, and no one can drink more than maybe
this is like 4 liters of water.
So maybe this is the actual distribution.
And then one standard deviation is going to be 0.7
liters away.
So this is 1, 0.7 liters is-- so this would be 1 liter, 2
liters, 3 liters.
So one standard deviation is going to be about that far
away from the mean.
If you go above it it'll be about that far, if
you go below it.
So let me draw.
This is the standard deviation.
That right there is the standard deviation to the
right, that's the standard deviation to the left.
And we know that the standard deviation is equal to-- I'll
write the 0 in front, 0.7 liters.
So that's the actual distribution of how much water
the average man needs when active.
Now what's interesting about this problem, we are planning
a full day nature trip for 50 men and will bring
110 liters of water.
What is the probability that you will run out?
So the probability that you will run out-- let
me write this down.
The probability that I will or that you will run out is equal
or is the same thing as the probability that we use more
than 110 liters on our outdoor nature day,
whatever we're doing.
Which is the same thing as the probability, if we use more
than 110 liters, that means that on average, because we
had 50 men, so 110 divided by 50 is what?
That's 2.-- let me get the calculator out just so we
don't make any mistakes here.
So this is going to be, the calculator out.
So on average, if we have 110 liters that's going to be
drunk by 50 men, including ourselves I guess, that means
that it's the-- so we would run out if on average more
than 2.2 liters is used per man.
So this is the same thing as the probability of the
average, or maybe we should say the sample mean-- Or let
me write it this way, that the average water use per man of
our 50 men is greater than, or we could say greater than or
equal to, greater-- well I'll say greater than because if
we're right on the money then we won't run out of water-- is
greater than 2.2 liters per man.
So let's think about this.
We are essentially taking 50 men out of a universal sample.
We got this data, who knows where we got this data from
that the average man drinks 2 liters and that the standard
deviation is this.
Maybe there's some huge study and this was the best estimate
of what the population parameters are.
That this is the mean and this is the standard deviation.
Now we're sampling 50 men.
And what we need to do is figure out essentially what is
the probability that the mean of the sample, that the sample
mean, is going to be greater than 2.2 liters.
And to do that we have to figure out the distribution of
the sampling mean.
And we know what that's called.
It's the sampling distribution of the sample mean.
And we know that that is going to be a normal distribution.
And we know a few of the properties of that normal
distribution.
So this is a distribution of just all men.
And then if you take samples of, say, 50 men, so this will
be-- let me write this down.
So down here I'm going to draw the sampling distribution of
the sample mean when n, so when our sample
size is equal to 50.
So this is essentially telling us the likelihood of the
different means when we are sampling 50 men from this
population and taking their average water use.
So let me draw that.
So let's say that this is the frequency and then here are
the different values.
Now the mean value of this, the mean-- let me write it--
the mean of the sampling distribution of the sample
mean, this x bar-- that's really just the sample mean
right over there-- is equal to, if we were to do this
millions and millions of times.
If we were to plot all of the means when we keep taking
samples of 50, and we were to plot them all out, we would
show that this mean of distribution is actually going
to be the mean of our actual population.
So it's going to be the same value, I'm going to do it in
that same blue.
It's going to be the same value as this
population over here.
So that is going to be 2 liters.
So we still have-- we're still centered at 2 liters.
But what's neat about this is that the sampling distribution
of the sample mean, so you take 50 people, find their
mean, plot the frequency.
This is actually going to be a normal distribution regardless
of-- this one just has a well-defined standard
deviation mean.
It's not normal.
Even though this one isn't normal, this one over here
will be, and we've seen it in multiple videos already.
So this is going to be a normal distribution.
And the standard deviation-- and we saw this in the last
video, and hopefully we've got a little bit of intuition for
why this is true.
The standard deviation-- actually maybe
put it a better way.
The variance of the sample mean is
going to be the variance.
So remember, it's going to be-- this is standard
deviation, so it's going to be the variance of the population
divided by n.
And if you wanted the standard deviation of this distribution
right here, you just take the square root of both sides.
If you take the square root of both sides of that we have the
standard deviation of the sample mean is going to be
equal to the square root of this side over here, is going
to be equal to the standard deviation of the population
divided by the square root of n.
And what's this going to be in our case?
We know what the standard deviation of
the population is.
It is 0.7.

And what is n?
We have 50 men.
So 0.7 over the square root of 50.
Now let's figure out what that is with the calculator.
So we have 0.7 divided by the square root of 50.
And we have 0.09-- well I'll say 0.098-- well it's pretty
close the 0.99.
So I'll just write that down.
So this is equal to 0.099.
That's going to be the standard deviation of this.
It's going to have a lower standard deviation.
So the distribution is going to be normal, it's going to
look something like this.
So this is 3 liters over here, this is 1 liter.
The standard deviation is almost a tenth, so it's going
to be a much narrower distribution.
It's going to look something-- I'm trying my best to draw
it-- it's going to look something like this.

You get the idea.
Where the standard deviation right now is almost 0.1, so
it's 0.09, almost a tenth.
So it's going to be something-- one standard
deviation away is going to look something like that.
So we have our distribution.
It's a normal distribution.
And now let's go back to our question that we're asking.
We want to know the probability that our sample
will have an average greater than 2.2.
So this is the distribution of all of the possible samples.
The means of all of the possible samples.
Now to be greater than 2.2, 2.2 is going to be right
around here.

So we essentially are asking we will run out if our sample
mean falls into this bucket over here.
So we essentially need to figure out what is-- you can
even view it as what's this area under this curve there?
And to figure that out we just have to figure out how many
standard deviations above the mean we are, which is going to
be our Z-score.
And then we could use a Z-table to figure out what
this area right over here is.
So we want to know when we're above 2.2 liters, so 2.2
liters-- we could even do it in our head-- 2.2 liters is
what we care about.
That's right over here.
Our mean is 2, so we are 0.2 above the mean.

And if we want that in terms of standard deviations, we
just divide this by the standard deviation of this
distribution over here.
And we figured out what that is.
The standard deviation of this distribution is 0.099.
So if we take-- and you'll see a formula where you take this
value minus the mean and divide it by the standard
deviation-- that's all we're doing.
We're just figuring how many standard deviations above the
mean we are.
So you just take this number right over here divided by the
standard deviation, so 0.099 or 0.099, and then we get--
let's get our calculator.
And actually we had the exact number over here.
So we can just take 0.2-- we could just take this 0.2
divided by this value over here.
On this calculator when I press second answer it just
means the last answer.
So I'm taking 0.2 divided by this value over
there and I get 2.020.
So that means that this value, or I should write this
probability is the same probability of being 2.02
standard deviations-- or maybe I should write it this way--
more than-- Let me write it down here
where I have more space.
So this all boils down to the probability of running out of
water is the probability that the sample mean will be more
than-- just the 50 that we happened to select-- remember,
if we take a bunch of samples of 50 and plot all of them
we'll get this whole distribution.
But the one 50, the group of 50 that we happened to select,
the probability of running out of water is the same thing as
the probability of the mean of those people, will be more
than 2.020 standard deviations above the mean of this
distribution, which they're actually the same
distribution.
So what is that going to be?
And here we just have to look up our Z-table.
Remember, this 2.02 is just this value right here.
0.2 divided by 0.09.
I just had to pause the video because there's some type of
fighter jet outside or something.
So anyway, hopefully they won't come back.
But anyway, so we need to figure out the probability
that the sample mean will be more than 2.02 standard
deviations above the mean.
And to figure that out we go to a Z-table, and you could
find this pretty much anywhere.
Usually it's in any stat book or on the internet, wherever.
And so essentially we want to know the probability-- the
Z-table will tell you how much area is below this value.
So if you go to z of 2.02-- that was the value that we
were dealing with, right.
You have 2.02, it was-- so you go for the first digit.
We go to 2.0, and it was 2.02.
2.02 is right over there.
So we have 2.0, and then in the next digit you go up here.
So 2.02 is right over there.
So this 0.9783-- let me write it down over here-- this
0.9783-- I want to be very careful.
0.9783, that Z-table, that's not this value over here.
This 0.9783 on the Z-table, that is giving us this whole
area over here.
It's giving us the probability that we are below that value.
That we are less than 2.02 standard
deviations above the mean.
So it's giving us that value over here.
So to answer our question, to answer this probability, we
just have to subtract this from 1 because these will all
add up to 1.
So we just take our calculator back out and we just take 1
minus 0.9783 is equal to 0.0217.
So this right here is 0.0217.
Or another way you could say it, it is a 2.17% probability
that we will run out of water.
And we are done.
Let me make sure I got that number right.
So that number it was, yeah, 0.0217, right.
So it's a 2.17% chance we run out of water.