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Hi. In this video we are going to continue

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to talk about inference.

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

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tests in R.

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So the general hypothesis
testing procedure

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is we always state hypotheses
about your parameter.

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We collect some data.

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We construct a test statistic.

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We then apply a decision
rule so we can either

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do that through a critical value

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

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Or with p-values.

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And then we will draw

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conclusions in context.

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

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

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And see like we're interested in one.

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And try to hypothesize
that we think that the average

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petal length for iris flowers

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is four centimeters. So,

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our null hypothesis would be

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that average,

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petal length is equal to four centimeters.

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And our alternative will be average

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petal length is not equal
to four centimeters.

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Okay.

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The data we are going to use

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is the iris petal length data.

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So it's from the iris dataset.

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And this is the petal length and variable.

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Just to kind of remind us, it is just 150

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observations of different irises.

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To construct our test statistic

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we will first need an xbar value,

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which we can find by taking

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the mean of our sample.

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So the mean of the iris of petal length.

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Which will be 3.758.

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We also are going to need
the hypothesized value

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

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

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

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We also need to know

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the sample standard deviation, s.

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And so you can get that by
doing this standard deviation

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of the variable.

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That value is 1.765.

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

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So, n. So we will reduce
the length function.

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

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which is 50.

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Now once we have all of those

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individual pieces we can
build the test statistic.

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

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

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

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And how we create

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that is we do xbar minus
mu in the numerator

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divided by I'm just gonna
put this in parentheses

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as well. S divided by
the square root of n.

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So thankfully we have all of these pieces
already xbar, mu,

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s, and n. S, n, and xbar
all come from the data.

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Mu is the value we specified
in our null hypothesis.

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And this will compute
our test statistic for us,

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which is

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-1.67897. So.

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Here we go.

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So our next step

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is to apply a decision rule.

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So we have two different 
ways we can do that. We'll-

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We will use a significance level

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or an alpha of 0.05.

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

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And then if we want to

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calculate a rejection region,

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

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Rejection region.

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We can find which critical value

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will give us a tail probability of 0.0 uh-

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

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we'll do our alpha divided by two.

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I'll kind of show you.

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So our rejection region

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

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

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is equal to alpha over two.

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

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

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

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

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of the mean on the curve,

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we will go ahead and say
lower.tail equals true.

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

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

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equals false because we 
want the upper tail.

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We want kind of the extremes.

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

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So what this will tell us

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is our, oh, alpha not found.

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I forgot to run that line. There we go.

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Okay.

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So our rejection value is one, -1.976.

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

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is equal to -1.976 or less,

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or if it's greater than positive 1.976,

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then we will reject our null hypothesis.

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And in this case, since our test

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

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we will fail to reject
our null hypothesis.

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So this is telling us that,

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we will fail to reject

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

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

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

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And since we are doing a

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two sided hypothesis test,

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we will, can do two times
whatever probability

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

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

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by two.

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

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in our test statistic that we get.

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The degrees of freedom again

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and again we're going to do lower.tail

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equals true because our
original test statistic is negative.

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So we want a lower tail
like the extreme value.

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

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p-va- two sided hypothesis test.

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And then this is the value
that we compare to

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our alpha, which is 0.05.

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So if our p-value is less than the alpha

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less than 0.05, we would
reject the null hypothesis.

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In this case our p-value 
is greater than 0.05.

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

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You should get the same conclusion.

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With either method, you should be

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coming to the same
reject or fail to reject.

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You should not be getting
different conclusions.

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So that's how you can kind of

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compute a hypothesis test by hand.

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

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So there is a function t.test

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which may be familiar from when we did.

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Confidence intervals for means.

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And this is actually you can

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do confidence intervals plus
hypothesis testing in here.

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So we still are going to
have the same null.

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And I turned it off
hypotheses from up here.

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

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

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which is the petal length of iris flowers.

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We need to specify what our

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null hypothesis new value is.

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

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petal length is four.

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So we will say mu is equal to four.

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And then we also need to specify that our,

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our, that our alternative hypothesis is a

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two sided hypothesis test.

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Okay.

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And if we go ahead and run that.

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

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We have one sample and a t-test.

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It gives us a t which
is our test statistic

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which should match what we got up here.

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And it does.

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The degrees freedom is pretty easy.

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150 minus one. And then here's a p-value

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

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And then

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you can kind of see
they have xbar right here.

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

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So this is an, quick and easy way

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that you can compute a t-test for me.

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

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

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by computing a p-value for you.

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If you wanted to change what your,

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your null hypothesis was.

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

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You could totally do that.

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

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Or if you wanted to 
change your alternative.

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

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maybe less or greater than. So

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you could do it like this.

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You can do less or

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greater and that'll tell you,

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which, that'll

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change the output of your hypothesis test,

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