BAM! Mr. Tarrou. Ok, we are going to continue our discussion about setting up confidence
intervals with using the t procedure. We are working with sample means be we only have,
like realistically, we only have the data that we collected from our sample or experiment.
So we only have the sample standard deviation, not the population standard deviation. And
in this video I am going to talk about a Pair t Procedure. When we talked about experimental
designs one of those experimental designs was called a matched pairs design. We are
going to learn how to make confidence intervals with data gathered from that Matched Pairs
design. Now in experiments that would be like where... Let me just read this, Analyze the
difference between the two treatments within a matched pairs of subjects. So maybe like
I want to see if water... If there is a difference recovery rate between those that drink water
and athletes drink say Gatorade. Well, I could matched up players that have the same weight,
the same age, have the same speed, have historically seemed to perform the same and let one drink
Gatorade during exercise and one drink water. Then compare their recovery time. That is
one way of designing Matched Pairs. But a lot of times, at least in my textbook, a lot
of times Matched Pairs design is where you put two treatments on the same subject. Maybe
you have somebody do something left handed or right handed. This example is going to
be talking about peopling taking, the same people, taking a pre and post test and then
comparing those numbers... and keeping them to some degree organized. That is what I have
said here. Analyze the mean difference between responses to two treatments applied the same
subject or unit. People don't like to be thought of numbers, so if you are collecting data
on people or if you are experimenting on people they like to be called subjects. It just feels
a little bit more personable. But it does not really mean anything different. Ok. So
you would like to analyze the effectiveness of a class you are teaching. Like Statistics.
You know I don't just teach just one period. I teach multiple periods and I have taught
it year after year after year. Well, if I would like to... um... analyze the effectiveness
of my class... So we are going to do this. We have each student take a pretest and a
post-test. And hopefully from that show an improvement in their grades. So we are going
to estimate the true mean increase in scores for all students with 95% level of confidence.
So maybe for some reason I did not have every class take a pre or post test, or maybe we
are saying I taught this for years from the same textbook... You know.. I have developed
my skills so now from year to year it is all effectively the same scores. Hopefully this
more with the added benefit of watching some videos as a review. But, the point is we are
going to allow a class to act as a sample and then use the results from that sample
to make an inference about the entire population or all these students taking this particular
class. We are going to do that by measure the grades of a pre and post tests. So let's
get on with the example, shall we. Alright. So we have got ten scores recorded. The pretest
grades are 43, 71, 50, 53, 41, 62, 59, 67, 47, and 53. The we take a posttest at the
end of the course. Now a Matched Pairs design is not just two groups of randomly selected
unrelated, or independent, sets of data. No, I want to track the pre and posttests for
each individual student. So it will very important if I am doing this actual study to keep these
numbers organized. So we have a pre and a posttest for each individual student and keeping
those organized. Because these scores are not independent of each other. It is the same
person taking both tests. Now when you have a Matched Pairs design it almost looks like
you have two samples, a group of data for a pretest and another group of data for the
posttest. Well we don't actually have two independent sets of data. We have not even
studied that yet as a class. Maybe you have watching these videos and you are just reviewing.
But we have not yet. So this must be a one sample design even though there are two treatments
and thus two sets of numbers. Indeed there are because in a matched pairs design we only
analyze the differences between the two responses. In this case the pretest and posttest scores.
So I am going to subtract each pair of these... and I have got my ten differences between
the pretest and posttests. And looks like a pretty effective class. All of those grades
have gone up. Nobody went down... or whatever. Hopefully every class is like that right if
it is effective. We are going to set up a Paired T interval. So again you have three
checks for confidence intervals. This is now my second or third video. So we need... Yes...
I did not put them down ahead of time. We need an SRS, we need Normality, and we need
Independence. So we need to assume these students represent an SRS of all students. Might not
actually be the case. Now if I am just a teacher going along with my job and I happen to have
a class of students, they may not be an actual Simple Random Sample of all the students that
I teach. But if I have no control over the scheduling this might be, you know, a bit
of a stretch to assume. But to be able to do the analysis at all we have to make that
assumption. If they are obviously not an SRS then maybe take the results with a grain of
salt. Normality... well... when you are working with averages or means you always have the
Central Limit Theorem helping you out. If you are collecting a set of numbers and averaging
them together and analyzing that average you have that Central Limit Theorem to help you
out. But when you are given the raw data make sure that during this Normality check that
you do actually show some graphical display for checking for that Normality. We are not
going to just assume it. We don't need to assume Normality. We have all the numbers
to look at. We can analyze that by making Dot Plots, or Stem Plots, or Histograms and
looking for that unimodal symmetric distribution. Unimodal and symmetric is not automatically
Normal. So we also have something called a Normal Probability Plot. I keep saying that
but it has been a while since we have done one as an example. I do however have videos
posted about it. Now with the Central Limit Theorem helping us make sure the sampling
distribution of averages, because we are not going to work with the raw individual data,
but the collection of all ten scores as an average... the Central Limit Theorem is going
to help make sure that sampling distribution of averages is approximately Normal. So I
don't really need to use a Normal Probability plot. I just need a histogram that is fairly
symmetric and unimodal and indeed we do. So with the graph here for us to help us out...
With the differences being fairly symmetric and unimodal the Central Limit Theorem will
ensure the sampling distribution of averages is approximately Normal with n=10. And at
the end of this video I will remind you again what the guidelines are for the Central Limit
Theorem of what constitutes a large enough sample to make sure that your sample averages
are indeed approximately Normal. So let's see what that confidence interval looks like...
shall we. Ok. So anyway... third check. Independence. Assuming the tests are taking individually
and not cooperatively and that we can assume that there are more than 100 students... ten
times the sample size... we should be ok with the idea of independence. So from One Var
Stats and that list of differences that we just had we get an average difference of 29.4.
The standard deviation of our sample is 8.09. The degrees of freedom is n-1 when you are
dealing with t-distributions. N being the sample size. 10-1 is 9. And then when you
go to the t-distribution go to the bottom of that chart, find where it says... What
level of confidence are we using here? 95 percent. Go to the bottom of that t score
chart, find where it says 95% confidence, and go up the chart until you get to a degree
of freedom line of 9. Now if your sample size is such that you don't have... like maybe
the degree of freedom is 131. Well, my t-table chart does not have a DF line of 131. So I
would go up the chart and use the degree of freedom line of 100 just to be conservative.
There is a df line of 9 though and so my critical value, my t star value, is 2.262. So we are
going to take the sample's average of 29.4 then add or subtract 2.262 Standard Errors
away from the mean... Actually I did not write that out in this list. 8.09 divided by the
square root of 10 is... I am calculating the standard error of the sample mean. I just
talked about that in my previous video, but you might not be watching both of these. When
you use a sample standard deviation to calculate the Standard Error of the sample mean it looks
just like it did with Standard Deviation of the Sample Mean. It is the standard deviation
divided by the square root of n. It is just that you are using the standard deviation
from the sample. So estimate plus or minus the critical value times the standard deviation
of the estimate if you are using the words that are off the formula sheet on the AP exam.
And I am 95 percent confident that the true mean increase is between 23.6 and 35.2 points.
Ok. The Robustness of a T Procedure. Robustness is the ability of the statistical test, or
this process in general like this confidence interval, to resist non-Normalities in the
data. Well, when we are working with averages we have the Central Limit Theorem to help
us resist or take care of... or handle.. any non-Normality that we have in the raw distribution
or data. So the Normality of the population becomes less important as sample size increases.
That is giving the Central Limit Theorem more data to work with and make sure the Sampling
Distribution of the Sample Averages is approximately Normal. And so we are working with means and
the Central Limit Theorem to ensure averages will be approximately Normal if n is large
enough. And again the guidelines, this will be probably the third time I have included
this in my videos... If n is less than 15 the t procedures is safe, and that could be
a t test which we have not learned yet or this t confidence interval, the t procedure
is safe if the data is approximately Normal. It will need to be pretty symmetric and unimodal.
If n is between 15 and 30, that is a pretty good size for the Central Limit Theorem to
work. So the data cannot have any really strong skewness or outliers. But if there is some
skewness and some Non-Normality, the Central Limit Theorem is still going to make sure
your sampling distribution of averages, in this case it was a sample of 10 pieces of
data, that sample distribution of averaged values will be approximately Normal. And then
if the raw data, or if your raw sample, has very strong skewness then it is only going
to be safe to do a t test or a t confidence interval if your sample size is over 30. So,
I am Mr. Tarrou. That is how to do a confidence interval with a Matched Pairs Design or a
matched t procedure like confidence. Go Do Your Homework. BAM!