So we've said that the standard normal distribution is the Z distribution, and that we can use the
Z table to look up probabilities associated with that Z distribution.
Now we want to talk about what's called the student's T distribution.
So it is also a probability distribution, like the standard normal distribution represented by Z.
So we're talking about the student's T distribution.
It is also associated with its own T table in the same way that the Z distribution is associated with
the Z table.
And of course the T table gives us probabilities associated with this T distribution.
Now the T distribution is similar to the Z distribution in the sense that it is symmetrical, it's bell
shaped and it's centered around this mean of zero.
The mean is always zero, but the T distribution is flatter and wider than the standard normal distribution,
which means that more of the area under the T distribution is pushed out toward the tails, whereas
more of the area in the Z distribution is collected around the mean.
What that means, of course, is that the standard deviation of the T distribution is larger than the
standard deviation of the Z distribution or the standard normal distribution.
Now, the reason that we have this T distribution is because if we're ever using a small enough sample,
the Z distribution becomes unreliable, as we've seen before, and we can kind of intuitively think
about this.
The smaller our sample size, the less reliably that sample is representative of the population.
And so because the sample is less reliable, of course, the standard deviation is going to be larger,
since we can't be as certain that we'll find a value around the mean.
In other words, a smaller sample means less certainty, which means a larger standard deviation, which
means more area is pushed out toward the tails of the distribution.
Now, it turns out that when we're taking small samples, like a sample size of ten or a sample size
of 15, that sample size is too small to reliably use the Z distribution.
So instead we'll use the T distribution.
But if we keep increasing that sample size, the magic number is a sample size of about 30.
A sample size of 30, at which point this T distribution matches almost exactly the standard normal
distribution.
In other words, you can almost imagine here with this T distribution, the smaller the sample is,
as we said before, the larger the standard deviation, which means the flatter this blue curve becomes.
So the tip of the blue curve, the tip of the T distribution moves down like this and the curve gets
flatter as our sample size gets smaller.
But as the sample size gets larger, the peak of this blue curve increases, it moves up, which pulls
in the area from the left and right side pulls in the area closer in toward the mean, pulling an area
away from the tails and in toward the center of the distribution.
And so the top of the distribution starts to rise and it's at this sample size and equals 30 where the
T distribution has risen enough to match almost exactly the standard normal distribution, the Z distribution.
And so what we can say is that for sample sizes of 30 or greater, in other words, and greater than
or equal to 30, the values we would get from the T distribution and Z distribution are virtually identical.
And so at that point we can revert to our Z distribution and use Z scores looking up values in the Z
table, but for samples that are smaller than 30.
So when NW is less than 30, that's when the T distribution is going to give us more accurate values.
And so we want to find T scores and look them up in the T table.
So we'll use values of T when N is less than 30, we'll use values of Z when N is greater than or equal
to 30.
Now, to go back to this point of the T distribution, we said that the peak of this distribution,
the height of it increases or moves up as sample size increases and moves down as sample size decreases.
Effectively, what we're saying there is that the shape of the T distribution changes based on the sample
size and more specifically, we say that it changes based on what we call the degrees of freedom of
the sample and the degrees of freedom is equal to n minus one where N is the sample size.
So if our sample size is 20, then we'll have 19 degrees of freedom.
If our sample size is ten, we'll have nine degrees of freedom.
Keep in mind that sample size isn't the only condition that determines when we should use a T score
versus a Z score.
For instance, whenever population standard deviation is unknown, we want to use a T score instead
of a Z score.
So we'll be using the T score and the T table very often, which is why it's critical that we look at
it here along with the Z table before we move.
Forward.
So when it comes to looking up values in the tea table, here is our tea table.
Notice that along this left hand side here, down this first column, we show degrees of freedom, starting
at one all the way up to 30.
And the reason that we stop there, you can find tea tables on line for larger degrees of freedom.
But the reason that the standard tea table stops here is because, remember, this is our magic number,
at which point the Z table can take over because the values from the T and Z tables are almost identical
for larger sample sizes, specifically sample sizes greater than or equal to 30.
So in our standard tea table, we just list degrees of freedom from 1 to 30.
So looking up values in the tea table requires us to know the degrees of freedom, which again is given
by n minus one where n is the sample size and to know either upper tail probability which we see here
across the top of the tea table or confidence level, which we see here along the bottom of the table.
Now, we'll talk more later about upper tail probability and confidence level.
All we want to focus on right now is just how to locate values in the tea table.
So we understand that at minimum.
And then later on, we'll dive into these two concepts further so that we can understand them, pair
them with degrees of freedom, and find the correct value within the body of the tea table.
So when it comes to just locating a value in the tea table, we look for the intersection in the body
of the table between degrees of freedom and either upper tail probability or confidence level.
So let's say we know that our upper tail probability is 0.15 or 15%.
That puts us in this third column here.
And let's say we know that we have six degrees of freedom.
Well, of course we would come down here to six degrees of freedom over to this third column and we
would locate this value right here, 1.134 as the intersection of six degrees of freedom, meaning a
sample size of seven and 0.15 upper tail probability or 15% of the area under the tea distribution in
the upper tail.
Now, we said that we could use upper tail probability or confidence level, and the reason is because
upper tail probability and confidence level are kind of two different ways of saying the same thing
or describing the same scenario.
So an upper tail probability of 0.15, if we come all the way down to the bottom of the column here,
we see this 70% figure for confidence level, a confidence level of 70% and an upper tail probability
of 0.15 mean the same thing.
What you can notice if you compare all of the confidence levels with all of the upper tail probabilities,
is that if we look here at confidence level, let's say we have 70% here.
If we subtract 70% from 100%, we get 30%.
If we divide that value by two, we get 15%.
And we see that that is the value here for upper tail probability 0.15 and that holds across the board.
So if we go over here to a 99% confidence level, if we subtract that from 100%, we get 1%.
If we divide 1% by two, we get half a percent.
And if we come up to the top of the column here, we can see that that upper tail probability is half
of 1%.
So really, we're looking up the same thing, whether we look up 0.15 upper tail probability or 70%
for the confidence level.
But sometimes we'll have confidence level and sometimes we'll have upper tail probability.
And so we include both of those in the tea table so that we can quickly find whatever value it is that
we're working with and then look up the corresponding value from the body of the tea table.
Now, the only other thing we really want to say at this point is that, of course all of these values
are important and useful, but the values that we will use by far the most often are given by 90 or
95% confidence or 99% confidence.
And of course, those are associated with upper tail probabilities of 5%, 2.5% and half of a percent.
Respectively.
Again, like we said earlier, we'll talk about these things later, but it is very common for us to
choose a confidence level of 90%, 95% or 99%.
Those are common confidence levels to work with, sort of think about them as industry standards.
And so these are the values we'll see the most, which means that we will very often find ourselves
within these columns of the tea table, not that we won't use the other columns ever, but these will
probably be the columns we'll use most often.
And so we just really want to be aware of these things.
In particular, 0.05 upper table probability is equal to a 90% confidence level.
0.0 to 5 upper tail probability is the same thing as 95% confidence level and 0.005 upper tail probability
is equivalent to a 99% confidence level.
If we can be very comfortable remembering these relationships, then that'll help us a lot moving forward.
So that's it for the student's T distribution.
We just wanted to introduce this idea that the tea distribution is a normal curve symmetric, centered
at the mean of zero, just like the standard normal distribution, but that the exact shape of the tea
distribution will change depending on sample size.
And we will rely heavily on the tea distribution, especially when we're taking small samples or when
population standard deviation is unknown, which it very, very commonly is unknown.
So we'll rely on the tea distribution and therefore the tea table very often.
But now that we understand the difference between the standard normal distribution and the tea distribution
going forward will have the foundation, we need to start investigating hypotheses and making statements
about our level of confidence surrounding the values of the population parameters, specifically the
actual value of the population mean.