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.