Small Sample Size Confidence Intervals

7 patients blood pressures
have been measured after

having been given a new
drug for 3 months.

They had blood pressure
increases of, and they give us

seven data points right here--
who knows, that's in some

blood pressure units.

Construct a 95% confidence
interval for the true expected

blood pressure increase for all
patients in a population.

So there's some population
distribution here.

It's a reasonable assumption
to think that it is normal.

It's a biological process.

So if you gave this drug to
every person who has ever

lived, that will result in some
mean increase in blood

pressure, or who knows, maybe
it actually will decrease.

And there's also going to be
some standard deviation here.

It is a normal distribution.

And the reason why it's
reasonable to assume that it's

a normal distribution
is because it's

a biological process.

It's going to be the sum of many
thousands and millions of

random events.

And things that are sums of
millions and thousands of

random events tend to be
normal distribution.

So this is a population
distribution.

And we don't know anything
really about it outside of the

sample that we have here.

Now, what we can do is, and this
tends to be a good thing

to do, when you do have a
sample just figure out

everything that you can
figure out about that

sample from the get-go.

So we have our seven
data points.

And you could add them up and
divide by 7 and get your

sample mean.

So our sample mean
here is 2.34.

And then you can also
calculate your

sample standard deviation.

Find the square distance from
each of these points to your

sample mean, add them up, divide
by n minus 1, because

it's a sample, then take the
square root, and you get your

sample standard deviation.

I did this ahead of time
just to save time.

Sample standard deviation
is 1.04.

And when you don't know anything
about the population

distribution, the thing that
we've been doing from the

get-go is estimating that
character with our sample

standard deviation.

So we've been estimating the
true standard deviation of the

population with our sample
standard deviation.

Now in this problem, this exact
problem, we're going to

run into a problem.

We're estimating our standard
deviation with an n of only 7.

So this is probably going to
be a not so good estimate

because-- let me just write--
because n is small.

In general, this is considered
a bad estimate if n

is less than 30.

Above 30 you're dealing
in the realm

of pretty good estimates.

So the whole focus of this video
is when we think about

the sampling distribution, which
is what we're going to

use to generate our interval,
instead of assuming that the

sampling distribution is normal
like we did in many

other videos using the central
limit theorem and all of that,

we're going to tweak the
sampling distribution.

We're not going to assume it's
a normal distribution because

this is a bad estimate.

We're going to assume that
it's something called a

t-distribution.

And a t-distribution is
essentially, the best way to

think about is it's almost
engineered so it gives a

better estimate of your
confidence intervals and all

of that when you do have
a small sample size.

It looks very similar to
a normal distribution.

It has some mean, so this is
your mean of your sampling

distribution still.

But it also has fatter tails.

And the way I think about why
it has fatter tails is when

you make an assumption that this
is a standard deviation

for-- let me take
one more step.

So normally what we do is we
find the estimate of the true

standard deviation, and then
we say that the standard

deviation of the sampling
distribution is equal to the

true standard deviation of our
population divided by the

square root of n.

In this case, n is equal to 7.

And then we say OK, we never
know the true standard, or we

seldom know-- sometimes you do
know-- we seldom know the true

standard deviation.

So if we don't know that the
best thing we can put in there

is our sample standard
deviation.

And this right here, this is the
whole reason why we don't

say that this is just a 95
probability interval.

This is the whole reason why
we call it a confidence

interval because we're making
some assumptions.

This thing is going to change
from sample to sample.

And in particular, this is going
to be a particularly bad

estimate when we have a
small sample size, a

size less than 30.

So when you are estimating the
standard deviation where you

don't know it, you're estimating
it with your sample

standard deviation, and your
sample size is small, and

you're going to use this to
estimate the standard

deviation of your sampling
distribution, you don't assume

your sampling distribution
is a normal distribution.

You assume it has
fatter tails.

And it has fatter tails because
you're essentially

underestimating-- you're
underestimating the standard

deviation over here.

Anyway, with all of that said,
let's just actually go through

this problem.

So we need to think about a 95%
confidence interval around

this mean right over here.

So a 95% confidence interval,
if this was a normal

distribution you would just
look it up in a Z-table.

But it's not, this is
a t-distribution.

We're looking for a 95%
confidence interval.

So some interval around
the mean that

encapsulates 95% of the area.

For a t-distribution you use
t-table, and I have a t-table

ahead of time right over here.

And what you want to do is use
the two-sided row for what

we're doing right over here.

And the best way to think
about it is that we're

symmetric around the mean.

And that's why they
call it two-sided.

It would be one-sided if it
was kind of a cumulative

percentage up to some
critical threshold.

But in this case, it's
two-sided, we're symmetric.

Or another way to think
about it is we're

excluding the two sides.

So we want the 95%
in the middle.

And this is a sampling
distribution of the sample

mean for n is equal to 7.

And I won't go into the details
here, but when n is

equal to 7 you have 6 degrees
of freedom, or n minus 1.

And the way that t-tables are
set up, you go and find the

degrees of freedom.

So you don't go to the n,
you go to the n minus 1.

So you go to the 6 right here.

So if you want to encapsulate
95% of this right over here,

and you have an n of 6, you
have to go 2.447 standard

deviations in each direction.

And this t-table assumes that
you are approximating that

standard deviation using your
sample standard deviation.

So another way to think of it
you have to go 2.447 of these

approximated standard
deviations.

Let me it right here.

So you have to go 2.447-- this
distance right here is 2.447

times this approximated
standard deviation.

And sometimes you'll see this
in some statistics book.

This thing right here,
this exact number,

is shown like this.

They put a little hat on top of
the standard deviation to

show that it has been
approximated using the sample

standard deviation.

So we'll put a little hat over
here, because frankly, this is

the only thing that
we can calculate.

So this is how far you have
to go in each direction.

And we know what
this value is.

We know what the sample
distribution is.

So let's get our
calculator out.

So we know our sample standard
deviation is 1.04.

And we want to divide that
by the square root of 7.

So we get 0.39.

So this right here is 0.39.

And so if we want to find
the distance around this

population mean that
encapsulates 95% of the

population or of the sampling
distribution, we have to

multiply 0.39 times 2.447,
so let's do that.

So times 2.447 is
equal to 0.96.

So this is equal to-- so this
distance right here is 0.96,

and then this distance
right here is 0.96.

So if you take a random sample,
and that's exactly

what we did when we found
these 7 samples.

When we took these 7 samples and
took their mean, that mean

can be viewed as a random
sample from the sampling

distribution.

And so the probability, and so
we can view it, we could say

that there's a 95% chance-- and
we have to actually caveat

everything with a confident,
because we're doing all of

these estimations here.

So it's not a true precise
95% chance.

We're just confident that
there's a 95% chance that our

random population, our random
sampling mean right here, so

that 2.34, which we can kind of
use-- we just picked that

2.34 from this distribution
right here.

So there's a 95% chance that
2.34 is within 0.96 of the

true sampling distribution mean,
which we know is also

the same thing as the
population mean.

Or we can just rearrange the
sentence and say that there is

a 95% chance that the mean, the
true mean, which is the

same thing as a sampling
distribution mean, is within

0.96 of our sample
mean, of 2.34.

So at the low end, so if you go
2.36 minus-- if you go 2.34

minus 0.96-- that's the low
end of our confidence

interval, 1.38.

And the high end of our
confidence interval, 2.34 plus

0.96 is equal to 3.3.

So our 95% confidence interval
is from 1.38 to 3.3.