>> Before we go much further, let’s discuss some notation. Let Y be a continuous random
variable describing a value of interest, for example, the height of adult males or the
time it takes for a postal carrier to complete her route. Random variable Y naturally has
a distribution. And that distribution, which describes the population of the phenomenon
described by the random variable, has a population mean, μ. We usually don’t know much about
this population. We know that it exists, and we know that μ exists, but we don’t know
what is the shape of the distribution looks like or what its parameters are. All we can
do is collect a sample from the population and do our best to estimate the population
parameter of interest, mean μ.
We collect a sample of size n consisting of realizations of Y, Y1 through Yn and calculate
the mean of the sample, y-bar, which is the sum of observations y-sub-i, for i equals one to
n, and divide by n. We are interested in an estimate of the population parameter, μ.
And our best estimator for μ is the sample statistic, y-bar. Note the difference: population
parameter and sample statistic. When we’re talking about means, the population mean μ
is the population parameter, and the sample mean y-bar is the sample statistic.
The branch of statistics that lets us take a sample from a population and draw inferences
about that population is called inferential statistics. The assignment of values to a
population parameter based on a value of the corresponding sample statistic is called estimation.
The sample statistic used to estimate a population parameter is called an estimator (our estimator
for μ is y-bar).
The same is true for other population parameters and sample statistics. Sample variance s^2
is the estimator of population variance σ^2, and likewise the standard deviation s from
the sample is the estimator of population standard deviation sigma. And if we’re interested
in learning more about the proportion of a population that falls into one of two categories
(for example, the proportion that will vote “yes” on a particular ballot initiative,
or the proportion of defective cornballers produced in the US), we use the proportion
found in a sample, p-hat, as an estimator of the population proportion, p.
So why do we estimate in the first place? It’s generally too difficult, too time consuming,
and too expensive to find all members of the population and collect the realization of
our random variable from every member of the population. Consider the US census, which
collects realizations of several discrete random variables (for example, household size)
and continuous random variables (for example, household income), along with several qualitative
variables. But the US Census Bureau doesn’t collect information from every single person
or household – some aren’t known, some move during the process, some don’t respond
at all. It’s simply too difficult to collect information from everybody. The census contacts
a sample of the population to collect information, and tools of statistical inference are used
to draw conclusions about the larger population. Another example, from an engineering perspective,
deals with acceptance sampling. In some industrial settings, we may accept a batch of input components
from a supplier by inspecting a few random components in the batch, making sure that
the components in that sample meet specifications, then drawing conclusions about accepting the
entire batch based on the sample. This saves time and resources that might otherwise be
required to inspect every component in the batch.
The estimation procedure has four straightforward steps
The first is to: Select a random and representative sample from the population of interest. “Random”
suggests that any one member of the population is just as likely to be chosen to be in the
sample as any other. Choosing a random sample hopefully eliminates any sort of bias that
may affect our estimate. And “representative” suggests that members of the population generally
look the same as the population we’re trying to understand. For example, if we’re looking
to learn about the views of “people who watch the Golden Girls,” we probably shouldn’t
perform our study at a bar frequented by college students. If you want to estimate the time
required to perform a task for trained operators, don’t include in your sample a new employee
that hasn’t fully gone through training.
The second step in the estimation procedure is to collect information from the members
of the sample. Namely, this means collecting the realizations of the random variable or
variables of interest from the members of that sample. Say we want to estimate the mean
height of students at a university. The population describes the students at the university,
and the random variable that we’re interested in is the “height of students.” We choose
a random and representative sample of those students, then we measure their individual
heights. From an engineering perspective we want to learn about the mean throughput per
day of a particular production line, the sample would be a “randomly collected set of days,”
and the realizations we measure each day would be the day’s throughput.
The third step would be to calculate the sample statistic of interest. For example, if we
want to know the mean throughput each day, then we should calculate the mean throughput
of our sample.
And finally, in the fourth step, we now assign a value to the population parameter that we
want to estimate based on the sample statistic. Our estimate of the overall mean throughput
for all days is the value we found for the mean of our sample. Or our estimate for the
mean height of all students would be the sample mean of our representative sample of students.
Another brief example to illustrate these four steps could involve designing a system
around the characteristics of operators expected to use the system. Say we collect a random
and representative sample of n equals forty- two system operators, and then for each member
of the sample, we measure their “usable forward reach.” Let’s assume that our
population parameter of interest isn’t the mean reach of operators, but the proportion
of those operators whose forward reach is fewer than twenty-nine inches. That is, we
want to estimate population proportion, p. From our sample, we found that six of those
forty-two operators had a forward reach of fewer than twenty-nine inches, so sample proportion
p-hat is six divided by forty-two or point one four three. Therefore, based on our sample,
our estimate of the population proportion, p, of all system operators whose reach is
fewer than twenty-nine inches would be point one four three. Or fourteen point three percent
of operators have a reach of fewer than twenty-nine inches.
However, the value of a sample statistic that is used to estimate a population parameter
is just a point estimate. We know that it’s highly dependent upon the sample, highly dependent
upon the size n of the sample (naturally, the larger the n, the better we feel about
the conclusions we can draw about the population), and highly dependent upon the variability
within the sample (high variability in the sample might mean that the population is highly
variable and might mean that our sample may not really represent the population). We take
Everything that we know about the sample, both good and bad, and reduce it to one number.
Then we estimate the larger population based on that number. We can do better than this.
We counter this point estimate problem with an interval. We state that this interval is
likely to contain the corresponding population parameter. Back to our throughput example.
Say our lower bound on the interval is a, and the upper bound is b. We believe that
the true population throughput lies somewhere, though not entirely sure where, inside the
interval between a and b.
We call these intervals confidence intervals, as we have some amount of confidence that
the true population parameter lies on the interval. For example, we say “we’re ninety-five
percent confident that the population mean throughput lies between 𝑎 and 𝑏 based
on our sample.” The wider the interval, the more confidence we have that the population
parameter lies in the interval.
So how do we find this interval? A natural starting place is our point estimate. We then
add to the point estimate and subtract from it a value called the margin of error, to
estimate the interval. That is, the point estimate plus and minus margin of error. If
we’re looking to find an interval for the population mean, then the interval is calculated
with y-bar plus and minus margin of error.
What size should the margin of error be? Three primary elements make up the margin of error:
variability, sample size, and confidence.
Naturally, you can imagine that if the population is highly variable (that is, has a large standard
deviation), then if our sample is representative, the sample should be highly variable as well.
The larger the variability in the sample, the more difficulty we might have in pinpointing
an estimate for the population parameter we’re interested in. As such, variability affects
the size of the margin of error: the larger the variability, the larger the margin of
error needs to be for us to have faith that our population parameter lies between bounds
𝑎 and 𝑏.
The effect of variability and sample size are related. Recalling from our discussion
of sampling distributions that our estimator of population variance is a function of sample
size. For example, the variance of the sampling distribution for sample means is sigma squared
over n. As such, sample size also affects the size of the margin of error: the larger
the sample size, the smaller the margin of error needs to be for us to have faith that
our population parameter lies between bounds 𝑎 and 𝑏. This should make intuitive
sense as well: the more we witness a phenomenon, the better we might feel about estimating,
for example, the mean performance of that phenomenon. The less you observe a phenomenon,
that is the smaller the sample size n, the less faith you have in your ability to estimate
what will happen.
Finally, the last element of the margin of error is a measure of the confidence we want
to have that the population parameter lies in our interval. “Confidence” is a term
that is used regularly in inferential statistics, and it refers to the probability that the
interval calculated from a sample will contain the population parameter. Typical confidence
amounts are ninety percent, ninety-five percent, and ninety-nine percent, with ninety-five
percent being the most common. Again, a ninety-five percent confidence interval suggests that
if an experiment is repeated one hundred times (same sampling approach, same sample size),
ninety-five of the resulting one hundred intervals calculated will contain the population parameter
of interest just based on sampling. In fact, that’s why we refer to these intervals as
“confidence intervals.” Why not a hundred percent confidence interval? The interval
would be too wide as to be unusable. For example, I’m one hundred percent confident that the
sun will rise between midnight and noon. With that wide of a confidence interval, I essentially
provide no useful information.
How is confidence calculated? The typical notation for a confidence interval is a one
minus alpha, one hundred percent confidence interval. For example, for a ninety-five percent
confidence interval, alpha equals point oh five. The value of alpha is often referred
to as the significance level. In this case, ninety-five percent confidence or point oh
five significance. As such, we want to choose a margin of error such that ninety-five percent
of repeated experiments will provide us with confidence intervals that contain the population
parameter. As we’re dealing with a percentage (or a probability), you might imagine that
we could relate this margin of error to a value on the probability distribution. For
the case of creating an interval for a population mean, we use the standard normal, or Z, distribution
to derive as confidence. In this case, the values of plus and minus z on the Z axis correspond
to a one minus alpha times one hundred percent confidence interval. Note in the distribution
that one minus alpha times one hundred percent of the realizations of Z lie between negative
z and positive z.
This concludes our introduction to statistical intervals. We’ll bring these concepts to
life with a discussion of specific confidence intervals for a population mean, population
proportion, and population variance, and we’ll apply these intervals to estimate population
parameters in several engineering contexts.