Sampling and Estimation. Here are the sections in
this reading. We'll talk about how to use sample data
to estimate population parameters and our main focus
will be on the population mean generally denoted
by mu. Let us understand the basic concept of sampling.
Say you have a large population such as the returns
on all stocks in the United States for last year, and
you are interested in the mean return or the average
return, that is an example of a parameter, generally
denoted by the symbol, mu, so a parameter such as the
mean return is used to describe a population. Population
here being the return on all stocks in the United
States last year. You might not have the time to go
through every single stock and come up with the average
return. So what you could do is pull a sample from
the population. So let's say you pull out a sample
of 100 stocks and then you find the average of these
100, that average is generally denoted by X bar,
and this is called a statistic so a statistic is used to describe
the sample. Simple random sampling a simple random
sample is a subset of a larger population such that
each element has an equal probability of being selected
to the subset. So let's say that this population
has a thousand items. So simplistically, let's
say that there are a thousand stocks in our overall
population. When we create a sample of hundred, if
every stock in this population has an equal probability
of being selected when we say that we have a simple
random sample. The next concept you need to know
is that of Sampling Error. Clearly this population
has a certain mean called mu, and when you come up
with a sample and you compute the sample mean, this
sample mean is not necessarily going to be exactly
the same as the population mean. The difference between
the sample mean and the population mean is called
the sampling error. So to put this formally, sampling
error is the difference between the observed value
of the statistic and the quantity it is intended
to estimate. Sample Distribution of a Statistic.
Sample Distribution of a Statistic is the distribution
of all the distinct possible values that the statistic
can assume when computed from a randomly drawn sample.
Let's continue with our example. So we have this
large population which consists of all stocks in
the United States and you are concerned about the
population mean. Let's say you draw the first sample
with a sample size equal to 100 and you come up with the
mean for sample 1 and let's say that that number
is 10%. Then you draw another sample, obviously the
mean here will not be exactly the same as the mean
for the first sample. Assume that you are drawing
the same sample size, let's say that the mean for
sample 2 turns out to be 11% then you draw a third sample,
and here the mean might be 9% and so on. You keep drawing
samples, and you will notice that there is a certain
distribution of the sample means and that is called
the sample distribution of a statistic. So now this
definition will make more sense. Sample distribution
of a statistic is the distribution of all the distinct
possible values that the statistic can assume when
computed from a randomly drawn sample. Let us now
take a look at Stratified Random Sampling. Say you
want to become a little more sophisticated in terms
of picking your sample from this large population
of American stocks and you recognize the fact that
they are different exchanges and you want to make
sure that your sample has a representation from
each exchange. So what you can do is divide the population
into strata based on one or more classification
criteria, in my example, the classification criteria
is the different exchanges so you can say there is
exchanged 1 exchange 2 exchange 3 and exchange
4 so now you have the different sub-populations
or strata next you pull a sample from each sub-population,
then you draw a simple random sample from each stratum
in sizes proportional to the relative size of each
stratum in the population. In other words, if exchange
2 is two times bigger than exchange 1 then clearly
the random sample that you draw from exchange 2
must be two times bigger than exchange 1 so here
you might have 20 stocks, here, you would pull a sample
of 10 and so on. In my simple example exchange 3
is the same size as exchange 2 so here you will
also have 20 and exchange 4 is the same size as exchange
1 so here you would have 10. And then you pool these
samples to form a stratified random sample so hear
your sample of 60 has representation from all four
sub-populations, and this would be called a Stratified
Random Sample. Let us look at a simple practice
question. Paul wants to categorize publicly listed
stocks for his research project. He first divides
the stocks into 15 industries. Then from each industry,
he categorizes companies into three groups.
Finally, he divides these into value vs growth
stocks. How many cells of strata does the sampling
plan entail? Now this is a little more complicated
than what I just talked about and here we need to recognize
that there are 15 Industries, for each there are three groups,
and then we have value vs growth so we multiply it by 2 so the correct
answer is simply a product of these three numbers,
which is 90. Time Series and Cross-Sectional Data. Time
Series is a sample of observations taken at specific
and equally spaced points in time. For example,
the monthly returns on Microsoft stock from January
1995 to January 2005 that is fairly self-explanatory.
Cross-sectional data is a sample of observations
taken at a single point in time. So for example, the
sample of reported earnings per share for all NASDAQ
companies for 2005, so t notice that here, the data
is for 2005 for a range of companies and this would
be an example of cross-sectional data. For both
time series and cross-sectional data the random
sample must be representative of the populations
we wish to study. Consider this practice question.
A researcher needs to make use of the 2012 household
budget data for Scandinavian countries. Is this
cross-sectional data, time series or panel data? Based
on what we've just talked about, this is cross-sectional
data. Distribution of the Sample Mean. Let's say
you draw several samples from the population and
for every sample you compute the mean. Clearly, there
will be a certain distribution for the sample means
and that is what we are going to talk about. For a population
with mean, mu, and variance, sigma squared, the sampling
distribution of the sample means so that is the
distribution of the X bars. Of all possible samples
of size n, each of these samples needs to be of size n
and let's say that n is 100 in our example,
so the distribution of the sample mean will be approximately
normal with a mean equal to mu and a variance equal
two sigma squared over n. So there are 3 core statements
being made. The first one is that the distribution
of x bar or the distribution of the sample mean is
going to be normal. The second point is that the mean
is mu which is essentially the mean of the population,
and the variance of this distribution is sigma squared
over n. Hopefully it is fairly obvious that the mean
of this distribution is mu because you are drawing
these samples from the population. So the expected
value of x bar or the sample mean should be the population
mean. In terms of the variance think of it this way, if the population has a very
high variance, so the population data is spread
out a lot, then you would expect x bar to also be spread
out. In other words, the distribution of x bar would
also have a high variance. That is why we have sigma
squared in the numerator, on the other hand if the
size of the sample is large then the distribution
of x bar become smaller because largest samples
mean that you are more likely to be close to the population
mean. In other words, larger sample means that you
are more likely to have X bar or a sample mean that is
closer to the population mean that's why n is in
the numerator. What we have just learned is the Central
Limit Theorem and I will repeat exactly what is
stated in the curriculum. Given a population described
by any probability distribution and this is critical
because the central limit theorem applies to a distribution
whether or not it is normal. So the population has
a mean mu and a finite variance sigma squared, this
is what we saw on the last slide, the sampling distribution
of the sample mean, the sample mean is the X bar and
we are talking about the distribution of x bar assuming
that you pull several samples from the mean computed
from samples of size n. Again, I'm repeating because
this is important, the sample size always has to be
the same. The distribution of x bar will be approximately
normal with a mean equal to mu and variance equal
to sigma squared over n. When the sample size is
large, and large means a sample size of 30 or more.
The standard error of the sample mean is the standard
deviation of the distribution of the sample means.
So there is a new piece of terminology but the concept
of straightforward. We just said that the variance
of the distribution of x bar is sigma squared over
n. The standard deviation, which is simply the square
root of this expression is essentially sigma over
root n, that is the standard error of the sample mean.
If we know the population variance then we simply
plug that here and we compute the standard deviation
of X bar or the sample mean using this expression.
If the population variance is not known then the
standard deviation for the distribution of x bar
is equal to s, this is the standard deviation of the
sample. Clearly, if the population variance is not known
then we use the variance or the standard deviation
of the sample as a proxy. And again divide that by root n so both
these expressions are similar, here we are using
the population standard deviation because it's
known. Here we are using the sample standard deviation
because we don't know the population standard deviation.
Let's look at this simple question. You need to use
the formula that we just discussed which is sigma
over root n, sigma is 3, n is 64 so 3 over root of 64 should
give you A, which is 0.375