# hypothesis test for proportions

#### Steve

##### New Member
I have two proportions p1 and p2. I believe p2 is not significantly less than p1 and I want to confirm by hypothesis.

Question: is p2 ≥ p1?
Opposite: p2 < p1.

Please confirm that I would then use a one-tailed z-test with:
H0 : p2 = p1
HA : p2 < p1

Is it a problem that I want H0 to not be rejected. I.e. my belief is aligned with H0 rather than HA? Should I set this up differently?

Any explanation as to why the hypothesis are setup the way they are would be appreciated. I am new to this domain.

Thanks,

#### Mean Joe

##### TS Contributor
I have two proportions p1 and p2. I believe p2 is not significantly less than p1 and I want to confirm by hypothesis.

Question: is p2 ≥ p1?
Opposite: p2 < p1.

Please confirm that I would then use a one-tailed z-test with:
H0 : p2 = p1
HA : p2 < p1
Yes, you'd want a one-tailed test. I think technically, your H is p2 >= p1.

Is it a problem that I want H0 to not be rejected. I.e. my belief is aligned with H0 rather than HA? Should I set this up differently?
That's not a problem. If cigarette companies want to do testing to see if smoking causes cancer (or has some kind of association? with cancer), they would do something like
H: cigarettes do not cause cancer
H[a]: cigarettes do cause cancer
And they would not want H to be rejected.

The null hypothesis is the hypothesis that "nothing out of the ordinary is happening". If you want to test: "grades at a school are lower than in the general district", then H: grades at school >= grades in general district.
etc.

Hope this helps, if not I can try again.

#### Steve

##### New Member
I think technically, your H is p2 >= p1.
In statistics references I have read that H should always be specified as "=" even if "≥" or "≤". I assume that in my case testing "=" is the same as testing "≥" since it is a one-tailed test?

#### Mean Joe

##### TS Contributor
In statistics references I have read that H should always be specified as "=" even if "≥" or "≤". I assume that in my case testing "=" is the same as testing "≥" since it is a one-tailed test?
This is just personally how I understand hypothesis tests; I'm not going to dispute your texts. But the way I determine if it's a one- or two-tailed test depends on H: If H specifies "=", then you must do two-tailed test. Do a one-tailed test if it specifies "≥" or "≤".

Hypothesis testing follows a procedure of
1) specify the null & alternative hypotheses
2) collect sample data
3) test the data, to determine if null hypothesis is accepted or rejected.

I think that hypothesis tests must present an either/or situation. So if H: p1 = p2, then H[a] must be: p1 not equal p2. So then you collect data, and when you do your test (which would be two-tailed, because of your H), it would be possible to reject the null hypothesis if your sample data has p2 > p1. Which is not quite what you want (you want to test if there is enough evidence to say that p1 > p2)

#### JohnM

##### TS Contributor
The null hypothesis is always stated as "=" since the term "null" implies "no difference." In other words, Ho: u1 = u2 could also be stated as Ho: u1 - u2 = 0.

In a non-directional test, the alternative hypothesis is "not =" and in a directional test, the alternative is "<=" or ">=".

#### Martingale

##### TS Contributor
for a one-tailed test using "≥" or "≤" is fine.

In a more abstract sense ...if we let $\Theta$ be the parameter space then the general form a test is...(written in latex)

$H_0:\theta \in \Theta_0$ and $H_1:\theta \in \Theta_0^c$

where $\Theta_0$ is a subset of the parameter space $\Theta$ and
$\Theta_0^c$ is the complement in $\Theta$ of $\Theta_0$