| Chapter.Page |
Comments |
| 1.2-1.6 |
Here are some of the problems
I have encountered in the past decade or so that can be considered in
the domain of environmental statistics:
 | Policy analysis |
| RCRA groundwater monitoring |
| Insurance cost recovery estimation and negotiation |
| Superfund (CERCLA) investigations (mainly soil and
groundwater, but sometimes surface water, too) |
| Controlling industrial processes |
| Air monitoring (ambient air and in the workplace) |
| Predicting future conditions (for monitoring,
cleanup, and land cover change, for example) |
| Surface water (river) monitoring, permitting, and
TMDL ("total maximum daily load") estimation |
| Monitoring land cover and land use, species
richness, ecological diversity, and so on |
| Waste discharge control |
| Land re-use and development |
| Determining when cleanup is over |
| Human and ecological risk assessment |
| Determining whether "waste" is
"hazardous" |
| Managing natural disasters (floods, for example) |
| PCB detection, assessment, and cleanup (as
regulated by TSCA) |
| Workplace monitoring and safety assurance (OSHA) |
|
| 1.7 |
The text asks, "what is
the probability of picking an ace out of a deck of 52 well-shuffled
standard playing cards?" Obviously it is 4/52 because there are
four aces in the deck. A variation of this question is to ask what
is the probability of picking an ace at least once in twenty draws of a
card from this deck (replacing each card in the deck at random after
each draw). This kind of question can model situations such as the
following: you (or a client) has monitored waste discharges to a river
quarterly for 13 years. Four times, a result exceeded the
permitted limit. The permit is about to be renewed for five years
(20 quarters of monitoring). If the waste process and the permit
limits remain the same, what is the probability there will be no
violation during these five years? (The answer makes many
assumptions about the process, but we will not go into that
here.) It is helpful to be able to estimate answers to
questions like this quickly. In this case, the probability of not
drawing an ace each time is 48/52 = 12/13 = 1 - 1/13. Therefore
the probability of not drawing an ace all 20 times is (1 - 1/13)20
. If you are familiar with the approximation (1 - k/n)n
~ exp(-k), you will recognize this is a good approximation in the
present instance, and so will compute (1 - 1/13)20 = ((1 -
1/13)13)20/13 ~ exp(-1)1.5 = exp(-1.5)
= somewhere between 0.2 and 0.25 (using approximations we will
learn). Whence the probability of drawing an ace (or of violating
the permit at least once) is around 70 to 80%. With a little
practice you can do this kind of computation in your head in a few
seconds. |
| 1.9 |
Make a copy of Figure
2.1. Write down as many additional "sources of
variability" as you can think of. Keep updating your copy
throughout the course. |
| 3.53 |
The admonition to
"thoroughly examine the data in as many ways as possible and
relevant" is very good advice. Consider this: one datum
typically costs several to several hundred dollars. (For example,
a large soil sampling program will gather data on 10 to 20 relevant
chemicals for around 1,000 soil samples: about 20,000 data. It
will cost hundreds of thousands of dollars, averaging maybe $25 per
number.) Despite this cost, careful examination of the data--which
might take only a few hours' to a few days' time with a good set of
computer software and therefore cost a few hundreds or thousands of
dollars--is rare indeed. It is unusual to encounter any
environmental dataset that does not yield useful information upon close
examination. This state of affairs should be
incomprehensible to any observer from the outside looking in: why should
we spend over 99% of the budget just gathering the numbers and less than
1% actually looking at them? Ideally, the proportions
should be reversed! |
| 3.53 |
You can add a lot to any
statistical analysis by understanding what you are analyzing.
Think about this: what properties of 1,2,3,4-tetrachlorobenzene might be
useful to know? Make a short list and look them up. It will
help you understand the rest of the chapter a little better. (You
may consult Aldrich
Chemical or Fisher
Scientific to get started.) |
| 3.57 |
The "MAD" displayed
in these summaries is not the MAD as defined on page 3.65
(formulas 3.14 and 3.15). The values shown on page 3.57 are the
MAD divided by 0.68, approximately. We will learn later why this
has been done. |
| 3.61 |
To test your understanding of
the formulas, demonstrate mathematically that formula 3.2 for the
trimmed mean gives formula 3.3 for the median when the trimming
proportion (alpha) is 0.5. If this causes you trouble, or you are
not sure what to do, then ask in class (or privately by e-mail). |
| 3.61 |
The remark "half of the
observations lie below the median and half of them lie above the
median" is not strictly true. What are the exceptions? |
| 3.62 |
Evidently, "log()"
in this text means natural logarithm, not logarithm to base 10.
See page 175. |
| 3.66 |
The unbiased estimator of
skewness (formula 3.18) is the one used by most software, including
Systat and Excel. |
| 3.67 |
The formula
is correct, but the text preceding the formula should say the 4th power
is divided by the square of the variance (or the fourth power
of the standard deviation). |
| 3.67 |
There are also unbiased
estimators of kurtosis. See the Excel help for "Kurt"
for one formula. |
| 3.69 |
Tufte
provides a graphical redesign of the dot plot. It is a shame
Tufte's ideas, even after 17 years, have not been accepted by software
designers: you will not see many of his improvements even in the S-Plus
software. |
| 3.81 |
Note that the vertical
positions of the dots in the strip plots of figure 3.9 are randomly
"jittered." This minimizes overlaps of dots from tied or
nearly-tied values. |
| 3.84 |
[Hoaglin
et al] provides some guidance on how to choose histogram interval
widths. |
| 3.85 |
A density plot is a kind of
"one dimensional contour plot." For details on how the
interpolation might be done (in one and two dimensions), see the Map Algebra- Resampling
pages, for instance. |
| 3.88 |
The assertion that "you
would expect to see at least one 'outside value' only about 0.7% of the
time if you were always looking at data from a normal distribution"
is misleading. The 0.7% applies to each datum.
For example, if you are looking at a batch of 20 values drawn
independently from a common normal distribution, then a simple
calculation suggests the chance of seeing at least one outside value is
around 20 * 0.7 = 14%. The simple calculation is incorrect,
however, as "Student" (W. Gosset) pointed out in his 1908
paper. The number 0.7% can be as high as about 22% (for a batch of
five) according to our calculations and as confirmed in [Hoaglin
et al]. |
| 3.96 |
"Bloom" in Table 3.7
should read "Blom". |
| 3.98 |
You can compute the
"normal quantiles" in Table 3.8 using Excel's NormSInv()
function (apply it to the "plotting position" values). |
| 3.98 |
The "Normal
Quantile" column can easily be reproduced using Excel's NormSInv()
function (as applied to the "Plotting position"). |
| 3.102 |
Note the different scales used
on the X and Y axes. This graphic should be adjusted to display
equal intervals on both the vertical and horizontal scales. |
| 3.103 |
The Tukey m-d plot is a Q-Q
plot with a change of coordinates. On a Q-Q plot, let x be the
horizontal coordinate and y the vertical coordinate. Compute new
coordinates X = (x+y)/2 and Y = y-x. The m-d plot
displays Y versus X. Geometrically, X is
proportional to the distance along the diagonal line (y=x) and Y
is proportional to the distance (in the Q-Q plot) to the diagonal line in
the perpendicular direction. This latter property
distinguishes the m-d plot from a plot of residuals (relative to the
diagonal line) because the residuals are vertical distances to
the diagonal line. Equivalently, by changing scale slightly
(multiply X by sqrt(2) and divide Y by sqrt(2)), the m-d
plot is obtained by rotating the Q-Q plot 45 degrees clockwise.
For example, rotate Figure 3.20 clockwise 45 degrees and compare it to
Figure 3.23. |
| 3.134 |
A partial set of answers
to exercise 3.1 is available. You should be able to do all but the
density plot by hand or with the aid of a spreadsheet; indeed, you
should do several exercises with manual and spreadsheet calculations
until you are comfortable with the formulas and techniques. Then
learning to use any statistical software will be easy. |
| 3.136 |
It is difficult to believe
that the arsenic concentrations shown in problem 3.8 are really
"ppm" (milligrams per liter), because they are so
spectacularly high. These values are most likely either (a) in ppb
(micrograms per liter) or (b) have been made up by somebody. The
"ppm" is present in the original EPA guidance document.
The data appear on page 21 in my copy, not on page 6. |
