Now that we understand the idea of the central limit theorem, we can use it in inferential statistics
to start testing hypotheses.
And the focus of this lesson is going to be the hypothesis testing procedure that will follow each time.
And more specifically, we'll look at this first step of our hypothesis testing procedure.
So in statistics, as we've already talked about, the whole idea is that we collect a sample from the
population, we investigate that sample and then we try to make a clear statement about whether or not
that sample supports some original statement that we've made about our population.
So this process is the hypothesis testing process or the process of inferential statistics, because
we're using information about the sample to make inferences about the population.
Now, the first step in this process is to state our hypothesis, which is a statement of expectation
about some population parameter that we develop for the purpose of testing it, and we can hypothesize
about anything.
So our hypothesis might be that mean order processing time is 17 hours from the time the customer places
the order to the time that we ship it out.
Or we might hypothesize that employees in one department of our company work longer hours than employees
in another department.
Our hypothesis can be anything we want it to be.
It's just that we hypothesize about whatever it is we want to test.
And what we need to realize is that in this hypothesis testing process, in the process of doing inferential
statistics, we actually always create a pair of opposing hypothesis statements.
One of those statements is the null hypothesis and the other is the alternative hypothesis.
We can indicate the null hypothesis as h sub zero.
Sometimes we call it h not and the alternative hypothesis as h sub A We can remember that if we think
about the A standing for alternative.
Now the alternative hypothesis is the abnormality we're looking for in the data.
It's the significance that we're hoping to find.
Once we have our alternative hypothesis, then we always state the opposite claim and that opposite
claim is the null hypothesis.
So as an example, let's say we want to test the hypothesis that employees in the billing department
of our company are working more than full time.
They're working more than 40 hours a week.
In that case, our alternative hypothesis would be that the mean number of hours worked is greater than
40.
Therefore, the null hypothesis is the opposite statement, which means that the null hypothesis is
that the employees work less than or equal to 40 hours a week.
Now, interestingly, when we think about hypotheses, we always tend to think in terms of the alternative
hypothesis, because that's the thing that we're really trying to test.
We're wondering here, do employees in the billing department work more than 40 hours a week?
So we sort of frame up our thinking in terms of the alternative hypothesis.
But when it comes to this hypothesis testing procedure, what we're actually going to be doing is testing
the null hypothesis.
So our focus going forward is going to be on the null hypothesis, even though when we think about what
it is that we want to test, we think about the alternative hypothesis.
That's not wrong.
It's just that we might frame up our thinking in terms of the alternative hypothesis state.
That state the null hypothesis is the opposite.
And then going forward we'll focus on the null hypothesis that opposite statement and work on testing
that hypothesis instead of really looking at the alternative hypothesis.
As we work through this hypothesis testing process.
Now when it comes to setting up the null and alternative hypothesis, there are only three ways that
we can set these up.
This second option here is the one we used in our example where we said that employees in the billing
department worked more than 40 hours a week.
The alternative hypothesis has the mean set greater than some value, which means that the null hypothesis
is the opposite.
Claim that the mean is less than or equal to that same value.
We can have the opposite situation where we set the alternative hypothesis as the mean less than some
value, in which case the null hypothesis is that the mean is greater than or equal to that same value
that we chose.
And then the third option is that we choose an alternative hypothesis where the mean is not equal to
some value.
So in this case, maybe we don't have a suspicion that the employees in the billing department work
more than 40 hours a week.
Maybe we just think that they work some number of hours other than 40.
We're not sure if they work fewer than 40 hours or more than 40 hours, but we don't think that they
work exactly 40 hours a week.
Our suspicion, our hypothesis, the thing we want to test is that the employees in the billing department
do not work 40 hours a week.
So our alternative hypothesis would be that the mean is not equal to 40, and therefore the null hypothesis
that goes along with that would be that the mean is equal to 40.
So again, in this example we said that the billing department were.
It's more than 40 hours a week, which means that we described a relationship between the mean and some
constant value.
We can also set up hypothesis statements that compare to groups.
So if we think that the billing department works more hours than the payroll department, we could say
something like this.
So the mean for the billing department is greater than the mean for the payroll department, in which
case the null hypothesis is the opposite statement that the mean for the billing department is less
than or equal to the mean for the payroll department.
And we could compare means like this with all three of these different statement types, these different
pairs of hypothesis statements.
But that being said, it's important for us to say that we should only ever use these second and third
types here when we have good reason to believe that there's this kind of directionality that we're indicating
here.
In other words, we only want to set up these kind of directional hypothesis statements where we use
either a greater than or less than simple in the alternative hypothesis, when we have a pretty strong
suspicion that that is the directionality that actually exists in the population to continue on with
this example we've been using here, maybe we have data from previous pay stubs for the employees in
our billing department to indicate that they do tend to work more than 40 hours a week.
And what we're doing here is just testing the current pay period.
We already have some kind of prior knowledge, some strong suspicion that they are, in fact, working
more than 40 hours.
And we're looking to test that here.
So we need to have some kind of suspicion for some good reason if we're going to use either of these
sets of hypothesis statements.
Otherwise, we really need to use this first one because this first pair of hypothesis statements has
no directionality to it at all.
And so it's sort of a more conservative approach to take what we're saying here with this alternative
hypothesis that the mean is not equal to some value is that we don't know if the mean is less than that
value or greater than that value.
We don't really have any suspicion at all about the directionality.
We're not sure which way it goes.
We're just saying maybe the mean is not equal to that particular value.
Whereas with these two sets of hypothesis statements, we're being more specific, we're indicating
directionality, we're saying no, we think the mean is greater than some value, not less than the
value, not equal to the value, but greater than the value.
Or here the mean is less than that value.
We're sort of presupposing that directionality.
And so we really need to have some kind of a good reason to think that this is the case.
If we're going to use either one of these pairs of hypothesis statements.
Otherwise we should use this first one.
It's just more conservative.
It doesn't suppose to much if we don't really have any evidence for supposing a particular direction.
So to summarize here, the first step in our hypothesis testing procedure is to restate the null and
alternative hypotheses.
Remember that we'll frame up our thinking around the alternative hypothesis, so we'll sort of think
about what we want to test.
That'll be our alternative hypothesis, and then the null hypothesis will be the opposite of that statement.
But that being said, the alternative hypothesis always needs to be that the mean is not equal to some
value greater than a value or less than a value.
We can't say in the alternative hypothesis that the mean is equal to some value less than or equal to
or greater than or equal to a value.
We need to stick with these symbols in the alternative hypothesis, which means that the null hypothesis
will be the opposite statement and therefore will always only see these values in the null hypothesis.
So we'll save equal to less than or equal to and greater than or equal to for the null hypothesis statement.
So we frame up our thinking around the alternative hypothesis.
We state that as the mean is not equal to some value greater than some value or less than some value.
Then we make the null hypothesis the opposite statement with the opposite sign, which means that we
will end up with either this first pair here of hypothesis statements, this second pair, or this third
pair.
And once we have our pair of hypothesis statements, we can move on to step two, which is about determining
the level of significance.
But keep in mind that as we're going through this hypothesis testing procedure, it's important to always
remember that the goal here of this whole process is to provide support for the claim that's being made
by our alternative hypothesis.
The goal is not to prove that our alternative hypothesis is true.
This hypothesis testing procedure can never actually prove the statement being made in our alternative
hypothesis.
We can never prove either of these hypothesis statements to be wrong or right.
All we can do is lend support or thought about the other way.
Fail to lend support to the statement that we're trying to make.
So if we go through this whole process, the best we can do is find support for or fail to.
Find support for the alternative hypothesis.
And the way that that's going to manifest itself in the end is with sort of this inverted language,
which can be a little confusing, but we will either reject the null hypothesis, and you can imagine
that if we reject the null hypothesis, that's going to give support to the alternative hypothesis since
these are opposite statements, or if this process doesn't give us enough evidence to reject the null,
then the conclusion that will make is that we failed to reject the null.
We didn't have enough evidence to reject the null.
We couldn't say that the null was wrong.
And if we can't say that the null is wrong, that means that we don't really have any support to throw
behind the alternative hypothesis.
Our original statement.
So just remember that this whole process is about support only, not proof, and that what we're going
to do is break down each of these steps to eventually get to the conclusion where we try to reject the
null in order to lend support to the alternative or fail to reject the null, which means that we've
failed to support the alternative.