>> Hi, and welcome back for our next discussion of statistics.
This time we're discussing the Language of Hypothesis Testing.
We start out with an introduction to hypothesis testing,
and an initial checklist to assess the hypothesis test.
Well, first, hypothesis testing can be a bit confusing.
As a result, we really want to emphasize this idea that we need
to read each question carefully and evaluate what is being asked before working
on the mathematical side of the hypothesis test, that is, make sure we take the time
to assess these questions correctly.
Leading into our introduction, a hypothesis is a statement
or claim regarding a characteristic of one or more populations.
We technically just call this a claim, and it leads into our hypothesis testing.
And our hypothesis testing is a process using sample data and probability
to test claims regarding some characteristic of the populations that we're discussing.
We test hypotheses using sample data because it's often impossible or unreasonable
to gather data for an entire population.
Well that takes us to the idea of the null hypothesis.
Well, the null hypothesis is a statement of no change or difference as compared
to a current situation or characteristic.
On the other hand, the alternative hypothesis is
a statement implicating that there is a change
when compared to the current situation or characteristic that we're discussing.
That is, the null hypothesis is saying that nothing has changed from the past.
It's some original claim that comes typically from the past.
The null hypothesis is assumed to be true, and the goal of hypothesis testing is
to find evidence that supports the alternative hypothesis.
So that takes us over to the idea of this assessment before we head
into the math of hypothesis testing.
Well, below are some initial questions to use in order to assess exactly what type
of hypothesis test we're presented.
We first want to ask about which parameter.
Am I being asked to conduct a hypothesis test?
Now this seems like it may be a little unnecessary,
but we cannot tell you how many times students do the wrong hypothesis test.
They do it mathematically correctly, but they do that test for the wrong parameter.
And what we mean by parameter, as a refresher, is we have three options.
They would be mean, proportion, and standard deviation.
Now, keep in mind we do have that little caveat
that is the variance from the standard deviation.
But that's a small little adjustment.
The next question asked would be,
is this hypothesis test comparing one population or two populations?
If the hypothesis test concerns only one population, we'll be comparing our sample
results
to some value that is a constant, as presented through the null hypothesis.
However, if the hypothesis test concerns two populations,
we'll be comparing the values obtained from two different samples.
Now that's a big deal if the hypothesis test is on one population,
we will only have one sample that we will be pulling results from.
But if it's on two populations, and that means we'll have a sample
from each population, or two samples altogether.
And heading over to our third question then, what are the null and alternative hypotheses?
First, we have some keywords to look at to determine the alternative hypothesis.
And you see the list would go on,
for the overall idea here is the alternative hypothesis is determined
by which direction we think the data has moved, if you will,
as compared to a previous value or another population.
We'll ellaborate on this though with some examples later.
To continue the explanation.
We have three options to set up the null and alternative hypotheses.
The first option presented would be for a two-tailed test.
Well, a two-tailed test leads us to a not-equal sign in our alternative hypothesis.
And this comes from keywords like is different or differs
within the story that we're presented.
And that's because the alternative hypothesis does not give us a specific direction
in which we think the data moved.
Our next option would be a left-tailed test,
and that's where the alternative hypothesis has a less than sign
in comparison to some value or population.
And this comes from words such as less than or decreased within the story presented.
And that takes us to our third option which would be a right-tailed test.
That's where everything for the data has increased or exceeds the former value
with keywords such as greater than as well.
And that wraps up our initial discussion on the language of hypothesis testing.
And now we're going to open up with the idea of being able
to identify the null and alternative hypotheses.
We jump right into an example.
Right now we're working with just one population hypothesis test.
We're told to determine the null hypothesis for each of the following,
then state whether the test is two-tailed, left-tailed, or right-tailed,
and then determine the alternative hypothesis.
This is the order in which we want to go in.
Of course, we're going to highlight the parameter
with which we're working for each of these tests as well.
So taking a look at this first one --
42% of American adults did not donate to charity despite the tax write-off.
The Chairman of the Senior Citizens Association thinks that society is going down the tube
and this percentage is greater today.
Well, the first word that we're going to highlight now would be percentage
because as we're presented this hypothesis test, we're comparing the percentage today
to a percentage that was presented back in 2017.
Now that we know we're working with percentage, which is the same as proportion,
we can see our null hypothesis, and that would be P equal to.
And typically, when we get to the math of things, we work with proportions
or comparing the old value that would be a status quo,
and that would be that 42% presented into 2017.
Continuing on.
Since the researcher believes that the percentage is greater today,
the alternative hypothesis is a right-tailed hypothesis.
Again, the keyword being greater there, leading us to the right-tailed test.
And that means the alternative hypothesis would be a P greater than a 0.42.
Well then, heading over to the second example, according to the study published in March
of 2016, the mean number of text messages sent by millennials was 227 per day.
A researcher believes that the mean has changed since then.
Well, like we did before, our research is on the mean of these text messages.
So we have a hypothesis test about the mean.
And working on our null hypothesis, the mean number in the past, March of 2016,
was 227 per day, a mu equal to the 227.
Now, working on the alternative hypothesis, the researcher believes
that the mean call length has changed since that 2016 result.
That keyword doesn't give us a specific direction.
That means we have a two-tailed test,
and that means our alternative hypothesis would be a mu not equal to that 227.
Again, just to reiterate, we have a two-tailed test.
We are not equal to sign here in our alternative hypothesis.
The keyword change didn't send us in a specific direction.
All we're saying at this point is that we think it's something different than the 227,
which it was back in March of '16.
And if that's okay, we'll go ahead and roll over to our next example.
We're told, using an old basket-weaving process, the standard deviation of the amount
of wicker used to make baskets under water was 0.83 feet.
Now with new full-faced snorkels,
a quality control manager believes the standard deviation has decreased.
Well, first, we see now that the manager is concerned about the standard deviation,
so the parameter we're working with would be sigma,
and that means our null hypothesis would be
that sigma equal to that 0.83 feet presented
by the problem using the old basket-weaving process.
Well, if we're okay with that, the quality control manager believes
that the standard deviation has decreased, and
that means we've got ourselves a left-tail test
because the direction we're being sent would be to the left, again,
referring to the idea of tying this to a number line.
And if that's okay, we end up with an alternative hypothesis
of a sigma being less than that 0.83 feet.
And we're all set with our examples dealing with one population.
Heading into our next set of examples here, we want to be able to identify the null
and alternative hypotheses, but now with two populations.
We're taking a look at our example.
We're told a grounded teenager want to determine whether teenagers spend more time on Instagram
or parents spend more time on Facebook, because she was grounded for being
on her phone too much, only to see her mom whip
out Facebook two minutes after she got in trouble.
Well, the teenager believes that parents actually spend more time, on average,
on Facebook than teenagers do on Instagram.
Working on our null hypothesis, of course, the
first thing we want to recognize though is,
and it's a little bit sneaky in this problem, which parameter are we working with.
They help us out a little bit as they tell us on average in our key sentence
that she believes parents actually spend more time on average
on Facebook than teenagers do on Instagram.
The average tells us that we're working with means even if it didn't say on average though.
And it skipped that part.
We'd have to infer that ourselves.
And that's where it can get a little bit tricky.
In this case, though, with that on average being
there for us, we know we're working with means,
and that means our null hypothesis is just a mu 1 equal to a mu 2.
Now this is where we have to be a bit careful though.
We have to make sure we know which population we're calling mu 1 and mu 2.
Now over here in our key sentence, the parents were mentioned first, so I'm going to call
that our population number one, and that makes the teenagers our population number
two.
Only because we're trying to work on this, let's go and label that over on a side.
Mu 1 is representing the parents, and mu 2 is representing the teens.
We highlight this because it can get a little bit messed up when we go to work
with our calculators, so we need to be a bit careful with it.
Now to work on our alternative hypothesis, the
teenager believes that parents spend more time
on Facebook than teenagers do on Instagram.
This is where, again, we just need to make sure we're careful with our organization.
Remember, we said that parents represent our mu 1, and we just used the word more,
leading us to a right-tail test, and that means we have an alternative hypothesis.
That would be a mu 1 greater than mu 2.
Again, being careful with that because we think that parents spend more time
than teenagers on their respective apps.
On the other hand, just thinking mathematically, we want to make sure this is clear,
that we can write this another way.
We can turn this into a left-tail test by reversing the populations.
That is, we can write the alternative hypothesis as mu 2 is less than mu 1
since those two statements are equivalent.
The point of this would be, make sure we're careful with our organization
when we're working with two populations.
Well, if that's okay, heading over to our next example, we have a grandpa and grandson
who are talking about classic American muscle cars and modern import cars.
The grandpa believes that people like classic muscle cars just
as much as the modern import cars.
However, the grandson wants to conduct some research on the topic
because he thinks the proportion of people who like muscle cars will be different
than the proportion of people who like modern import cars.
Well, the first keyword we want to highlight would be proportion so that we know
which parameter we're working with.
And that means our null hypothesis would be a P1 equal to a P2.
And just like we did before though, now we just need to make sure we're careful
with which one we're calling the first population and the second population.
We'll just stay consistent with the problem.
The muscle car is represented first, so I'm going to call those population number one.
And the import cars represented second, so there's our population number two.
Heading over to the work on the alternative hypothesis, the grandson thinks the proportion
of people who like muscle cars will be different than the proportion
of people who like the import cars.
Different being our keyword again.
No specific direction given for us to commit to.
That means we're working with a two-tail test,
and that means the alternative hypothesis is a
P1 not equal to a P2, which means we're all set
with our null and alternative hypotheses.
And as one more little conversation though,
we showed how we could reverse our one directional test
on the previous example just speaking mathematically,
thinking of this in terms of algebra and variables.
If doesn't cause confusion, I can subtract P2 from both sides of the equation here,
and that would lead me to a P1 minus P2 not equal to zero.
And what we're just trying to say is another way to write this hypothesis would be
as the difference between these two populations would be not equal to zero.
Now this is less popular way of writing the alternative hypothesis.
We're just showing you that it's an option.
Reverting back to the original writing, typically we just write this as a P1 not equal
to P2, or we're just trying to be as clear as we can.
Pause the video and try these problems.
Now we're jumping into our next objective where we want to make sure we know how
to state conclusions to hypothesis tests.
Our first conversation though concerns this big technicality that is the conclusion
of a hypothesis test, so we have to be very careful with the language.
The first note we want to make is that the null hypothesis is never accepted.
Absolutely never is.
We can lean on that.
We never accept the null hypothesis.
Instead there are only two possible outcomes to a hypothesis test.
The null hypothesis is either rejected or not rejected.
Now, I may mess with your head a little bit.
The idea of, well, if it's not rejected, that means it's accepted.
Again, this is language stuff as opposed to just logic.
Not rejected is very different, technically speaking, than accepting the null hypothesis.
The logic behind that is since we only have sample data we never truly know the value
of the parameter in discussion.
Remember, the parameter would be the true mean or proportion or standard deviation.
Since we'll never know the true value of that, we can't accept a null hypothesis.
We can only say that it is or is not enough evidence to reject the null hypothesis.
But we'll get into that in a minute in case it's still a little bit confusing.
The analogy we have is just like the court system.
We never declare a defendant innocent, meaning when a verdict is read,
we don't hear the word innocence coming out from that jury.
Instead a defendant is either guilty or not guilty, and those are the only two options.
This is because the system is designed that the defendant is innocent until proven guilty,
so the whole case is presented trying to prove guilt.
If it's not proven, the verdict is simply not guilty,
and we can't say that a defendant is innocent
because the system wasn't designed to prove his innocence.
Hopefully that makes sense.
We elaborate more with some examples, of course.
But before we do that, let's discuss the wording of these conclusions
since those are also quite technical.
Our first scenario is, let's say we're going to reject the null hypothesis.
Well, if rejecting the null is the result of the test, then we can say something like,
there is sufficient, which is a fancy word for enough, evidence to conclude that --
and what you see in this generic template here is we just tie
in the alternative hypothesis in context with the problem.
Again, we'll elaborate with some examples.
But that's our key phrase -- there is sufficient evidence to conclude;
whereas scenario number two, if we don't get to reject the null hypothesis,
that means there is insufficient evidence to conclude that --
and, again, we tie in the alternative hypothesis.
Trying to sum this up.
We basically make the same sentence for each problem,
the only difference being whether we have sufficient or insufficient evidence
to conclude whatever the alternative hypothesis would be.
Well, that takes us over to our example, and hopefully this clarifies any confusion.
Let's jump right into it.
We're back to 2017 and the 42% of American adults that did not donate
to charity despite the tax write-off.
The Chairman of the Senior Citizens Association still thinks that society is going right
down the tube, and this percentage is greater today.
Well, now that we're done with the null and alternative hypotheses
from our previous conversations, let's go ahead
and suppose that the sample evidence indicates
that the null hypothesis should be rejected.
We're told to state the wording of the conclusion.
Well, if we're rejecting the null hypothesis, then that means there is sufficient evidence
to conclude, and now we bring in that alternative hypothesis.
So we have sufficient evidence to conclude that the percentage of American adults
who do not donate to charity is greater than the originally stated 42% back in 2017.
And that's what we mean by tying in the alternative hypothesis.
Now the next part says, suppose the sample evidence indicates the null hypothesis should
not be rejected.
Well, now we are not rejecting the null hypothesis,
and that's because there is insufficient evidence to conclude and notice
that the sentence is about the same,
just changing those keywords, sufficient versus insufficient.
There is insufficient evidence to conclude that the percentage of American adults
who do not donate to charity is greater than 42%.
Now notice one more time, the only word that changed was insufficient.
The alternative hypothesis of being greater than 42% is still the end of our conclusion.
Well, if that's okay, let's go ahead and jump into our next example.
We're back to the text messages sent by millennials.
Remember, according to a study published in March of 2016,
the mean number of text messages sent by millennials was 227 per day.
Any researcher believes that the mean has changed since then.
Well, now we're told suppose the sample evidence indicates
that the null hypothesis should be rejected.
And, again, we need to state the wording of our conclusion.
Well since we have the evidence to reject the null hypothesis,
that means there is sufficient evidence to conclude that the mean number
of text messages sent by millennials is different than the 227 per day.
Now just highlighting those keywords.
Remember our researcher believed that the mean has changed
since then, so we had a two-tail test.
And that means our alternative hypothesis did not send us in a specific direction.
That's why we're saying in our conclusion that the mean number of text messages sent
by millennials is different than the 227 per day as opposed to committing
to something like greater than or less than.
Heading over to Part B then, now we're told again, suppose the sample evidence indicates
that the null hypothesis should not be rejected.
All that means for the wording of our conclusion,
there is insufficient evidence this time to conclude that the mean number
of text messages sent by millennials is -- and just like we had before,
different than the 227 per day telling you that
alternative hypothesis is on the two-tail test.
We're hoping that's okay.
Let's head into one more example.
Using an old basket-weaving process, the standard deviation of the amount of wicker
used
to make baskets under water was 0.83 feet with new full-faced snorkels.
A quality control manager believes the standard deviation has decreased.
Well, then, suppose the sample evidence no indicates
that the null hypothesis should be rejected again.
Well, hopefully, we're getting it now.
Trying to emphasize the fact that it doesn't change a whole lot
since we are rejecting the null hypothesis.
That means there is sufficient evidence to conclude that the standard deviation is --
and tying in our alternative hypothesis.
Since the Quality Control Manager believed the standard deviation has decreased,
that means we have sufficient evidence to conclude
that the standard deviation is less than our 0.83 feet.
And heading over to Part B, suppose the sample evidence this time, like we've done before,
indicates that the null hypothesis should not be rejected.
And that means one more time we have insufficient evidence to conclude
that the standard deviation is less than the 0.83 feet.
And just reiterate one more time that less than
0.83 feet comes from the alternative hypothesis
where we think that the standard deviation has decreased.
Pause the video and try these problems.
Well, that wraps up our conversation.
Thank you one more time for joining us in our discussion
of The Language of Hypothesis Testing.