Hypothesis Testing for Proportions

Hi. This video is a continuation
of our hypothesis testing video.

We just finished an example of
doing hypothesis testing for means.

One sample.

Now we're going to talk about a one
sample hypothesis test for proportions.

Up here, in our previous example,

I talked about the major
steps that you take.

You state your hypotheses, collect
data, construct a test statistic.

Apply a decision rule,
either a critical region

or a p value, and draw
conclusions in context.

So we're going to follow those
exact same steps again here.

So I'm going to say this
is for proportions now.

Our null hypothesis is well

I guess the research question is- is
we're going to be interested in seeing

if, the proportion of red haired

people is equal to point one five. So,

proportion of red

haired people is point one five.

And the alternative we're just
going to do, it's simply a two sided.

So proportion of red haired people is not

point one five, okay.

The data we are going to be using
is from the hair eye color dataset.

Same dataset we used in our
confidence interval section.

So as you can see here,
it has a table of different hair

colors, eye colors and
different sexes for some students.

So to get the total number of red
haired people, you just sum up

all of the values of red haired people.

So ten plus ten plus seven plus
seven, and then sixteen plus seven

plus seven, plus seven.

will give you seventy-one.

Okay.

I'm just gonna put a note here.

seventy-one red haired people.

Okay.

So a test statistic for
a proportion is given.

You're going to need to have a,

a p hat value.

You're going to need to
have a hypothesized value.

And you're going to need to know
the number of observations.

So I'm going to do the total
number of observations first.

So we're going to

just we're not actually going to use
the length function in this case.

Since this is in a table,
I'm going to just sum up

all of the numbers in this whole table.

And that'll give us the total number
of observations in this data set,

which is five-hundred-and-ninety-two.

And then our, p hat is
our sample proportion.

So our sample we had seventy-one

red- red haired people.

And in order to get a proportion,

we need to divide that by
the total number of people.

So that'll be seventy-one
divided by five ninety-two,

which is also equal to
point one one nine nine.

And then the last thing
we will need is wherever

our hypothesized proportion is,
usually denoted like with p naught.

So I'm going to call it p zero.

And this is chosen by us the
researcher or the statistician.

And it is point one five.

Okay.

So now since we usually use

a standard normal table or a
normal distribution for proportions,

I'm going to call this
the r z test statistic.

And for a proportion, it's

we will calculate it as being p hat

minus p naught,
the hypothesized value

divided by the square root of

Now there's a numerator and denominator

inside this.

Inside this square root.

So it's going to have
a lot of parentheses.

But if you just are very aware of all
your parentheses, it'll be just fine.

So this is

p naught times
one minus p naught.

Then outside of

But let's see.

All right.

So we have closed the
parentheses for this one.

And then.

Close

the parentheses for that p naught
times one minus p naught.

So it's still inside the square root.

We will also divide that by n
and then this parentheses

will close off the end of the square root.

And that'll close off the
end of the denominator.

And that will give us.

A value of negative two
point zero four eight eight

So that is our test statistic.

For this hypothesis test.

Now to apply a decision rule

we're going to start off
with our rejection region.

Before we do that, we always have

the researcher set our preferred
or a set significance level,

usually it's point zero five,
but obviously that is up to you.

So we are going to
find a rejection region.

And we are going to be
using the Q norm functions

because we use the normal
distribution when doing

any kind of inference
with proportions.

And we want to find a tail
probability that is our significance

level divided by two because we are
doing a two sided hypothesis test.

So we want the two tails
to each have alpha over two.

And since our test statistic is
negative means--meaning that

it's on the left hand side of the
curve, we want to get a lower or a,

left hand probability.

Tail probability.

We're going to say lower
dot tail equals true.

There we go, and then our value

where we would reject is
negative one point nine six.

So this tells us that if we got a test,

if we ever get a test statistic that is
less than negative one point nine six

or greater than positive
one point nine six,

we will reject our null hypothesis.

Our test statistic is smaller

than negative one point nine six.

So we would reject our null
hypothesis and say that

the true proportion of red haired
people is not point one five.

Our sample we got was point one two.

So you know, point two, point one two,
point one five, you know, pretty close.

But it's not exactly point one five.

It's probably something different

then if you want to do, a finding

you're doing your decision rule.

You can also do it with p values.

So we are doing a two sided test again.

So we will also multiply this and p

the p like tail value
probability value by two.

And the critical value
that we will plug in here

to find the probability
is z dot test dot stat.

The one we got from our data.

And again we will say
lower dot tail equals true

because we are finding a
extreme or tail probability.

So our two sided p value is
point zero four zero five.

And so this is where we compare it to
our alpha value which is point zero five.

So if our p value is less
than alpha point zero five

that means we reject
our null hypothesis.

This value is just barely, but it
is still less than point zero five.

So in this case we would
reject our null hypothesis again.

And say that the true proportion of
red haired people is not point one five.

Where, whatever decision rule
you applied, whether it's rejection

region or a p value, you should be
coming to the same conclusion.

On whether to reject or fail
to reject your null hypothesis.

You shouldn't be getting
different conclusions with these.

And there are other
functions, kind of like

how I talked about the confidence
intervals for proportion video.

There are, functions
out there that can do

a one sample proportion test,

because, you know, for means
you can easily just use t dot test.

But there are ones for
proportions out there,

but they do require you to
download other packages

and kind of do the research and
find out those other functions.

And so just to kind of make it as
simple as possible for you guys

so you don't have to go find
that all that information.

We thought we would just show you
how to do it by hand here, and,

you'll get the same conclusions.

So--

Anyway, hopefully that this
these videos were helpful for you

in helping out with hypothesis
testing for means and for proportions.

All right.

We will see you later in the next r video.

Have a good one.