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.
