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>> Do college students get
less sleep?
To answer this question we're
going to use SPSS
to calculate a sample mean,
do a single sample T test,
as well as a
confidence interval.
We're going to try
to answer the
following questions.
First, approximately how much
sleep do college students get?
Second, do college students
sleep less
in the general population?
And third,
what's a ballpark range
for how much sleep college
students get?
That's 95 percent likely it
contained the 2
population mean.
Let's focus
on our first question.
To answer that we'll use a
sample mean
as a point estimate
for the population mean.
Now it is based on our sample
of 40 randomly selected
college students.
We'll go ahead
and estimate how much sleep
the typical college
student needs.
We'll keep in mind
that if we were
to sample repeatedly,
each of our sample means would
be a little bit different.
So we know going into this
that there's going
to be some error
that will be involved.
OK, with that
in mind we'll go ahead
with this scenario
and use SPSS.
We'll go ahead and click
on analyze,
and then from the drop
down menu select Compare
Means, and then 1 sample
T test.
When a dialogue box comes
up we go ahead
and we take our sleep variable
and move it
over to the test variables.
For the test value we're going
to put in what we're going
to compare college
students to.
In this case we're going
to compare them
to the United States adult
population,
and recent research shows
that they're getting 7 hours
of sleep each night
on average.
Here's our SPS output.
In looking at it we'll focus
on the sample mean of 6.36.
So we know now,
in terms of our estimate,
that on average college
students get about 6.36 hours
of sleep each night.
There may be some error,
in fact there certainly is
some error involved,
but that's our best estimate
based upon our sample.
Second, do college students
sleep less
than the general population?
Well we're going
to focus specifically on,
can we reject our
null hypothesis?
The null hypothesis is
that college students are just
like everybody else,
needing 7 hours
or perhaps even more of sleep.
We're looking
at a distribution here
for the amount
of sleep required
by the adult population
in the U.S., it's centered
on 7 hours.
Looking at the left tail,
because we're going to do this
as a 1 tail hypothesis test,
we can see our .05 shaded
region, this will be our
reject zone,
so that if any sample mean
falls in this reject zone it
will go ahead and be rejected.
Coming back
to our distribution
of our adult population,
let's assume for a moment
that the null hypothesis
is true.
If that's the case then .95
of our sample mean should fall
in the none reject zone,
or the retain the null
hypothesis area.
So we have a decision.
Will we reject the null
hypothesis,
or retain the null hypothesis?
If the null hypothesis is true
we know that due
to error we'll reject the null
.05 of the time.
The .95 of the time we'll make
the correct decision.
So again if the null
hypothesis is true,
retaining the null hypothesis
will be a correct decision.
Rejecting the null hypothesis
would be a type 1 error.
On the other hand,
it could be the case
that college students really
do need less sleep
than the typical U.S. adult.
And in that case,
if we were to reject the null,
that would be a correct
decision, and if we retain the
null that would be a type
2 error.
OK, that was a brief review
of the decision matrix.
Let's come back
to our SPS output.
Remembering that we get
to reject a hypothesis
if our P value is .05 or less.
Here's kind
of like the big picture.
You have the null hypothesis
and you have some data.
If they conflict,
then they both can't be true
and we can't throw
out the data,
assuming we think it's good
data, so rejecting a
hypothesis instead
when that P value is .05
or less.
Looking at our SPS output,
we have our T value of -2.452,
our degrees of freedom of 32,
and our significance .02.
Let's focus first
on the T value,
keeping in mind,
how did SPSS calculate
that T value?
Remember our formula
for single sample T test is
sample mean minus the
population mean,
divided by the standard error.
We can get this information
from our SPS output.
Our sample mean is 6.36,
our population mean,
assuming that a hypothesis is
true, that college students
are just like everyone else,
would be 7,
and our standard error is .26.
And we calculate
that we get a T value
of -2.46, pretty close
to what SPS gave us.
Any difference is likely
to be due to a rounding error
on our part.
What does that T value
of -2.45 actually mean?
It means that our sample mean
is approximately 2.45 standard
errors below the mean.
That is, we would have
to go 1, 2,
and half approximately
standard errors below the mean
before we get
to our sample mean.
That's pretty far away
if this null hypothesis
is true.
This would be a very unlikely
sample mean.
So again, that sample mean is
2.45 standard errors below
the mean.
OK the next thing
in our SPS is the degrees
of freedom, which is 32,
and that's simply calculated
as our sample size minus 1.
And then finally we have our
significance
for a two-tailed test.
Now remember,
we were interested
in a one-tail test.
We just wanted to show
that college students sleep
less than the
general population.
But SPSS doesn't ask us
if we're doing a one-tail
or two-tail,
it always gives us
the two-tail.
Given that it gave us a
two-tail of .02,
think about it as two-tails,
we take that .02 and we know
that has to mean .01 is below
the mean, and .01 is
above the mean.
So if we added them together
we'd get our two-tail value
of .02.
We're doing a one-tail test,
then is just half the amount
of our two-tail test.
So for a one-tail test our P
value is .01.
So keep this in mind.
Any time you're doing a
one-tail test,
and you get the SPSS output
for a two-tail test,
you can usually make
the correction.
Just take the value,
divide it by 2,
and that will give you the P
value for a one-tail test.
OK, remember we get
to reject the null hypothesis
if P is less than
or equal to .05.
In this case we'll be able
to reject the null hypothesis
since our P value is less than
or equal to .05.
And we'll conclude
that college students sleep
less than the
average population.
Onto our 3rd question.
What's a ballpark range
for how much sleep college
students get?
That's 95 percent likely
to contain the true
population mean.
Specifically we're going
to determine a 95 percent
confidence interval.
Here you can see some examples
of what a confidence interval
might look like.
We have our sample mean
in the middle,
and then we have a range
around it.
We're 95 percent confident it
contains the true
population mean.
Keep in mind though
that not all confidence
intervals will contain the
true population mean.
If you're doing a 95 percent
confidence interval,
that means about 5 percent
of your confidence intervals
will not contain the
population mean.
Here's an example
of 100 confidence intervals
generated by a program
where for each one it took a
sample mean,
and that will let you know
whether the confidence
interval's more to the left
or to the right,
based upon that sample mean.
And also notice
that there are different
widths, depend upon the sample
standard deviation.
So here are some confidence
intervals,
and then generated another
100, and another 100.
And the ones in red are
where the confidence interval
does not contain the true
population mean
at the 95 percent level.
So you can see sometimes you
might get a few more than 5
out of every 100.
So to review.
A confidence interval is a
range that's likely
to contain the
population mean.
And now let's look
at how do you determine a 95
percent confidence interval
for how much sleep college
students get each night?
Let's return back
to our SPS output,
and we'll focus
on the 95 percent confidence
interval of the difference.
To create our confidence
interval we'll begin
with our test value of 7,
that is that's our comparison
point of the adult population
needing 7 hours.
But what about for
college students?
Well we take that 7 and we add
to it our lower bound value
of -1.1655, which comes
out to be 5.83.
So that's our lower bound
of our confidence interval.
Then for upper bound we take
again that 7,
and we add
to it our upper bound
of -.1079 that SPS gave us,
and that comes out to be 6.89.
So we have a confidence
interval for how much sleep
college students require.
That's 95 percent likely
to contain the true
population mean.
It goes from 5.83 to 6.89.
Now we could also have done
this by hand, and I just want
to take a moment
to show you that,
just so you understand
where is SPS getting
these values?
So our formula is sample mean
plus or minus T critical times
the standard error.
T critical is the only perhaps
new thing to you,
so let's just go
through this step by step.
Our sample mean we get
from the SPS output,
it's 6.36.
Our standard error,
we also get
from the SPS output, it's .26.
And for the T critical,
well for that we can go
to a T table.
If our confidence interval is
95 percent,
that leaves 5 percent
left over.
So for a confidence interval
of 95 percent,
at the T table we look up .05
as a column.
Now our degrees of freedom,
as you may recall
from the SPS output, was 32.
So in our T table,
when we look up row 32 degrees
of freedom,
and where our column
and row intersect, 2.0369,
that is our T critical value.
So let's plug that then
into our confidence
interval formula.
So we have confidence interval
for 95 percent is equal
to our sample mean, 6.36,
plus or minus the T critical,
2.03, times the standard
of error .26.
And we'll simplify there,
so we get that the confidence
interval of 95 percent is
equal to 6.36 plus
or minus .53,
and that gives us our
confidence interval range
of 5.83 to 6.89,
which matches what SPSS
gave us.
Alright now,
let's say you wanted a
different confidence interval.
With the T table,
if you wanted a confidence
interval of 99 percent you
would use an alpha .01.
If you want a confidence
interval of 90 percent you use
an alpha of .1.
Everything else should be
the same.
With SPSS what you'll do is
first you'll again select the
single sample T test,
then when the dialogue box
comes up for the one sample T
test you click on Options,
and for the confidence
interval percentage you put
in the percentage you're
interested in,
for example 99 percent.
OK, so to recap.
For 99 percent confidence
interval there's a 95 percent
probability
that the confidence interval
contains the population mean,
keeping in mind
that approximately 5
out of every 100
of them will not.
Let's say we want a more
shorter confidence interval.
Well the confidence interval
will become smaller
if we're willing
to be wrong more often.
For example,
up above we see the confidence
interval for 99 percent.
If we instead want a
confidence interval
for 95 percent,
well the confidence interval
is now shorter.
It's less likely
to contain the true
population mean.
If we had a confidence
interval of 50 percent,
well our confidence interval
would become even smaller,
and it would be even less
likely to contain the true
population mean.
OK, what would be another way
to go to a smaller confidence
interval that would imply
reader accuracy?
Well, if we increase our
sample size
that will also result
in a decrease
in our confidence interval.
So if we increase our sample
size, for example,
from 10 to 20
that would make a smaller
confidence interval.
And over here are another 100
confidence intervals
that I ran,
this time increasing
that sample size.
So in this case there's still
a 95 percent confidence
interval, but we're now
working with a smaller range
which can be helpful.
So a larger sample size gives
us more information,
allowing us
to have a smaller confidence
interval size.
OK, we've looked
at how SPSS can help answer
our questions regarding do
college students get
less sleep.
Based on our sample,
approximately how much sleep
do college students get?
And we said college students
get approximately 6.36 hours
of sleep.
Do college students sleep less
than the general population?
And we concluded they sleep
less than the population P
less than or equal to .05.
And finally,
what's a ballpark range
for how much sleep college
students get?
That's 95 percent likely
to contain the true
population mean.
And we concluded college
students get little sleep,
that at a 95 percent
confidence interval ranging
from 5.83 to 6.89.
I hope you've found this
tutorial helpful,
and my appreciation to those
who provide materials
on the web that helped me
in creating this presentation.
Thank you.