Hypothesis Testing for Means

Hi. In this video we are going to continue

to talk about inference.

But now we're going to be talking about
how you can conduct hypothesis

tests in AR.

So the general hypothesis
testing procedure

is we always state hypotheses
about your parameter.

We collect some data.

We construct a test statistic.

We then apply a decision rule
so we can either

do that through a critical value

or with p values
or like a critical region, excuse me.

Or with p values.

And then we will draw

conclusions in context.

So the first research question
we're going to talk about

today is we're going to continue
using the idea of iris flowers.

And see like we're interested in one.

And try to hypothesize
that we think that the average

petal length for iris flowers

is four centimeters. So,

our null hypothesis would be

that average,

petal length is equal to four centimeters.

And our alternative will be average

petal length is not equal
to four centimeters.

Okay.

The data we are going to use

is the iris petal length data.

So it's from the iris dataset.

And this is the petal length and variable.

Just to kind of remind us it is just 150

observations of different irises.

To construct our test statistic

we will first need an exposure value,

which we can find by taking

the mean of our sample.

So the mean of the iris of petal length.

Which will be 3.758.

We also are going to need
the hypothesized value

that we are wanting to hypothesize,
which is four centimeters.

So I'm going to just call that mu
because that's the parameter of interest.

We're going to say it's equal to four.

We also need to know

the sample standard deviation s.

And so you can get that by doing this
standard deviation

of the variable.

That value is 1.765.

And then we also need to know
the number of observations.

So and so we will reduce
the length function.

And then I'll count
how many observations are in your data set

which is 50.

Now once we have all of those

individual pieces
we can build the test statistic.

Since we are doing a hypothesis test
for a mean, we will be constructing

what is known as like a
t, a test statistic for a t distribution.

So I'm going to call it t test stat.

And how we create

that is we do x bar
minus mu in the numerator

divided by I'm
just gonna put this in parentheses

as well S divided by the square root of n.

So thankfully we have all of these pieces
already x bar mu

as an n, s and an x bar
all come from the data.

Mu is the value
we specified in our null hypothesis.

And this will compute
our test statistic for us,

which is

-1.67897. So.

We'll.

So our next step

is to apply a decision rule.

So we have two different ways
we can do that. Will

we will use a significance level

or an alpha of 0.05.

So I'm just going to go ahead
and set that.

And then if we want to

calculate a rejection region

because there's two different
kinds of decision rules we can do.

Rejection region.

We can find which critical value

will give us a tail probability of 0.0.

Or since we're doing a two sided
hypothesis test,

we'll do our alpha divided by two.

I'll kind of show you.

So our rejection region

is we're going to try
we're going to find the critical value

that, fits the t distribution,
where the probability in the tail

is equal to alpha over two.

Because we're doing a two sided interval
hypothesis test.

Our degrees of freedom is needed
for the t test, which is n minus one.

And since we are our test statistic
with a negative value,

meaning that it's on the left side of the,

of the mean on the curve,

we will go ahead and say
lower that tail equals true

because we want the lower tailed like
or the smaller the tail end probability.

If this is a positive number 1.67
we would then do lower dot

tail equals false
because we want the upper tail

we want kind of the extremes.

So anything from where
our test statistic is and more extreme.

So what this will tell us

is our oh Alpha not found.

I forgot to run that line. There we go.

Okay.

So our rejection value is one -1.976.

So what this is telling us
is that if our test statistic

is equal to -1.976 or less,

or if it's greater than positive 1.976,

then we will reject our null hypothesis.

And in this case, since our test

statistic is not in the extreme,
it's actually greater than this value,

we will fail
to reject our null hypothesis.

So this is telling us that,

we will fail to reject

our null, meaning that
we do not have enough evidence to conclude

that the average petal length
is not equal to four centimeters.

The other way you can apply a decision
rule is with a p value.

And since we are doing a

two sided hypothesis test,

we will can do two times
whatever probability

we get because we're going
to be calculating it for one tail.

But since we're doing two sided
we'll just need to multiply it

by two.

And so what we're going to put in here
is we're going to put

in our test statistic that we get.

The degrees of freedom again

and again we're going to do lower tail

equals true because our original test
statistic is negative.

So we want a lower tail
like the extreme value.

And then we're going to multiply by
two again because we are doing a two sided

piece two sided hypothesis test.

And then this is the value
that we compare to

our alpha which is 0.05.

So if our p value is less than the alpha

less than 0.05,
we would reject the null hypothesis.

In this case
our p value is greater than 0.05.

So we would fail to reject our null
hypothesis again as well.

You should get the same conclusion.

With either method, you should be

coming to the same reject
or fail to reject.

You should not be getting
different conclusions.

So that's how you can kind of

compute a hypothesis test by hand.

But as always, usually in R
there is an easier way to do it.

So there is a function t test

which may be familiar from when we did.

Confidence intervals for means.

And this is actually you can

do confidence intervals
plus hypothesis testing in here.

So we still are going to have
the same null.

And I turned it off
hypotheses from up here.

And so what we're going to do
is we're going to just say t dot test,

give it the data
that we are doing the t test on,

which is the petal length of iris flowers.

We need to specify what our,

null hypothesis new value is.

We're saying that we are hypothesizing
that the true, average

petal length is four.

So we will say mu is equal to four.

And then we also need to specify that our

our that our alternative hypothesis is a

two sided hypothesis test.

Okay.

And if we go ahead and run that.

And notice it shows it is a one
sample t test which is perfect.

We have one sample and a to t test.

It gives us eight t
which is our test statistic

which should match what we got up here.

And it does

the degrees freedom is pretty easy.

150 minus one. And then here's a P.

Same exact P-value
we got here by doing a by hand.

And then

you can
kind of see they have X bar right here.

And then it also gives you
that 95% confidence interval.

So this is an, quick and easy way

that you can compute a t test for me.

You can this is kind of showing you
how to do it all by hand.

And then this will show you kind of
how to just do it in one simple step

by computing a p value for you.

If you wanted to change what your,

your null hypothesis was.

So say, like you were testing,
is the mean equal to two instead?

You could totally do that.

And then you can see that
this p value is way, way smaller.

Or if you wanted
to change your alternative.

So it's not that it's
just not equal to four and it's, you know,

maybe less or greater than. So

you could do it like this.

You can do less or

greater and that'll tell you,

which that'll

change the output of your hypothesis test,

kind of depending on if you're doing a
one sided or two sided test.