We're going to wrap up our discussion on working with one unknown

population proportion by talking about doing a hypothesis test for a proportion.

Let's go through the steps for doing a hypothesis test,

these are going to look very similar to what we've seen before.

First, we set our hypothesis.

In this case, our unknown population parameter is denoted at with p,

as opposed to mu for means, so our null hypothesis sets that p

equal to some null value, and our alternative hypothesis says that p

can be less than, greater than, or not equal to that null value.

Next, we calculate our point estimate.

In this case, that's the sample proportion, a p-hat.

Then we check our conditions.

The first condition is independence.

We want to make sure that the

sampled observations are independent of each other.

We, this could either be ensured

through random sampling or random assignment.

Depending on whether you are doing an observational study or an experiment.

And if you're sampling with a replacement, we want the

sample size to be less than 10% of the population.

In terms of sample size ad skew, we want to make sure we

have at least ten expected successes and ten expected failures in our sample.

Note that here I've used p,

instead of p hat, and that is because in a hypothesis

test, we have to assume that the null hypothesis is true.

If you think about the definition of a p value, it says, probability

of observed or mare, more extreme outcome, if the null hypothesis is true.

So, when going through the conditions, or any other portion of the

hypothesis test, we must assume that the null is true, and therefore,

wherever we see a p, we plug in whatever the null

value for that p is, that's set forth in the null hypothesis.

So, we could read this as not ten observed successes and ten observed failures,

but instead as ten expected successes and ten expected failures.

Next step is to draw the sampling distribution.

Remember, we always,

always, always want to draw our curve before we calculate our p

value and we want to shade where the p value belongs to.

Either is it in one tail and if so, is it the upper tail or

the lower tail or is it a two tail test and we calculate our test statistic.

The test statistic is always of the form

observed minus null divided by the standard error.

That's observed sample proportion p hat minus

the null value p that comes from the

null hypothesis divided by the standard error, and

we calculate that standard error as the square root of p times 1 minus p over n.

Note again that I've said p and not p-hat,

because we are again assuming that the null hypothesis

is true and therefore we are using what the

null hypothesis has set forth as our true population parameter.

We don't know if that's the case, but we must assume

that the null is true as we proceed through the hypothesis test.

Lastly, we make a decision and interpret it in context of the research question.

If the p value is less than our significance level, we reject the

null hypothesis and decide that the data

provide convincing evidence for the alternative hypothesis.

If, in fact the p value

is greater than our significance level, we fail to reject the null hypothesis

and conclude that the data do not

provide convincing evidence for the alternative hypothesis.