Hypothesis Test for Difference of Means

Hypothesis Test for Difference of Means

In the last video, we came up
with a 95% confidence interval

for the mean weight loss between
the low-fat group and

the control group.

In this video, I actually want
to do a hypothesis test,

really to test if this data
makes us believe that the

low-fat diet actually does
anything at all.

And to do that let's set up
our null and alternative

hypotheses.

So our null hypothesis
should be that this

low-fat diet does nothing.

And if the low-fat diet does
nothing, that means that the

population mean on our low-fat
diet minus the population mean

on our control should
be equal to zero.

And this is a completely
equivalent statement to saying

that the mean of the sampling
distribution of our low-fat

diet minus the mean of the
sampling distribution of our

control should be
equal to zero.

And that's because we've seen
this multiple times.

The mean of your sampling
distribution is going to be

the same thing as your
population mean.

So this is the same
thing is that.

That is the same
thing is that.

Or, another way of saying it is,
if we think about the mean

of the distribution of the
difference of the sample

means, and we focused on this
in the last video, that that

should be equal to zero.

Because this thing right over
here is the same thing as that

right over there.

So that is our null
hypothesis.

And our alternative hypothesis,

I'll write over here.

It's just that it actually
does do something.

And let's say that it actually
has an improvement.

So that would mean that we
have more weight loss.

So if we have the mean of Group
One, the population mean

of Group One minus the
population mean of Group Two

should be greater then zero.

So this is going to be a one
tailed distribution.

Or another way we can view it,
is that the mean of the

difference of the distributions,
x1 minus x2 is

going to be greater then zero.

These are equivalent
statements.

Because we know that this is the
same thing as this, which

is the same thing as this,
which is what I

wrote right over here.

Now, to do any type of
hypothesis test, we have to

decide on a level
of significance.

What we're going to do is, we're
going to assume that our

null hypothesis is correct.

And then with that assumption
that the null hypothesis is

correct, we're going to see
what is the probability of

getting this sample data
right over here.

And if that probability is below
some threshold, we will

reject the null hypothesis in
favor of the alternative

hypothesis.

Now, that probability threshold,
and we've seen this

before, is called the
significance level, sometimes

called alpha.

And here, we're going to decide
for a significance

level of 95%.

Or another way to think about
it, assuming that the null

hypothesis is correct, we want
there to be no more than a 5%

chance of getting this
result here.

Or no more than a 5% chance of
incorrectly rejecting the null

hypothesis when it
is actually true.

Or that would be a
type one error.

So if there's less than a 5%
probability of this happening,

we're going to reject
the null hypothesis.

Less than a 5% probability given
the null hypothesis is

true, then we're going to reject
the null hypothesis in

favor of the alternative.

So let's think about this.

So we have the null
hypothesis.

Let me draw a distribution
over here.

The null hypothesis says that
the mean of the differences of

the sampling distributions
should be equal to zero.

Now, in that situation, what
is going to be our critical

region here?

Well, we need a result, so
we're going to need some

critical value here.

Because this isn't a
normalized normal

distribution.

But there's some critical
value here.

The hardest thing is statistics
is getting the

wording right.

There's some critical value here
that the probability of

getting a sample from this
distribution above that value

is only 5%.

So we just need to figure out
what this critical value is.

And if our value is larger than
that critical value, then

we can reject the
null hypothesis.

Because that means the
probability of getting this is

less than 5%.

We could reject the null
hypothesis and go with the

alternative hypothesis.

Remember, once again, we can
use Z-scores, and we can

assume this is a normal
distribution because our

sample size is large for either
of those samples.

We have a sample size of 100.

And to figure that out, the
first step, if we just look at

a normalized normal distribution
like this, what

is your critical Z value?

We're getting a result
above that Z value,

only has a 5% chance.

So this is actually
cumulative.

So this whole area right
over here is

going to be 95% chance.

We can just look
at the Z table.

We're looking for 95% percent.

We're looking at the
one tailed case.

So let's look for 95%.

This is the closest thing.

We want to err on the side of
being a little bit maybe to

the right of this.

So let's say 95.05
is pretty good.

So that's 1.65.

So this critical Z value
is equal to 1.65.

Or another way to view it is,
this distance right here is

going to be 1.65 standard
deviations.

I know my writing
is really small.

I'm just saying the standard
deviation of that

distribution.

So what is the standard
deviation of that

distribution?

We actually calculated it in
the last video, and I'll

recalculate it here.

The standard deviation of our
distribution of the difference

of the sample means is going to
be equal to the square root

of the variance of our
first population.

Now, the variance of our first
population, we don't know it.

But we could estimate it with
our sample standard deviation.

If you take your sample standard
deviation, 4.67 and

you square it, you get
your sample variance.

And so this is the variance.

This is our best estimate
of the variance of the

population.

And we want to divide that
by the sample size.

And then plus our best estimate
of the variance of

the population of group two,
which is 4.04 squared.

The sample standard deviation
of group two squared.

That gives us variance
divided by 100.

I did before in the last. Maybe
it's still sitting on my

calculator.

Yes, it's still sitting
on the calculator.

It's this quantity
right up here.

4.67 squared divided
by 100 plus 4.04

squared divided by 100.

So it's 0.617.

So this right here is
going to be 0.617.

So this distance right
here, is going to

be 1.65 times 0.617.

So let's figure out
what that is.

So let's take 0.617
times 1.65.

So it's 1.02.

This distance right
here is 1.02.

So what this tells us is, if
we assume that the diet

actually does nothing, there's a
only a 5% chance of having a

difference between the means of
these two samples to have a

difference of more than 1.02.

There's only a 5%
chance of that.

Well, the mean that we
actually got is 1.91.

So that's sitting out
here someplace.

So it definitely falls in
this critical region.

The probability of getting this,
assuming that the null

hypothesis is correct,
is less than 5%.

So it's smaller probability than
our significance level.

Actually, let me
be very clear.

The significance level,
this alpha right

here, needs to be 5%.

Not the 95%.

I think I might have
said here.

But I wrote down the
wrong number there.

I subtracted it from
one by accident.

Probably in my head.

But anyway, the significance
level is 5%.

The probability given that the
null hypothesis is true, the

probability of getting the
result that we got, the

probability of getting that
difference, is less than our

significance level.

It is less than 5%.

So based on the rules that we
set out for ourselves of

having a significance level of
5%, we will reject the null

hypothesis in favor of the
alternative that the diet

actually does make you
lose more weight.