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.

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 this 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 0.15. So,

proportion of red

haired people is 0.15.

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

So proportion of red haired people is not

.15 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 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 16 plus seven

plus seven,

will give you 71.

Okay.

I'm just gonna put a note here.

71 red two.

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 592.

And then our, p
hat is our sample proportion.

So our sample we had 71

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 71 divided by five, 92,

which is also equal to 2.1199.

And then the last thing we will need
is whatever

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 station.

And it is point one of 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, not 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 what.

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 -2.0488.

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 0.05,
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

and 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
mean 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 up
tail equals true.

To and then our value

where we would reject is -1.96.

So this tells us that if we got a test,

if we ever get a test statistic
that is less than -1.96

or greater than positive 1.96,

we will reject the null hypothesis.

Our test statistic is smaller

than -1.96.

So we would reject our null hypothesis
and say that

the true proportion of red haired
people is 0.15.

Our sample we got was 0.12.

So you know, .2.1, 2.15,
you know, pretty close.

But it's not exactly 0.15.

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 of stat.

The one we got from our data.

And again
we will say lower tail equals true

because we are finding a
extreme or tail probability.

So our

two sided p value is 0.0405.

And so this is where we compare it
to our alpha value which is 0.05.

So if our p value is less than alpha .05

that means we reject our null hypothesis.

This value is just barely,
but it is still less than 0.05.

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

And see that the true proportion of red

haired people is 0.15.

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 the 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 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 out
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 video.

Have a good one.