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The Great Lakes Environmental Research Laboratory of the U.S. National Oceanic and Atmospheric Administration (NOAA-GLERL) routinely monitors the western basin of Lake Erie from May to October every year. The data file LakeErie1.csv includes all available data up to the end of 2014. Two variables are of particular interest in studying lake eutrophication. They are the total phosphorus (TP) and chlorophyll a (chla) concentrations. These are concentration variables and we often assume that their distributions are approximately normal.

• Read the data into R and use the graphics to evaluate whether TP and chla are normally distributed.

• Western Lake Erie’s nutrient concentrations are largely associated with Maumee River input, which varies from year to year due to variation in weather conditions. As a result, we expect that TP, as well as chla, concentration distributions vary by year. Use the function qqmath (from package lattice) to draw normal Q-Q plots of TP and chla concentrations by year. Are the annual concentration distributions closer to being normal?

 
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Carl Sagan’s intelligence data. In his book, The Dragons of Eden, Carl Sagan presented a graph showing the brain and body masses, both on log scale, of a collection of animal species. The purpose of the graph was to describe an intelligence scale: the ratio of the average brain weight over the average body weight to the power of 2/3.

(a) Read the data (in file Intelligence.csv) into R and graph brain weight against body weight, both in logarithmic scales. The brain weight is in grams and body weight is in kilograms. Can you tell which species has the highest intelligence from this figure?

(b) Calculate the intelligence measure (call it Int) and add the result as a new column to the data frame.

(c) Use the function dotplot from package lattice to plot the intelligence measure directly: dotplot (Species~Int, data=Intelligence)

(d) The dot plot orders the species alphabetically. Reorder the column based on the intelligence scale using function ordered and redraw the dot plot so that the species are sorted based on their intelligence scales. Is there a problem in the data? If so, what might be the cause of the problem?

 
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In a normal Q-Q plot, we expect to see data points line up to form a straight line when they are random samples from a normal distribution. As in almost all statistical rules, this expectation is literally an “expectation” or what we expect to see on average. When the sample size of the data is small, a normal Q-Q plot may not entirely resemble a straight line even when the data points are truly random samples from a normal distribution. Use my.qqnorm (rnorm (20)) repeatedly to draw several normal Q-Q plots, each using 20 random numbers drawn from the standard normal distribution to see the likely departure from a straight line.

 
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The 1971 edition of Dr. Seuss’s The Lorax included lines describing the dire fate of the “humming fish” after their pond was polluted:

“They’ll walk on their fins and get woefully weary in search of some water that isn’t so smeary. I hear things are just as bad up in Lake Erie.”

The last line was removed in the 1985 edition after Dr. Seuss realized that Lake Erie is “the happy home of smiling fish” again after the input of a key culprit of lake eutrophication, phosphorus, was successfully reduced, particularly in the Maumee River basin. The input of phosphorus from the Maumee River has since stablized. However, since the late 1990s, harmful algal blooms have returned to western Lake Erie. Some suggested that the widespread use of an inorganic form of phosphorus fertilizer is to blame. Because much of the phosphorus in western Lake Erie are from Maumee River, we can use the long-term monitoring data from Heidelberg University to evaluate whether this hypothesis is supported by data.

• Plot daily SRP (soluble reactive phosphorus) concentration against time. Can you see an increasing trend over time?

• Input of nutrient to a lake is better measured by the mass loading rate (the product of flow and concentration). Plot the daily SRP loading rate against time. Is there a temporal trend in loading rate?

• If a temporal trend is not obvious, it is often because of the large daily fluctuations. Calculate the annual total of SRP loading rates and plot them against the respective years. Is there a trend?

• If we repeat the above steps for TP, we will see that the annual TP loadings are more or less the same in the last 20 years. Could the return of harmful algal blooms in the last 20 years be caused by increased proportion of SRP in the total phosphorus? Plot the ratio of SRP over TP over time (both at the daily and annual scales).

 
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