>>And now we explore the last population parameter we might be interested in for two different
populations, the variance. Generally speaking we said that variability is often viewed as
a bad thing. Variable processes and systems make predicting system behavior or depending
on particular behavior difficult. And so we as engineers may often want to know how the
variability of two processes or systems compare with each other. Is the variability in the
diameter of output from this machine the same as it is for that machine? Is the variability
in the weight of component parts from supplier one the same as from supplier two? Is the
variability in measurements from one bridge inspector the same as another bridge inspector?
And so on. We start off with confidence interval for the ratio variances of population one
and population two. This confidence interval is a little tricky. Because of the nature
of the F-distribution it doesn’t follow the pattern of point estimate plus and minus
a margin of error. We know that s-one squared over sigma-one squared divided by s-two squared
over sigma-two squared follows an F-distribution with n-one minus one degrees of freedom in
the numerator and n-two minus one degrees of freedom in the denominator. To find a one
minus alpha times one hundred percent confidence interval for this ratio we bound it by an
F-value with alpha over two in the upper tail and an F-value with alpha over two in the
lower tail. Of course the alpha over two in the lower tail corresponds to one minus alpha
over two in the upper tail. What we’re ultimately interested in a confidence interval for the
ratio of sigma-one squared to sigma-two squared. The ratio of these two variances gives us
an idea about how different they are. If the ratio is large then the variance of population
one is likely greater than that of population two. If the ratio is small then population
one is less variable than population two. With some algebraic manipulation we isolate
the ratio of sigma-one squared to sigma-two squared and find bounds for this ratio as
a function of sample variances in F-values. Note that the degrees of freedom are reversed
from the bounds of the previous ratio. That’s due to the algebra required to isolate the
sigma-one squared to sigma-two squared. Working out this algebra would make a great homework
problem. So let’s put this confidence interval to work with an example. Say that two different
chemical companies can supply raw material. The concentration of nitrogen in this material
is important. The mean concentration of both suppliers is the same but we have reason to
believe that the variability in nitrogen concentration may be different between the two companies.
The standard deviation of nitrogen concentration in a random sample of ten batches produced
by company one is four point nine grams per liter. For company two a random sample of
sixteen batches yields five point eight grams per liter. Let’s find a ninety-five percent
confidence interval for the ratio variances of nitrogen concentration for the two companies.
Let’s first look up the F-values. The degrees of freedom for population one are ten minus
one or nine. For population two we have sixteen minus one or fifteen degrees of freedom. For
the upper bound we need a value of F for fifteen degrees of freedom in the numerator, nine
degrees of freedom in the denominator and an alpha over two of point oh two five in
the upper tail. We go to the table for point oh two five in the upper tail. We go over
to the column associated with fifteen degrees of freedom in the numerator and down to row
nine. That intersection is three point seven seven. To find the F-value in the lower bound
we have fifteen degrees of freedom in the numerator. Nine in the denominator and we
want one minus alpha over two in the upper tail. That’s equal to the inverse of the
F-value with degrees of freedom reversed and just alpha over two in the upper tail. We
go to the F-table for point oh two five in the upper tail over to the column for nine
degrees of freedom and down to the row for fifteen degrees of freedom. And we find that
the F-value is three point one two. One divided by three point one two is three point one.
So the upper bound on the confidence interval is four point nine squared divided by five
point eight squared times three point seven seven. Which is two point six nine. The lower
bound is four point nine squared divided by five point eight squared times point three
two one, which is point two three. So the ratio of the variance of nitrogen concentration
in the material produced by company one to that produced by company two lies between
point two three and two point six nine with ninety-five percent confidence. We could also
say that we’re ninety-five percent confident that the true ratio of standard deviations
lies between point four eight and one point six four by taking the standard deviation
of both sides. This suggests that company one is as little as half its variable up to
roughly one and a half times it’s variable as company two. Based on this result we can’t
really draw strong conclusions about which company produces a more highly variable product.
And of course we must also talk about hypothesis testing. We might want to know that the variance
from one process or system is equal to that of another. We could have an equality statement
in the null hypothesis with less than in the alternative. Or we could have greater than
in the alternative. The same three options as always. The test statistic is F-naught.
The ratio of S-one squared to s-two squared. If the ratio is large enough maybe the variance
in population one is larger than that of population two. Likewise that the ratio is small enough
maybe the variance of population two is bigger. And of course large enough and small enough
is determined by the rejection region of size alpha. The critical value and p-value approaches
still apply. The critical value approach is a little easier with tables and either approach
is made easy with Excel. Some textbooks suggest to always put the larger of the two sample
variances on top to force an upper tail test. But accounting for two tailed upper tail and
lower tail tests lead to a better understanding of what the test statistic is really telling
us. Let’s take a look at an example. There’s a field of Mathematics that studies waiting
in lines. From people standing in line for an attraction at an amusement park to raw
materials waiting to be machines on a factory floor. The fancy word for a line is a queue.
In the field of mathematics that studies queues is called queuing theory. One thing we learned
from queuing theory is how to design lines. Most service counters, say at a bank or a
fast food restaurant, have one of two kinds of queue designs. The first is where there’s
a number C of servers behind a counter and C, corresponding lines form for customers
to wait for an individual server. When a customer enters, here she would rationally try to get
in the shortest of the C lines. But of course there’s no guarantee that this leads to
the shortest waiting time. As a service time for customers ahead of them could get lengthy.
In the second queue design a single queue is formed usually with a barrier rope. When
the server finishes with a customer the next customer in the front of the single queue
approaches the counter. As it turns out the average waiting time for these two designs
is roughly the same. However the variability of waiting time can be much larger for the
first design. Waiting in line behind an occasional especially lengthy customer can lead to periodic
waiting time values that are longer than others. Thus this leads to variability in waiting
time. Let’s say we set up an experiment to explore the difference in variances in
waiting times resulting from these two queue designs and we want to know is the variability
of the individual line, which we’ll call population one greater than that of the variability
of a single line, which we’ll call population two. Say we collected a sample of twenty-five
individuals serviced in individual lines under conditions of average busyness. And we found
a standard deviation of waiting time of two point three minutes. And we collected a sample
of sixteen individuals serviced from a single queue and we found that they had a waiting
time standard deviation of one point one minutes. The test statistic is two point three squared
divided by one point one squared. Which is four point three seven. We have twenty-five
minus one or twenty-four degrees of freedom in the numerator and sixteen minus one or
fifteen degrees of freedom in the denominator. Let’s say we want to implement the p-value
approach using the F-tables. The smallest area in the tail which we’re given in the
table is point oh one. And going over to the column associated with twenty-four degrees
of freedom in the numerator and down to row for fifteen in the denominator we find three
point two nine. Our F-naught statistic of four point three seven lies beyond three point
two nine. Suggesting that the p-value is smaller than point oh one. A small p-value, smaller
than most any alpha we’d ever choose, leads us to reject the null hypothesis. Based on
our samples we would conclude that the variability of waiting time resulting from multiple lines
is indeed longer than that resulting from a single line. We can use Excel to get a better
estimate of a P-value with the fdist function we find that the p-value is point zero zero
two four. Again a very small number. And so that sums up statistical inference for differences
in population variances. Recall that in statistical inferences for differences in means we have
two different approaches depending on whether we can assume the variances of the two populations
are equal or unequal. As you might imagine we can first perform a statistical comparison
of variances with an F-test to determine whether they are equal or unequal before we then proceed
to the appropriate t-test for comparing means. It’s common practice to perform statistical
tests for various assumptions before performing the ultimate statistical analyses that you
might want to address.