Tip:
Highlight text to annotate it
X
>>This narrative PowerPoint
that is designed
to help integrate your
understanding
about probability,
distribution of sample means,
and hypothesis testing.
This presentation will make
the most sense
if you've already had the
opportunity to read workbook
on these topics.
To help us to go through
and see how these topics are
related, I'd like us
to address the question,
does lead impair IQ?
Here's the dilemma,
if you want to find
out whether dead impairs IQ,
let's say your theory is
that it will effect brain
development,
there's an ethical issue.
It would not be appropriate
to give people,
or children lead to see
if it harms their IQ, right,
we wouldn't want
to do a treatment
or manipulation
that would actually cause harm
to someone, and yet,
it may be very important to us
to find out if lead impairs
cognitive development,
because if it does,
it would be very important
to take lead substances
that could get into the air,
like, that could give off lead
dust, to take those substances
out of classrooms.
Okay, so we have a situation
where we really would want
to know the answer
to the question,
but it would be ethical
to actually test people to see
if it has an effect.
Our solution,
you are inadvertently exposed
to lead, and compare mental
building to the population.
And, because paints used
to be lead-based,
this would actually not be
as difficult to do
as you might think.
Research has now shown
that lead does negatively
affect brain development,
and there was late paying all
blinds that kids would suck
on his little kids were lead
dust in the air,
were lots of lead paint was
used, and this is actually
been shown to decrease IQ.
So, let's say our researcher
has found the group
of children they are
in an environment
that had a lot
of lead present,
and that these children,
for all other purposes,
were normal except
for this one unusual
difference, that they were
in lead-based environment.
When you're ready
to develop your hypothesis
concerning the relationship
between lead
and mental development,
you have to decide whether
you're going
to have a directional
hypothesis, which is known
as a one tail test,
or if you're going
to have a nondirectional
hypothesis, which is known
as a two-tailed test.
With a directional hypothesis,
you are specifying whether the
sample mean should be below a
world of the population mean,
and so you see here two
illustrations
of that distribution sample
means and, in one case, the,
in the top case,
the right tail is shaded in.
This would be a directional
hypothesis
where you would expect the
sample mean to be greater
than the population mean.
For example,
we want to test whether a food
diet will actually improve
your weight gain
for sumo wrestlers, you know,
some major carbo diet
to really blow them up,
you would do a directional
hypothesis
where you expect the sample
mean to be
above the population mean.
Take a look
at the bottom illustration
normal distribution,
and there you will see
that the left tail is shaded
on the bottom side,
and that directional
hypothesis corresponds
to where you would expect the
sample mean
to be below the population
mean, that might be a diet
where you are actually hoping
to lose weight,
and we would hope
that the sample mean is below
the population mean.
So what researchers are going
to do, when he or she writes
up the search hypothesis,
they're going to say,
in my expect them a sample
mean to be
above the population mean
or below it?
In our case,
the research hypothesis is
going to be that children
who would help dust,
lead dust that is,
will have a lower IQ scores
than the general population,
so we're going
to have a directional
hypothesis, specifically going
to predict
that are sample mean will be
below the population mean.
Now in addition
to a one tail test,
that is a directional test,
you can also do a two-tailed
test, which is known
as a
nondirectional hypothesis.
A nondirectional hypothesis is
where you want
to cover both bets, you say,
hey I think this sample mean
is going be different
than the population mean,
maybe it'll be above,
maybe it'll be below,
I don't know,
I just think it will
be different.
Okay, so that would be a
nondirectional hypothesis,
otherwise known
as a two-tailed test.
In our case, we're going with
but one tail hypothesis,
that is a directional
hypothesis, we're going
to predict
that the sample mean will be
below the population mean.
We have, in essence,
placed our bets
for what the results will
look like.
We have decided to do
that over the other
possibilities saying
that lead would improve IQ,
if we did this other one tail
hypothesis test, or saying,
hey I think lead would just
have an effect, I don't know
if it will improve it
or harm it, and that would be
that two-tailed
hypothesis test.
So we're going
with the one tail hypothesis
that lead would harm IQ.
Okay, now, what we're going
to be doing is work comparing
a sample to a population mean,
so we need
to pick the right inferential
statistical tests,
so we can make this comparison
and see what the probability
of our getting an outcome due
to chance.
For our particular research
design, our deterrent variable
isn't IQ score, which is a way
to measure intelligence;
we believe it will be affected
by lead.
IQ score is a scale variable,
it is an integral scale
variable, so all in the scale,
it's normally distributed,
that is true,
and we notice
standard deviation.
The people
who design IQ tests,
actually designed
and set the standard deviation
will be of particular value;
for example, 16.
And that is going to allow us
to do a Z test, so a Z test,
we've done many
of these the past,
only it was known
as a Z score,
when you're comparing a single
value to a population made,
now we're, kind of,
going up to the next level
and were saying,
okay let's compare a sample
mean to a population mean,
let's find
out what's the probability
of a whole sample having some
average of value compared
to the population.
If we're going to do a Z test,
there are four
basic requirements.
Number one,
are you comparing a single
sample to the population,
if so you're
at the right place.
Number two is your dependent
variable scale.
It has to be scaled
because that way you can see
what is the shape
of the distribution.
If it is nominal ordinal,
they were just dealing
with categories that, at most,
can be arranged,
so we actually need a scale
variable where there's equal
intervals between the values.
Number three,
only look at our distribution,
the distribution scores must
be normal,
or the distribution scores,
if the shape is unknown,
or definitely not normal,
were still okay,
as laws are sample size was
at least 1000
or more people in it.
Okay, so if the distribution
of population scores is
normal, were good to go,
doesn't really matter our
sample size.
On the other hand,
if we know
that the distribution
of our individual scores isn't
normal, and we need a sample
size of at least 1000,
and they were okay
to use the Z test.
And finally,
the standard deviation
for the population also needs
to be known.
And you may say why?
Why are all these requirements
in place?
Well, remember
that with the Z test,
we're going to be looking
up our proportion using a Z
table, and just
like with scores,
just as with sample means,
the Z table is only going
to give us useful information
if the distribution is normal,
if it is not normal,
we can still look up something
in the Z table,
it will give us a proportion,
but it would be wrong, right,
because the Z table is based
upon that assumption we have a
normal distribution.
We also need
to know the standard deviation
to know how wide is
our distribution.
Interestingly,
you don't always have
to know the population mean,
in fact, later
on this semester,
will talk about when we're
going to compare a sample mean
again so hypothesized
population mean,
but don't worry about that,
just know that for our Z test
requirements,
you've got a have a normal
distribution,
you need to know the standard
deviation for it,
and if you got those two,
you're well on your way.
Also, of course,
we'd only do a Z test
if work comparing a single
sample to the population
and likewise
if our dependent variable is
scale, so that covers all
four requirements.
When I was talking
about the Z test requirements,
that illustration on the left
that would be a distribution
of sample means,
so this is a distribution
where each value,
instead of being a single
score, actually represents
a sample.
Okay, now you may wonder why,
if the Z table requires the
distribution be normal,
why did I previously say,
you know what,
it doesn't matter the shape
of the distribution
of individual scores,
you're still okay
if the sample size of 1000
or more, why is that true?
Well, there is one key theorem
covered in statistics,
and anyone
who takes a statistics class
is expected
to know this key theorem,
is known as a central limit
theorem, and you can read
about it in that workbook,
but the main idea behind it is
that it doesn't matter what
the shape of the distribution
individuals scores looks like,
if you work, for example,
at that bimodal distribution
that shown at the top,
if you randomly sample 1000
people from that bimodal
distribution and you plot
that single point,
then you sample another 1000
people, and then you plot
another single point represent
in their average,
and you keep doing this,
sampling 1000 people,
recording a single point
representative average,
by the time you're done
with hundreds and hundreds
of the samples,
your distribution
of all the sample means will
appear normal,
guaranteed every time,
and that's why you can use the
Z table, because there are
guaranteed
that the distribution sample
means will be normal.
So if the individual scores
are already normal,
you're good to go,
you don't need the sample size
with thousand or more.
If the individual scores are
normal, the distribution a
sample means will be normal,
but if you're distribution
of individual scores is not
normal, manager sample size is
at least 1000,
you're still okay,
that distribution sample mean
will be normal.