ES 25: Midterm Practice Problems
Background: Mississippi Tributary Nitrate Concentrations
Nutrients (fertilizers) applied to croplands in the Mississippi River drainage basin often
make their way into rivers and streams, and ultimately are discharged into the Gulf of
Mexico. Fertilizers in the Gulf of Mexico increase the production of marine algae, thereby
decreasing the oxygen content of coastal waterways. Massive algal blooms following wet
season runoffs can cause severe hypoxia, threatening a wide variety of marine organisms,
including commercial fisheries.
Compounds of nitrogen (such as nitrates) are one of the most important fertilizers used
on croplands in the central United States. As a result, about 1 million metric tons of nitrate
are carried into the Gulf of Mexico every year by streams draining the Mississippi River
basin. This value of nitrate flux is about 3 times higher than the flux 30 years ago,
corresponding to increasing use of fertilizers to boost or maintain crop yields.
In order to quantitatively evaluate the effect of agricultural practices on water quality,
Goolsby et al (1999) made a comprehensive analysis of 42 sub-basins within the
Mississippi-Atchafalaya drainage system. Individual interior basins range in size from 7000
km2 to 237,000 km2. Each basin was measured for the percentage of land in row crops
(corn, soybeans and/or sorghum), which ranged from less than 0.1% to 74% (see data
table). [NOTE: I have taken a representative sample of 10 sub-basins from the larger
dataset of 42 sub-basins to simplify graphing]. The concentration of nitrate in river water
discharging from the "outlet" of each of these basins was also measured repeatedly (from
40 to 300 times), and the mean nitrate concentration was calculated (see data table).
Concentrations are reported in milligrams of nitrate per liter of water (mg/L), which is
essentially equivalent to parts per million (ppm). Concentrations range from near zero to
almost 7 mg/L.
Data:
Basin #
cropland
(percentage)
1
2
3
4
5
6
7
8
9
10
0.1
1.7
4
11.8
17.5
45.6
56.6
56.8
63.6
73.9
nitrate (mg/L)
0.05
0.17
0.15
0.32
0.70
3.56
4.19
0.83
4.12
6.67
o Create a scatterplot of the data.
o Assuming that the underlying relationship is linear, estimate the equation of the
line y = mx + b. Show your work clearly.
o Define each term in the equation, with units:
y=
m=
x=
b=
o Explain what the y-intercept (b) tells you (i.e., what is its physical significance?).
o According to the linear equation that you wrote, a 20% decrease in cropland will
correspond to a ___________ increase/decrease (circle one) in nitrate level.
o Using your linear equation, predict the nitrate level for a basin that has:
a. 20% cropland
b. 60% cropland
o Identify a datapoint that is an outlier by CIRCLING it on the graph.
o Does this sub-basin contribute more, or less, than the ‘average’ amount of
pollution? [your trendline is assumed to represent the ‘average’] ___________
EXPLAIN:
o Calculate the residual, or error, associated with this outlier._______________
o What physical factor(s) could explain the observed nitrate level associated with this
outlier?
www.gfdl.noaa.gov/~tk/images/slp_all_w_cat.png
Backgroud:
The strongest hurricanes in the present climate may be upstaged by even more intense
hurricanes over the next century as the earth's climate is warmed by increasing levels of
greenhouse gases in the atmosphere. Although we cannot say at present whether more or
fewer hurricanes will occur in the future with global warming, the hurricanes that do occur
near the end of the 21st century are expected to be stronger and have significantly more
intense rainfall than under present day climate conditions. This expectation (Figure 1) is
based on an anticipated enhancement of energy available to the storms due to higher
tropical sea surface temperatures.
The results shown in Figure 1 are based on a simulation study carried out by Thomas R.
Knutson and Robert E. Tuleya at NOAA's Geophysical Fluid Dynamics Laboratory
(GFDL). In this study hurricanes were simulated for a climate warming as projected to
occur with a substantial build-up of atmospheric CO2. An increase of intensity of about
one-half category on the Saffir-Simpson scale was simulated for an 80 year build-up of
atmospheric CO2 at 1%/yr (compounded).
Questions:
o Looking at the diagram, we can infer that a higher ‘minimum central pressure’
corresponds to a _____________ (lower or higher?) intensity hurricane.
o Which curve has a larger standard deviation? (Note: the number of occurrences is
the same for both the ‘Control’ and the ‘High C02” histograms)
Assume that hurricane intensities (measured by minimum central pressure) presently
follow a normal distribution, with a mean of 940 mb and a standard deviation of 20.
o Calculate the z-score associated with 900 mb.__________________
o Interpret the z-score:
:
o Draw a picture of the distribution described above. Be sure your diagram
follows the 68/95/99.7% rule.
o What is the (approximate) probability of a hurricane occurring with a pressure
of 900 mb, or less, by chance alone?_________________. Shade in the area
corresponding to this value, on your diagram above.
o At a 5% significance level, would you reject the hypothesis that a 900 mb, or
less, hurricane occurred purely by chance? Explain (briefly)
o What other explanation (aside from chance) could you offer for the occurrence
of a 900 mb hurricane?
o Assume that the distribution of hurricanes during a historical warming trend
followed a normal distribution with a mean of 920 mb and a standard deviation
of 40 mb. Add this new distribution, on the same x-axis to your drawing,
above.
o What is the probability of having a hurricane with a min. central pressure of
960, or more, under “present” conditions________________________? What
about under “historic” conditions)?______________________________?
Shade in each probability (and label) on your diagram.
THIS IS JUST A TASTE FOR THE MATERIAL/DIFFICULTY OF THE TEST. I reserve
the right to test on anything that is on the “key concepts” pages for statistics and calculus
guide. I encourage you to study with someone else in the class, and ask any of the
facilitators if you get stuck. Be sure to be able to explain what key terms (see those
“boxed” in workbook) for specific exercises that you are given.