An example of the t-test
A sports physiologist suspected
that bouillon cubes may improve
the performance of runners. She
got this idea when she read a
1996 experiment of mine in the
Journal of Physiology and
Behavior. She randomly selected
12 U of I male undergraduates
and randomly assigned them to
either group 1 who were given a
cup of soup with a bouillon cube,
or group 2 who were given a cup
of chamomile.
She subsequently asked the
students to run a 100 meter
course and recorded the time they
took to reach the finish line.
Here is the layout of the
experiment.
Note that a subject belongs to only
one group, so the groups are
independent. If subject 1 is John,
and he belongs to the first group,
he does not participate in the
second group.
When the data were collected,
she analyzed them by using the
t-test. We say she ran a t-test.
The data and analysis are
presented in the next table:
What do you mean by df? you
say. You look unhappy too.
In order to develop the concept of
degrees of freedom, df, we will run
a tought experiment (thought
experiment) as Einstein used to
say. The benefit will be that you
will not need to memorize any of
the many formulas for degrees of
freedom. Ever.
Drama
Mean Prophet
I take a sheet of paper from my printer
and cut it into four equal pieces. I take a
pencil and write the number 3 on the first
piece, I write number 4 on the second
piece, number 8 on the third piece, and
lastly number 5 on the fourth piece of
paper. I place all four pieces of paper in a
shoe box and put the cover on it so that
the pieces of paper cannot be seen. On
another sheet of paper I keep a record of
these four numbers. Then I add them up
3+4+8+5=20. Then I calculate the
average or mean.
20 divided by 4 equals 5. The mean is 5.
I destroy this piece of paper.
I take another sheet of paper and I write
on it the following:
Mean=5 n=4
I stick this paper on the shoe box for
anyone to see.
I ask Jerry, the English major who is
sitting in the lounge, to come into the
room. I explain to him that there are 4
pieces of paper in the box, each having a
number on it. The mean of those numbers
is 5. That is all he is told.
I begin by asking Jerry:
Jerry, close your eyes, stick your
hand in the shoe box and pick one
of the four pieces of paper.
Jerry does that. Before he draws
his hand out of the box I ask him
to guess the number he has picked.
Jerry giggles.
I am not a magician, he says.
Open your eyes and read out the
number.
Eight, he says.
I lay the piece of paper with the
number 8 down on the table so
that it can be seen at glance at
anytime.
Now Jerry, stick your hand again in
the shoebox and pick another piece
of paper. Jerry does so. Before he
draws his hand out of the box, I
ask him to guess the number he
has picked.
Again, Jerry giggles and mumbles:
No way!
Open your eyes and read the
number out loud.
Five, he says.
I lay the piece of paper on the
table next the first one that Jerry
picked, the one that has the
number 8 on it. It can easily be
seen. There are two pieces of
papers on the table now with the
numbers, 8 and 5.
I repeat the procedure for the third time.
Jerry, can you guess what the number that you drew?
He smiles faintly and simply shrugs his
shoulders.
Open your eyes and read the number out loud.
Four, he says.
Again, I lay the piece of paper with the number 4 on the table, next to the other two. There are three numbers on the table now:
8 and 5 and 4.
We are ready to repeat the procedure, I say. Close your eyes and stick …
Before I can finish my sentence, Jerry says:
Number 3.
Great! Jerry knows simple
arithmetic. He added up the
numbers he had already drawn.
8+5+4=17. There is only one
number which, if added to 17, will
give 20. That is the number 3.
There is only one number that
divided by 4 will give us a mean of
5. That number is 20.
What am I to get out of this story?
you say.
Without knowing the mean and the
n (how many numbers) you could
not guess any of the numbers.
They were free to vary. Several
arrays of four numbers could give
us a sum of 20. For example
11+1+2+6=20, or 5+ 6+8+1=20.
Because I calculated the mean
and showed it to you, and I also
told you how many numbers there
are in the box, one of the numbers
is not free to vary.
In statistical language we say: We
lose one degree of freedom every
ime we calculate a mean.
mean we calculate.
Remember this.
The degrees of freedom in this
case is 3, that is
4-1=3.
Formally we write it as follows:
df=3.
In another situation that we have
two groups of 10 subjects each, 20
total, and we calculate two means
the degrees of freedom are 18. That
is, we subtract 1 for each mean
we calculate. This is simple to
remember. I am confident that you
understand this concept at the gut
level, not just repeating my words
like a parrot.
As I promised you, we will push
aside almost all the formulas that
otherwise you would have to
memorize.
It is logical, isn’t it. If you know the
concept and you know what you
are doing, you do not need a
formula to tell you what to do step
by step.
Back to our discussion of the
t-test.
Definition: t obtained.
That is the result of solving for t. In
other words when you run a t-test
you find a t value. A third way of
saying this is: the result of
analyzing your data using the t
formula.
Definition: t required.
The required t is the value
contained in the t-table which is
found in the end of every statistics
book, including the one you are
reading now (see Appendix).
Remember, the t-table lists the
modified 1.96 that Gosset
published. It lists the recalculated
values for 1.96 depending on
degrees of freedom.
To find the required t, you first
calculate the degrees of freedom.
You are an expert in calculating
degrees of freedom (df). No
formulas needed, not for us who
learn statistics by acting in soap
operas.
In the present example of the
t-test (page 114) we have two
groups of 6 subjects each, total of
12 subjects. Since in computing
the t we need to first compute the
mean of each group, two means,
we lose 2 degrees of freedom.
How many scores go into the
calculation of the t? All of the
scores. That is,12 scores, minus 2
equals 10. Therefore, df=10.
Now we go to the t table in
Appendix and run our finger
down the left column which is
labeled df. We stop at 10. Then we
draw out finger horizontally until
we reach the column that is
labeled 5% or 0.05. We copy the
value we find at the tip of our
finger. This is the required t.
We compare this with the obtained
t, i.e., the one that we calculated. If
the obtained t is larger than the
required t, we have significance.
We say that our finding (the
difference between the two
means) is reliable or significant.
This means that we trust that, if
we run the same experiment
again, we will find a difference
again.
We formally write this as follows:
The difference between the means
of the two groups is significant
(p<0.5).
By that we mean that the finding
we are reporting is reliable, but
there is still a chance that it may
not be “real”. That chance is less
than five per cent. Scientists
around the world have agreed to
accept findings for which the
probability of being chance events
and not “real“ is less than five
percent. You understand correctly,
there is no absolute certainty in
experimental natural science.
Findings are taken to be “true” on
a probability basis. You see that
boring, compulsive statistics
borders on philosophy if
approached from the correct
angle.
One last remark. It is really
unwarranted to speak of truth in
dealing with phenomena in the
empirical, material world. We can
only speak of truth in the formal,
logical and mathematical
sciences.
Two plus three equals five. This
is true. Two plus three is
six, is false. It makes no sense to
say that the statement: “Valium at
doses of 2, 5, 10, and 20 mg
reduces anxiety” is true. It is
simply reliable and there is a
probability attached to it, no matter
how small, that it may not be so.