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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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The Student’s t-test

The famous t-test was initially illustrated by “Student” in a paper published in 1908 (available at: http://biomet.oxfordjournals.org/ cgi/reprint/6/1/1). In the paper, Student used several examples to illustrate the process. One example (Illustration 3) discussed the yield of barley of seeding plots with two different kinds of seed. Each type of seed (kiln-dried and not kiln-dried) was planted in adjacent plots in two different years, accounting for 11 pairs of “split” plots. The data listed below are from the table on page 24 of the 1908 paper.

The Student’s t-test The famous t-test was initially illustrated by “Student” in a paper published...

The statistical method used in Student (1908) is very different from the ones we use now. On page 24, Student concluded that the odds that kiln-dried seeds have a higher yield is 14:1. Conduct the t-test using the “head corn” yield data shown above. Can you guess where the 14:1 odds come from?

 
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In a study of water quality trends, EPA compiled stream biological monitoring data in the Mid-Atlantic region before and after 1997. They are interested in whether there was a shift in biological conditions in streams in the area. The indicator they used is EPT taxa richness (number of taxa belong to three genera of flies commonly known as mayfly, stonefly, and caddisfly). As distributions of count variables are typically skewed, log transformation was used. The log mean and standard deviation of EPT taxa richness before 1997 are 2.2 and 6.9 (n = 355), and are 1.8 and 5.4 after 1997 (n = 280). Is the difference in log mean statistically significant? How do you report the difference in terms of EPT taxa richness?

 
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