Statistics Review - Mathematics and Statistics for the Environment - page 108

Unless the entire population is examined, an estimate of a parameter is likely to differ to some
degree from the population value. The unique contribution of statistics to research is that it pro-
vides ways of evaluating how far off the estimate may be. This is ordinarily done by computing
confidence limits, which have a known probability of including the true value of the parameter.
For example, the mean diameter of the trees in a pine plantation may be estimated from a sample
as being 9.2 inches, with 95% confidence limits of 8.8 and 9.6 inches. These limits (if properly
obtained) tell us that, unless a 1-in-20 chance has occurred in sampling, the true mean diameter is
somewhere between 8.8 and 9.6 inches.
The second basic objective in statistics is to test some hypothesis about the population parame-
ters. A common example is a test of the hypothesis that the regression coefficient in the linear model
Y = a + bX
has some specified value (say zero). Another example is a test of the hypothesis that the difference
between the means of two populations is zero.
Again, it is the research worker who should formulate meaningful hypotheses to be tested, not
the statistician. This task can be tricky. The beginner would do well to work with the statistician
to be sure that the hypothesis is put in a form that can be tested. Once the hypothesis is set, it is up
to the statistician to work out ways of testing it and to devise efficient procedures for obtaining the
data (Freese, 1969).
7.1.1 p robability and s tatistiCs
Those who work with probabilities are commonly thought to have an advantage when it comes to
knowing, for example, the likelihood of tossing coins heads up six times in a row, the chances of
a crapshooter making several consecutive winning throws (“passes”), and other such useful bits of
information. It is fairly well known that statisticians work with probabilities; thus, they are often
associated with having the upper hand, so to speak, on predicting outcomes in games of chance.
However, statisticians also know that this assumed edge they have in games of chance is often
dependent on other factors.
The fundamental role of probability in statistical activities is often not appreciated. In putting
confidence limits on an estimated parameter, the part played by probability is fairly obvious. Less
apparent to the neophyte is the operation of probability in the testing of hypotheses. Some of them
say with derision, “You can prove anything with statistics” (remember what Disraeli said about
statistics). Anyway, the truth is, you can prove nothing; you can at most compute the probability of
something happening and let the researcher draw his own conclusions.
Let's return to our game of chance to illustrate this point. In the game of craps, the probability
of a shooter winning (making a pass) is approximately 0.493—assuming, of course, a perfectly bal-
anced set of dice and a honest shooter. Suppose now that you run up against a shooter who picks
up the dice and immediately makes seven passes in a row! It can be shown that if the probability
of making a single pass is really 0.493, then the probability of seven or more consecutive passes is
about 0.007 (or 1 in 141). This is where the job of statistics ends; you can draw your own conclusions
about the shooter. If you conclude that the shooter is pulling a fast one, then in statistical terms you
are rejecting the hypothesis that the probability of the shooter making a single pass is 0.493.
In practice, most statistical tests are of this nature. A hypothesis is formulated and an experi-
ment is conducted or a sample is selected to test it. The next step is to compute the probability of
the experimental or sample results occurring by chance if the hypothesis is true. If this probability
is less than some preselected value (perhaps 0.05 or 0.01), then the hypothesis is rejected. Note that
nothing has been proved—we haven't even proved that the hypothesis is false. We merely inferred
this because of the low probability associated with the experiment or sample results.