Download The T Distribution

The T Distribution
©Dr. B. C. Paul 2005
Wasn’t the Herby Assembly Line
Problem Fun

But there is one little problem


We knew that our mean value could have
been all over the map relative to the real true
mean
We calculated our standard deviation from
the same sample

How come our mean could be anything and
yet our standard deviation is God’s own value
for the standard deviation?
It Isn't


When our value for the standard deviation is just
an estimate we have another chance for things
to be way out in the tails
Sadisticians – woops I mean statisticians figured
out probability distribution for what would
happen then

Called it the T distribution


First published in 1908 perfected in 1926
We look up values for areas under the curve of a
T distribution just like we did with a normal
distribution.
Let’s Redo Herby’s Problem Right
This Time

We will use the T distribution
X 
t
s
n
S is the estimated standard deviation
The test statistic has a T distribution (assuming the underyling population
Really is normally distributed)
The distribution has n-1 degrees of freedom
Degrees of Freedom! What are you
talking about? – this isn’t an
Amnesty International Class

Consider # of equations and # of unknowns


Each sample is like an equation



To uniquely solve 3 unknowns you need 3 independent
equations
If I have one sample I first use it as an estimate of the mean.
I can’t calculate a standard deviation – I don’t have enough data
If I have two samples

I can estimate std deviation and still have one degree of
freedom to measure something else


Happens to be the mean
How much extra data do I have above the bear
minimum?
So How Do I Use This?
(I have a really bad feeling your going to tell me)
Note that this table is set up
Different from Z values for normal
Distribution.
Area under the curve comes from
The top line.
Degrees of Freedom from the
side
Value in the middle is the T value
(equivalent to the Z value)
Remember in the normal table
The Z value was on the edge
And the area under the curve
In the middle of the table
Lets Do the Problem
s
  X  t *
n
X = 3.8
S= 0.73
N= 7
OK – So What Is t?
Finding t
If we do this as a two tailed test
(ie we would be concerned if our
Balls were to hard or to soft) we
Can only have 2.5% in each tail
Pick 97.5
We have 7 samples hence n-1 or
6 degrees of freedom
Read into the table
2.45
Plug and Chug
0.73
UpperLimit  3.8  2.45 *
7
4.48
We can still reject the null hypothesis with an
Alpha Level of 5% but it is now much closer
Than before
Some Observations About Degrees
of Freedom and the T statistic






95% of a normal distribution is within 1.96 standard
deviations of the mean
95% of a T distribution is within 2.45 estimated standard
deviations of the mean if the standard deviation estimate
came from 7 samples
With 20 samples it is 2.09 estimated standard deviation
units
With 50 samples it is 2.01
With 100 samples it is 1.98
With 500 samples it is 1.96

Note that as the number of samples increases the T distribution
converges to a normal distribution
So When Do I Use a T Distribution


The underlying population must be realistic to model as
having a normal distribution
The standard deviation of the population must have
been estimated from a standard deviation calculation
using a sample of the population


You can get out of using the T distribution and pretend that God
gave you the standard deviation if you used about 100 or more
samples to calculate your estimate of the standard deviation
People with a lot of experience with a distribution often
ignore the T distribution completely because they have
seen results from hundreds of samples

They are not “doing it wrong” using a simple normal distribution
if they have that kind of data supporting their standard deviation
value
Why Did You Do a Two Tailed Test?

Herby was going Bananas because he thought the line
might be putting out soft balls


That sounds to me like he is only concerned about 1 side of the
distribution.
We may be upset about one particular thing but that
doesn’t mean nothing else is important



One problem with things that are too hard is that they are often
brittle
Premature ball failure could be due to the balls being too soft or
breaking up because they are too hard
We have to ask our own case specific question about what we
are concerned about – You plan a one tailed test only if you are
only concerned about events on just one tail
Common Cheating on Random
Samples

Experiments should be planned before we look at the
data

If we look at the data and then decide what the experiment
should have been we are “political spin doctors” not scientists



Often we had a theory that made us want to look deeper



A spin doctor looks at a result and then tries to make it say what he
wants
A scientist sets up the test and lets the truth be what ever it is
Many theories are based on observations
But the scientific method causes you to then plan an experiment
and go out and get the data you need to test the theory
It’s a subtle difference but its often ignored

The doctrine of “political correctness” is causing us all to loose
our integrity
Back to Herby and the Two Tailed
Test


If it is true that hard balls make no difference – only soft
ones then the test should have been set up as one tailed
only
If the concern was the line being out of spec and that
causing unhappy customers we could not know the
sample would come out below 4.5 unless we peaked first



If at that point we decided we only cared about soft balls we
distort the reliability of our analysis
The data would have not only determined what the values of the
test statistics were – it would have determined the test
Normal distribution theory only accounts for the data
determining the test statistic

We in fact do not have good models for exactly what the
consequences are if we let the data set up the test – we can say we
are taking a chance of something bad happening
My Choice

So why did I do this example as a two tailed test




1- because that sample size analysis I did is nastier to
explain if I’m only working on one side
2- Because it sets up a great discussion on random
samples and peaking and cherry picking data
3- Because it allowed me to discuss when I should
run one and two tailed tests
The story problem told is inconclusive about
whether Herby was vulnerable to the line being
out of spec on one side only or on both sides
Look at the Problems We Have
Run So Far

We looked at a storm washing out the drainage system in a
subdivision

Only too much rain would create the disaster – we really only were
worried about too big rain events


We looked at a Mine and the amount of ore below cut-off grade that
would go to the dump

We aren’t going to dump our high grade ore – we really only care about
how much stuff is on the lower end


(And we ran a one tailed test on the lower side)
We looked at tolerance on a machined part

The spec said we had to be plus or minus so our customer would be
upset if the pegs were too big or too little


(And we ran a one tailed test on the upper side)
(And we ran a two tailed test)
Determine whether to run a one or two tailed test based on the
concerns for the process or design you are working on – not from
peaking at the data.